PSP-UFU site i18n

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diminuem a corre\xe7\xe3o."),Object(m.b)("div",{className:"admonition admonition-info alert alert--info"},Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(m.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"}),Object(m.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"})))),"Informa\xe7\xe3o")),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(m.b)("p",{parentName:"div"},"Valores menores que ",Object(m.b)("span",Object(s.a)({parentName:"p"},{className:"math 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",Object(m.b)("em",{parentName:"p"},"podem")," auxiliar na converg\xeancia, por\xe9m aumentam o n\xfamero de itera\xe7\xf5es e consequentemente o tempo de processamento do m\xe9todo."))),Object(m.b)("p",null,"Essa op\xe7\xe3o \xe9 habilitada somente para o m\xe9todo de ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerFlow#newton-raphson"}),"Newton-Raphson"),"."),Object(m.b)("h4",{id:"toler\xe2ncia-do-gauss"},"Toler\xe2ncia do Gauss"),Object(m.b)("p",null,"Define a toler\xe2ncia do Gauss-Seidel para o m\xe9todo h\xedbrido. 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",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"https://pt.wikipedia.org/wiki/Desenho_assistido_por_computador"}),"Wikipedia"))),Object(n.b)("h2",{id:"ferramentas-do-menu"},"Ferramentas do menu"),Object(n.b)("p",null,"Todos os objetos criados cont\xeam v\xe1rios atributos gr\xe1ficos, como: posi\xe7\xe3o, orienta\xe7\xe3o, cor, tamanho, etc. Para modific\xe1-los foram inseridas algumas ferramentas intuitivas e de f\xe1cil utiliza\xe7\xe3o, as quais s\xe3o: ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"cadTools#selecionar"}),Object(n.b)("strong",{parentName:"a"},"selecionar")),", ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"cadTools#mover"}),Object(n.b)("strong",{parentName:"a"},"mover")),", ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"cadTools#girar"}),Object(n.b)("strong",{parentName:"a"},"girar")),", ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"cadTools#arrastar"}),Object(n.b)("strong",{parentName:"a"},"arrastar")),", ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"cadTools#zoom"}),Object(n.b)("strong",{parentName:"a"},"zoom")),", ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"cadTools#encaixar"}),Object(n.b)("strong",{parentName:"a"},"encaixar")),", ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"cadTools#copiar-e-colar"}),Object(n.b)("strong",{parentName:"a"},"copiar, colar"))," e ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"cadTools#excluir"}),Object(n.b)("strong",{parentName:"a"},"excluir")),"."),Object(n.b)("p",null,"Essas ferramentas podem ser acessadas por suas teclas de atalho assim como pelo ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"submenu Ferramentas"),"."),Object(n.b)("p",null,"Tais ferramentas permitem a ",Object(n.b)("strong",{parentName:"p"},"personaliza\xe7\xe3o"),", ",Object(n.b)("strong",{parentName:"p"},"navega\xe7\xe3o")," e auxiliam a ",Object(n.b)("strong",{parentName:"p"},"cria\xe7\xe3o")," de redes el\xe9tricas de pot\xeancia, assim como diagramas de controle na plataforma."),Object(n.b)("h3",{id:"selecionar"},"Selecionar"),Object(n.b)("p",null,"A sele\xe7\xe3o de um \xfanico objeto \xe9 realizada clicando com o bot\xe3o esquerdo do mouse sobre o elemento, que ser\xe1 identificado pelo contorno em ",Object(n.b)(d,{color:"#80bfff",mdxType:"Highlight"},"azul")," do objeto."),Object(n.b)("p",null,"Para a sele\xe7\xe3o de v\xe1rios objetos simultaneamente \xe9 necess\xe1rio clicar com o bot\xe3o esquerdo do mouse em um local vazio da tela e arrast\xe1-lo, criando assim um ",Object(n.b)("strong",{parentName:"p"},"ret\xe2ngulo de sele\xe7\xe3o")," que ir\xe1 acompanhar o ponteiro do mouse enquanto o bot\xe3o esquerdo estiver pressionado. Ao solt\xe1-lo todos os objetos que intercederem o ret\xe2ngulo ser\xe3o selecionados."),Object(n.b)("p",null,"Mantendo pressionado a tecla Control (Ctrl) do teclado pode-se manter sele\xe7\xf5es anteriores enquanto novos elementos s\xe3o selecionados."),Object(n.b)("p",null,"Para desmarcar todos os objetos basta clicar em uma \xe1rea sem objetos."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/select.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/select.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"mover"},"Mover"),Object(n.b)("p",null,"Um elemento poder\xe1 ser movido para qualquer local por meio da a\xe7\xe3o ",Object(n.b)("em",{parentName:"p"},"drag-and-drop")," (clicando e arrastando) ou pela ferramenta mover."),Object(n.b)("p",null,"Ao habilitar o \u201cmodo mover\u201d, movendo todos os componentes selecionados \xe0 medida que a posi\xe7\xe3o do mouse \xe9 alterada. Para desabilitar esse modo basta pressionar a tecla Escape do teclado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/move.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/move.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"girar"},"Girar"),Object(n.b)("p",null,"Alguns objetos poder\xe3o ser rotacionados atrav\xe9s da ferramenta girar, acessado pela tecla de atalho \u201cR\u201d, girando o objeto no sentido hor\xe1rio, ou \u201cShift + R\u201d, para o sentido anti-hor\xe1rio (essa op\xe7\xe3o ir\xe1 afetar todos os objetos selecionados). O elemento tamb\xe9m pode ser rotacionado pelo ",Object(n.b)("a",Object(t.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"submenu Ferramentas")," ou ao acessar o menu de contexto clicando com o bot\xe3o direito sobre o elemento."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",Object(t.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(t.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(t.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(n.b)("path",Object(t.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(n.b)("div",Object(t.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"O menu de contexto somente ser\xe1 exibido se o elemento estiver selecionado."))),Object(n.b)("p",null,"Ao utilizar essa ferramenta os objetos ir\xe3o girar 45\xba a partir do seu ponto de origem. Para atingir a angula\xe7\xe3o pretendida basta repetir o processo de rota\xe7\xe3o."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/rotate.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/rotate.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"arrastar"},"Arrastar"),Object(n.b)("p",null,"Tr\xeas ferramentas muito \xfateis na navega\xe7\xe3o do circuito, principalmente em grandes redes, s\xe3o: arrastar, zoom e encaixar."),Object(n.b)("p",null,"A ferramenta arrastar possibilita mover todo o circuito ",Object(n.b)("strong",{parentName:"p"},"preservando as posi\xe7\xf5es relativas entre os elementos"),". Esta ferramenta pode ser acessada pelo submenu \u201cFerramentas\u201d ou ",Object(n.b)("strong",{parentName:"p"},"mantendo pressionada o ",Object(n.b)("em",{parentName:"strong"},"scroll")," do mouse"),", ativando assim o \u201cmodo arrastar\u201d. Ao ativ\xe1-lo todo o circuito ser\xe1 movido clicando e arrastando na dire\xe7\xe3o desejada."),Object(n.b)("p",null,"Para sair do \u201cmodo arrastar\u201d basta pressionar a tecla Escape do teclado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/drag.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/drag.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"zoom"},"Zoom"),Object(n.b)("p",null,"O zoom ser\xe1 aplicado ao utilizar o ",Object(n.b)("strong",{parentName:"p"},"scroll do mouse")," para mais ou menos zoom. A ferramenta zoom ir\xe1 aproximar ou distanciar o circuito a partir da posi\xe7\xe3o do ponteiro do mouse na tela para encontrar objetos ou destaca-los."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/zoom.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/zoom.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"encaixar"},"Encaixar"),Object(n.b)("p",null,"A ferramenta encaixar ir\xe1 mover o circuito e aplicar o zoom necess\xe1rio para que todos os componentes da rede sejam exibidos na tela. Esta ferramenta pode ser acessada atrav\xe9s do submenu \u201cFerramentas\u201d ou pela combina\xe7\xe3o de teclas \u201cShift + F\u201d."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/fit.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/fit.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"copiar-e-colar"},"Copiar e Colar"),Object(n.b)("p",null,"Duas ferramentas importantes durante o processo de cria\xe7\xe3o da rede el\xe9trica de pot\xeancia no ",Object(n.b)("em",{parentName:"p"},"software")," s\xe3o copiar e colar. Qualquer circuito criado no PSP-UFU poder\xe1 ser duplicado completamente ou parcialmente no mesmo projeto ou em m\xfaltiplos atrav\xe9s dessas ferramentas."),Object(n.b)("p",null,"Para copiar basta selecionar os elementos que se deseja duplicar e acessar a ferramenta no submenu \u201cFerramentas\u201d ou pressionando a combina\xe7\xe3o das teclas \u201cCtrl + C\u201d, ent\xe3o os dados do circuito ser\xe3o copiados para a \xe1rea de transfer\xeancia. Ao copiar um circuito ele poder\xe1 ser colado no mesmo projeto ou em um projeto distinto, utilizando o mesmo submenu ou pressionando a combina\xe7\xe3o das teclas \u201cCtrl + V\u201d."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/copyPaste.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/copyPaste.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"excluir"},"Excluir"),Object(n.b)("p",null,"Qualquer objeto poder\xe1 ser exclu\xeddo do projeto por meio da ferramenta deletar, presente no menu Ribbon, pela tecla de atalho Delete ou pelo acesso ao menu de contexto clicando com o bot\xe3o direito sobre o elemento selecionado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/delete.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/delete.webm"),type:"video/webm"})),Object(n.b)("h2",{id:"personaliza\xe7\xe3o-gr\xe1fica-os-elementos"},"Personaliza\xe7\xe3o gr\xe1fica os elementos"),Object(n.b)("p",null,"Alguns elementos permitem sua personaliza\xe7\xe3o gr\xe1fica, como as barras e linhas."),Object(n.b)("h3",{id:"barra"},"Barra"),Object(n.b)("p",null,"No caso dos barramentos, pode-se alterar o ",Object(n.b)("strong",{parentName:"p"},"comprimento da barra")," clicando e arrastando no ",Object(n.b)("em",{parentName:"p"},"pickbox")," exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado."),Object(n.b)("h3",{id:"linha"},"Linha"),Object(n.b)("p",null,"As linhas podem ser inseridas com ",Object(n.b)("strong",{parentName:"p"},'"n\xf3s" (pontos de ancoragem)'),", personalizando a disposi\xe7\xe3o das linhas no diagrama unifilar. Esses n\xf3s s\xe3o inseridos durante o processo de cria\xe7\xe3o do elemento, anteriormente \xe0 sele\xe7\xe3o da segunda barra."),Object(n.b)("p",null,"Ap\xf3s a sua inser\xe7\xe3o, novos n\xf3s podem ser anexados pelo menu de contexto acessado ao clicar com o bot\xe3o direito sobre a linha selecionada. Sua remo\xe7\xe3o tamb\xe9m \xe9 alcan\xe7ada por uma op\xe7\xe3o do menu de contexto."),Object(n.b)("p",null,"Assim como nos barramentos, a altera\xe7\xe3o da posi\xe7\xe3o dos n\xf3s \xe9 obtida clicando e arrastando o ",Object(n.b)("em",{parentName:"p"},"pickbox")," exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/graphicPropsChange.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/graphicPropsChange.webm"),type:"video/webm"})),Object(n.b)("h2",{id:"conex\xe3o-e-desconex\xe3o-de-elemetos-na-barra"},"Conex\xe3o e desconex\xe3o de elemetos na barra"),Object(n.b)("p",null,"Os elementos podem ser desconectados e reconectados \xe0s barras ap\xf3s sua inser\xe7\xe3o somente pela ferramenta ",Object(n.b)("em",{parentName:"p"},"drag-and-drop"),". 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A estabilidade \xe9 reconhecida como um importante problema para a seguran\xe7a da opera\xe7\xe3o desde a d\xe9cada de 1920. Muitos ",Object(m.b)("em",{parentName:"p"},"blackouts")," de grande porte causados pela instabilidade do SEP t\xeam ilustrado a import\xe2ncia desse fen\xf4meno."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stability.svg"),alt:"Estabilidade",title:"Estabilidade"}))),Object(m.b)("p",null,"Normalmente s\xe3o realizados dois tipos de estudos de estabilidade (uma classifica\xe7\xe3o mais aprofundada pode ser observada ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"https://doi.org/10.1109/TPWRS.2004.825981"}),"nesse artigo"),"). A recupera\xe7\xe3o de uma r\xe1pida e brusca perturba\xe7\xe3o \xe9 chamada de ",Object(m.b)("strong",{parentName:"p"},"estabilidade transit\xf3ria"),", e sua solu\xe7\xe3o \xe9 geralmente obtida no dom\xednio do tempo. Outro estudo \xe9 denominado ",Object(m.b)("strong",{parentName:"p"},"estabilidade din\xe2mica"),", utilizado para descrever a resposta do sistema frente a pequenas perturba\xe7\xf5es ao longo de muito tempo, o qual pode ser resolvido tanto no dom\xednio da frequ\xeancia quanto no dom\xednio do tempo."),Object(m.b)("div",{className:"admonition admonition-info alert alert--info"},Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(m.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"}),Object(m.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"})))),"Informa\xe7\xe3o")),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(m.b)("p",{parentName:"div"},"No PSP-UFU, a estabilidade din\xe2mica \xe9 tratada como uma extens\xe3o da estabilidade transit\xf3ria, em que os efeitos de pequenas constantes de tempo devem ser ignorados."))),Object(m.b)("p",null,"O modelo para representa\xe7\xe3o do SEP em um estudo de estabilidade transit\xf3ria \xe9 feito por meio do sistema de equa\xe7\xf5es alg\xe9brico-diferenciais (EADs), em que o estado inicial \xe9 considerado est\xe1vel e originado de um estudo de ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerFlow"}),"fluxo de carga"),". A solu\xe7\xe3o das equa\xe7\xf5es se d\xe1 por meio de um ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"https://en.wikipedia.org/wiki/Numerical_integration"}),"m\xe9todo de integra\xe7\xe3o num\xe9rica")," e, ent\xe3o, \xe9 observada a resposta do sistema de pot\xeancia."),Object(m.b)("p",null,"Caso a solu\xe7\xe3o no dom\xednio do tempo, ap\xf3s alguma perturba\xe7\xe3o, leve um ou mais par\xe2metros do sistema a valores fisicamente imposs\xedveis ou, at\xe9 mesmo, a solu\xe7\xe3o das equa\xe7\xf5es diferenciais n\xe3o obtenha converg\xeancia em processos iterativos, o sistema \xe9 dito ",Object(m.b)("strong",{parentName:"p"},"inst\xe1vel"),". Em contrapartida, caso as vari\xe1veis de estado dos elementos do sistema oscilem ap\xf3s uma perturba\xe7\xe3o e se estabele\xe7am em um novo ponto de opera\xe7\xe3o, mesmo que distinto dos valores iniciais, o sistema \xe9 considerado ",Object(m.b)("strong",{parentName:"p"},"est\xe1vel"),"."),Object(m.b)("h2",{id:"execu\xe7\xe3o-do-estudo-de-estabilidade-no-psp-ufu"},"Execu\xe7\xe3o do estudo de estabilidade no PSP-UFU"),Object(m.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerEditor"}),"editor de pot\xeancia"),", assim como os ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"controlEditor"}),"controles")," das ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"syncGenerator"}),"m\xe1quinas s\xedncronas"),", a execu\xe7\xe3o da estabilidade \xe9 realizada no ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," clicando no \xedcone do bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"Executar Estabilidade"),'. Ao clicar na parte inferior do bot\xe3o "Executar Estabilidade" ser\xe1 exibido um menu suspenso com a op\xe7\xe3o ',Object(m.b)("strong",{parentName:"p"},"Lista de eventos de estabilidade"),"."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/menuSimulationStability.svg"),alt:"Menu Simula\xe7\xe3o",title:"Menu Simula\xe7\xe3o"}))),Object(m.b)("p",null,"Ao executar o c\xe1lculo de estabilidade, uma caixa com o status do processo de simula\xe7\xe3o ser\xe1 exibida, indicando primeiramente a inicializa\xe7\xe3o do estudo e posteriormente o tempo de simula\xe7\xe3o calculado."),Object(m.b)("p",null,"Ao clicar na lista de eventos de estabilidade, ser\xe1 exibida uma janela com a descri\xe7\xe3o dos eventos de estabilidade inseridos."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabList.png"),alt:"Lista de eventos de estabilidade",title:"Lista de eventos de estabilidade"}))),Object(m.b)("h3",{id:"inserindo-um-evento-de-estabilidade"},"Inserindo um evento de estabilidade"),Object(m.b)("p",null,"Os dist\xfarbios mais comuns aplicados em estudos de estabilidade transit\xf3ria s\xe3o faltas e chaveamentos. Tais opera\xe7\xf5es s\xe3o facilmente realizadas no PSP-UFU:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Faltas"),": Curtos-circuitos trif\xe1sicos podem ser inseridos nos ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"bus"}),"barramentos")," do sistema por meio da inclus\xe3o da imped\xe2ncia de falta na matriz admit\xe2ncia de barras da mesma maneira realizada em um elemento ",Object(m.b)("em",{parentName:"p"},"shunt"),". Tal valor \xe9 definido pelo usu\xe1rio e caso seja um curto-circuito franco, um valor n\xe3o nulo, mas suficientemente pr\xf3ximo de zero, \xe9 aplicado, de forma que a tens\xe3o no barramento \xe9 levada a zero durante o dist\xfarbio.")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Chaveamento de ramo"),": Da mesma forma que as faltas, o chaveamento de ramos \xe9 realizado por meio da altera\xe7\xe3o na matriz admit\xe2ncia, removendo ou inserindo os par\xe2metros do elemento a ser chaveado.\nCada ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerEditor"}),"elemento de pot\xeancia"),' possui um bot\xe3o de "Chaveamento" ou "Estabilidade", em que pode ser inserido os tempos de remo\xe7\xe3o e/ou inser\xe7\xe3o do componente.')),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Chaveamento de m\xe1quinas"),": A remo\xe7\xe3o de uma m\xe1quina s\xedncrona \xe9 efetivada com a retirada de sua participa\xe7\xe3o no vetor de correntes, al\xe9m da remo\xe7\xe3o de sua admit\xe2ncia fict\xedcia.\nAssim como os ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerEditor"}),"elementos de pot\xeancia"),", as ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"syncGenerator"}),"m\xe1quinas s\xedncronas"),' possui um bot\xe3o "Chaveamento", em que pode ser inserido os tempos de remo\xe7\xe3o e/ou inser\xe7\xe3o do componente.'),Object(m.b)("div",Object(s.a)({parentName:"li"},{className:"admonition admonition-caution alert alert--warning"}),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(m.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(m.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(m.b)("p",{parentName:"div"},"Mesmo que removida da barra, os par\xe2metros das ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"syncGenerator"}),"m\xe1quinas s\xedncronas")," continuam a ser calculados com a corrente do estator nula, podendo fornecer resultados em uma eventual reconex\xe3o."))))),Object(m.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(m.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(m.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"})))),"Dica")),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(m.b)("p",{parentName:"div"},"Eventos nos ",Object(m.b)("strong",{parentName:"p"},"sistemas de controle")," podem ser facilmente introduzidos com o bloco de ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"mathExpression"}),"express\xe3o matem\xe1tica"),". Nesse caso, tais eventos ",Object(m.b)("strong",{parentName:"p"},"n\xe3o")," ser\xe3o exibidos na lista de eventos de estabilidade."))),Object(m.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-estudo-de-estabilidade"},"Erros comuns na execu\xe7\xe3o do estudo de estabilidade"),Object(m.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao estudo de estabilidade."),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-n\xe3o-foi-poss\xedvel-construir-a-matriz-admit\xe2ncia"},'A seguinte mensagem de erro \xe9 exibida: "N\xe3o foi poss\xedvel construir a matriz admit\xe2ncia"'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel construir a matriz admit\xe2ncia de barras. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-erro-ao-inicializar-os-valores-de-satura\xe7\xe3o-do"},'A seguinte mensagem de erro \xe9 exibida: "Erro ao inicializar os valores de satura\xe7\xe3o do..."'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel calcular os fatores de satura\xe7\xe3o da ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"syncGenerator"}),"m\xe1quina s\xedncrona"),". As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O fator de satura\xe7\xe3o \xe9 menor que 1,2"),". Esse valor deve ser maior que 1,2, ou ir\xe1 gerar erros na simula\xe7\xe3o. Caso n\xe3o seja informado, a satura\xe7\xe3o da m\xe1quina n\xe3o \xe9 considerada nos c\xe1lculos."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros da m\xe1quina s\xedncrona est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o das m\xe1quinas s\xedncronas sejam inseridos, o c\xe1lculo dos fatores de satura\xe7\xe3o pode divergir. Verifique se os dados foram inseridos corretamente.")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-erro-ao-inicializar-o-avr--regulador-de-velocidade"},'A seguinte mensagem de erro \xe9 exibida: "Erro ao inicializar o AVR / regulador de velocidade..."'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"controlEditor#inicializa%C3%A7%C3%A3o-do-sistema-de-controle"}),"inicializar o sistema de controle")," de uma ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"syncGenerator"}),"m\xe1quina s\xedncrona"),". As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O sistema de controle est\xe1 vazio"),". Caso esteja habilitado o AVR e/ou o regulador de velocidade e o controle n\xe3o foi inserido, esse erro pode ser acionado. Insira o controle da m\xe1quina ou desmarque a op\xe7\xe3o de utiliza\xe7\xe3o do AVR e/ou regulador de velocidade."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O sistema de controle n\xe3o possui ao menos uma entrada e uma sa\xedda"),". O sistema de controle deve ter ao menos uma ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"io"}),"entrada e uma sa\xedda"),", caso contr\xe1rio apresentar\xe1 erro de execu\xe7\xe3o."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O passo de integra\xe7\xe3o est\xe1 muito pequeno"),". Caso o passo de integra\xe7\xe3o esteja muito pequeno, os c\xe1lculos gerar\xe3o erros e ir\xe3o divergir. Reduza o passo de integra\xe7\xe3o nas ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),".")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-imposs\xedvel-resolver-as-m\xe1quinas-do-sistema"},'A seguinte mensagem de erro \xe9 exibida: "Imposs\xedvel resolver as m\xe1quinas do sistema"'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel resolver as EADs das m\xe1quinas s\xedncronas inseridas no sistemas. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros da m\xe1quina s\xedncrona est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o das m\xe1quinas s\xedncronas sejam inseridos, o c\xe1lculo das EADs pode se tornar imposs\xedvel. Verifique se os dados foram inseridos corretamente."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O passo de integra\xe7\xe3o est\xe1 muito pequeno"),". Caso o passo de integra\xe7\xe3o esteja muito pequeno, os c\xe1lculos gerar\xe3o erros e ir\xe3o divergir. Reduza o passo de integra\xe7\xe3o nas ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),"."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-erro-ao-inicializar-o-escorregamento-do-motor"},'A seguinte mensagem de erro \xe9 exibida: "Erro ao inicializar o escorregamento do motor..."'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel calcular o valor de escorregamento inicial do motor de indu\xe7\xe3o. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros de estabilidade do motor est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o dos motores de indu\xe7\xe3o sejam inseridos, o c\xe1lculo do escorregamento pode se tornar imposs\xedvel. Verifique se os dados foram inseridos corretamente.")),Object(m.b)("h2",{id:"estrutura-da-ferramenta-de-estabilidade"},"Estrutura da ferramenta de estabilidade"),Object(m.b)("p",null,"A estabilidade de um SEP \xe9 um problema din\xe2mico e necessita de modelos mais elaborados de elementos de pot\xeancia comparados \xe0queles apresentados nos outros estudos. Esses modelos s\xe3o descritos individualmente, com destaque \xe0s ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"syncGenerator#gerador-s%C3%ADncrono-no-estudo-de-estabilidade"}),"m\xe1quinas s\xedncronas"),", ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"load#carga-no-estudo-de-estabilidade"}),"cargas ZIP")," e ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-estudo-de-estabilidade"}),"motores de indu\xe7\xe3o"),"."),Object(m.b)("p",null,"A representa\xe7\xe3o dos demais componentes do sistema el\xe9trico: ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"line"}),"linhas de transmiss\xe3o"),", ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"transformer"}),"transformadores")," e elementos ",Object(m.b)("em",{parentName:"p"},"shunt")," (com exce\xe7\xe3o de cargas ZIP), que formam a rede de transmiss\xe3o ou distribui\xe7\xe3o balanceada, \xe9 realizada utilizando os mesmos modelos do ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerFlow"}),"fluxo de carga"),"."),Object(m.b)("p",null,"Os modelos din\xe2micos para a an\xe1lise no dom\xednio do tempo s\xe3o na forma de um sistema de equa\xe7\xf5es alg\xe9brico-diferenciais (EADs), descritas a seguir:"),Object(m.b)("div",{className:"math 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estado"),Object(m.b)("li",{parentName:"ul"},Object(m.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(m.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"y")),Object(m.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"y")))),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.625em",verticalAlign:"-0.19444em"}})),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord 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permanecem constantes durante todo o processo de c\xe1lculo. Para o problema de estabilidade, as vari\xe1veis alg\xe9bricas iniciais correspondem \xe0s tens\xf5es e \xe2ngulos das barras calculadas no ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerFlow"}),"fluxo de carga"),". As vari\xe1veis de estado iniciais s\xe3o estimadas ap\xf3s a converg\xeancia do fluxo de pot\xeancia (inicializa\xe7\xe3o dos elementos din\xe2micos)."),Object(m.b)("p",null,"Na sequ\xeancia \xe9 apresentado, na forma de fluxogramas, a estrutura do m\xf3dulo de estabilidade implementado no PSP-UFU. O fluxograma abaixo mostra a estrutura geral da ferramenta de estabilidade."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabGeneral.svg"),alt:"Estrutura geral da ferramenta de estabilidade",title:"Estrutura geral da ferramenta de estabilidade"}))),Object(m.b)("p",null,"A inicializa\xe7\xe3o da rede el\xe9trica e dos elementos din\xe2micos, assim como o precesso iterativo para solu\xe7\xe3o das EADs s\xe3o apresentados nos dois fluxogramas conseguintes."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabInit.svg"),alt:"Inicializa\xe7\xe3o da ferramenta de estabilidade",title:"Inicializa\xe7\xe3o da ferramenta de estabilidade"}))),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabIterative.svg"),alt:"Processo iterativo para c\xe1lculo dos elementos din\xe2micos",title:"Estrutura geral da ferramenta de estabilidade"}))),Object(m.b)("h3",{id:"integra\xe7\xe3o-num\xe9rica"},"Integra\xe7\xe3o num\xe9rica"),Object(m.b)("p",null,"Para resolver as equa\xe7\xf5es diferenciais da m\xe1quina \xe9 necess\xe1rio um m\xe9todo de integra\xe7\xe3o num\xe9rica. Tais m\xe9todos s\xe3o classificados em dois grupos: m\xe9todos expl\xedcitos e m\xe9todos impl\xedcitos. Os m\xe9todos expl\xedcitos, devido \xe0 sua formula\xe7\xe3o, calculam diretamente o estado do sistema em um instante de tempo posterior, enquanto m\xe9todos impl\xedcitos envolvem estados atuais e posteriores em suas equa\xe7\xf5es, exigindo, portanto, um processo iterativo."),Object(m.b)("p",null,"As constantes de tempo presentes no estudo de estabilidade t\xeam uma grande varia\xe7\xe3o em seu valor (podem variar de ",Object(m.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(m.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"1"),Object(m.b)("msup",{parentName:"mrow"},Object(m.b)("mn",{parentName:"msup"},"0"),Object(m.b)("mrow",{parentName:"msup"},Object(m.b)("mo",{parentName:"mrow"},"\u2212"),Object(m.b)("mn",{parentName:"mrow"},"3"))),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"s")),Object(m.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"10^{-3}~s")))),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8141079999999999em",verticalAlign:"0em"}})),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8141079999999999em"}}),Object(m.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.063em",marginRight:"0.05em"}}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"\u2212"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"3"))))))))),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mspace nobreak"}),"\xa0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"s")))))," a ",Object(m.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(m.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"10"),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"s")),Object(m.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"10~s")))),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}})),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mspace nobreak"}),"\xa0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"s"))))),"). Isso torna o sistema de equa\xe7\xf5es diferenciais da m\xe1quina s\xedncrona como um sistema r\xedgido (",Object(m.b)("em",{parentName:"p"},Object(m.b)("a",Object(s.a)({parentName:"em"},{href:"https://en.wikipedia.org/wiki/Stiff_equation"}),"stiff equation")),"). Caso a an\xe1lise da estabilidade num\xe9rica tanto das equa\xe7\xf5es diferenciais r\xedgidas quanto do m\xe9todo de integra\xe7\xe3o obtenham o mesmo comportamento, o m\xe9todo \xe9 chamado de absolutamente est\xe1vel, ou A-est\xe1vel."),Object(m.b)("p",null,"M\xe9todos de integra\xe7\xe3o num\xe9rica expl\xedcitos, como por exemplo o ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods"}),"Runge-Kutta")," de quarta ordem, n\xe3o podem ser A-est\xe1veis e, portanto, normalmente possuem comportamento ruim em problemas com equa\xe7\xf5es diferenciais r\xedgidas. Por outro lado, m\xe9todos impl\xedcitos podem ser A-est\xe1veis. 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AIEE Transactions, v. 48, n. 3, jul 1929. doi: ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1109/T-AIEE.1929.5055275"}),"https://doi.org/10.1109/T-AIEE.1929.5055275")),Object(m.b)("li",{parentName:"ol"},"ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1109/9780470545577"}),"https://doi.org/10.1109/9780470545577")),Object(m.b)("li",{parentName:"ol"},"SAUER, P. W.; PAI, M. A. Power System Dynamics and Stability. Pretience Hall, Upper Saddle River, 1998."),Object(m.b)("li",{parentName:"ol"},"KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994."),Object(m.b)("li",{parentName:"ol"},"DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1109/TPAS.1972.293341"}),"https://doi.org/10.1109/TPAS.1972.293341"))))}i.isMDXComponent=!0}}]);
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Retorna o \xe2ngulo em radianos.")),Object(b.b)("tr",{parentName:"tbody"},Object(b.b)("td",Object(n.a)({parentName:"tr"},{align:null}),Object(b.b)("inlineCode",{parentName:"td"},"sinh(A)")),Object(b.b)("td",Object(n.a)({parentName:"tr"},{align:null}),"Mesmo que ",Object(b.b)("inlineCode",{parentName:"td"},"sin()"),", mas para seno hiperb\xf3lico.")),Object(b.b)("tr",{parentName:"tbody"},Object(b.b)("td",Object(n.a)({parentName:"tr"},{align:null}),Object(b.b)("inlineCode",{parentName:"td"},"sqrt(A)")),Object(b.b)("td",Object(n.a)({parentName:"tr"},{align:null}),"Raiz quadrada de ",Object(b.b)("inlineCode",{parentName:"td"},"A"),".")),Object(b.b)("tr",{parentName:"tbody"},Object(b.b)("td",Object(n.a)({parentName:"tr"},{align:null}),Object(b.b)("inlineCode",{parentName:"td"},"tan(A)")),Object(b.b)("td",Object(n.a)({parentName:"tr"},{align:null}),"Tangente de ",Object(b.b)("inlineCode",{parentName:"td"},"A"),". 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Retorna o n\xfamero inteiro de ",Object(b.b)("inlineCode",{parentName:"td"},"A")," sem a parcela fracionada.")))),Object(b.b)("p",null,"\xc9 poss\xedvel atribuir novas vari\xe1veis utilizando a seguinte sintaxe: "),Object(b.b)("pre",null,Object(b.b)("code",Object(n.a)({parentName:"pre"},{className:"language-cpp"}),"<nome da vari\xe1vel> := <express\xe3o>; <fun\xe7\xe3o>\n")),Object(b.b)("p",null,"Por exemplo:"),Object(b.b)("pre",null,Object(b.b)("code",Object(n.a)({parentName:"pre"},{className:"language-cpp"}),"comprimento := sqrt(x*x+y*y); 2*comprimento*sin(comprimento)\n")),Object(b.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(b.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-heading"}),Object(b.b)("h5",{parentName:"div"},Object(b.b)("span",Object(n.a)({parentName:"h5"},{className:"admonition-icon"}),Object(b.b)("svg",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(b.b)("path",Object(n.a)({parentName:"svg"},{fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"})))),"Dica")),Object(b.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-content"}),Object(b.b)("p",{parentName:"div"},"O caractere de espa\xe7o e de nova linha s\xe3o ignorados na interpreta\xe7\xe3o da express\xe3o, portanto para maior organiza\xe7\xe3o, o c\xf3digo anterior pode ser escrito da seguinte forma:"),Object(b.b)("pre",{parentName:"div"},Object(b.b)("code",Object(n.a)({parentName:"pre"},{className:"language-cpp"}),"comprimento := sqrt(x*x+y*y);\n2*comprimento*sin(comprimento)\n")))),Object(b.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(b.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-heading"}),Object(b.b)("h5",{parentName:"div"},Object(b.b)("span",Object(n.a)({parentName:"h5"},{className:"admonition-icon"}),Object(b.b)("svg",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(b.b)("path",Object(n.a)({parentName:"svg"},{fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"})))),"Cuidado!")),Object(b.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-content"}),Object(b.b)("p",{parentName:"div"},"Note que a express\xe3o que determina o valor de sa\xedda do bloco ",Object(b.b)("strong",{parentName:"p"},"n\xe3o possui")," o caractere ",Object(b.b)("inlineCode",{parentName:"p"},";")," em seu final."))),Object(b.b)("h2",{id:"formul\xe1rio-de-edi\xe7\xe3o-de-dados-do-bloco-de-express\xe3o-matem\xe1tica"},"Formul\xe1rio de edi\xe7\xe3o de dados do bloco de Express\xe3o Matem\xe1tica"),Object(b.b)("p",null,"O formul\xe1rio de inser\xe7\xe3o e edi\xe7\xe3o das entradas do bloco e da express\xe3o matem\xe1tica gen\xe9rica, assim como ferramentas de aux\xedlio de sua constru\xe7\xe3o \xe9 apresentado na figura abaixo."),Object(b.b)("div",null,Object(b.b)("center",null,Object(b.b)("img",{src:Object(m.a)("images/mathExpressionForm.png"),alt:"Formul\xe1rio de edi\xe7\xe3o de dados do bloco de express\xe3o matem\xe1tica no PSP-UFU",title:"Formul\xe1rio de edi\xe7\xe3o de dados do bloco de express\xe3o matem\xe1tica no PSP-UFU"}))),Object(b.b)("p",null,"No campo de \u201cVari\xe1veis de entrada\u201d \xe9 inserida uma lista com os nomes das entradas separados por espa\xe7os. Um n\xfamero qualquer de entrada pode ser definido nessa lista esses nomes s\xe3o apresentados no \xedcone gr\xe1fico presente no editor de controle, al\xe9m de serem destacados na express\xe3o inserida pelo usu\xe1rio. O n\xfamero de entradas e sa\xeddas se comporta de forma semelhante aos ",Object(b.b)("a",Object(n.a)({parentName:"p"},{href:"sum"}),"blocos somadores"),"."),Object(b.b)("p",null,"Abaixo do campo das vari\xe1veis de entrada est\xe1 presente o local para inser\xe7\xe3o da express\xe3o matem\xe1tica. A sintaxe da express\xe3o inserida pelo usu\xe1rio possui realce (por meio de diferentes formas e cores da fonte) para n\xfameros, operadores, vari\xe1veis de entrada, fun\xe7\xf5es e constantes, facilitando a cria\xe7\xe3o, manipula\xe7\xe3o e identifica\xe7\xe3o de erros de digita\xe7\xe3o e l\xf3gica."),Object(b.b)("div",{className:"admonition admonition-info alert alert--info"},Object(b.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-heading"}),Object(b.b)("h5",{parentName:"div"},Object(b.b)("span",Object(n.a)({parentName:"h5"},{className:"admonition-icon"}),Object(b.b)("svg",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"}),Object(b.b)("path",Object(n.a)({parentName:"svg"},{fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"})))),"Informa\xe7\xe3o")),Object(b.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-content"}),Object(b.b)("p",{parentName:"div"},"Como ferramenta de aux\xedlio ao usu\xe1rio foi desenvolvida uma verifica\xe7\xe3o da express\xe3o inserida."))),Object(b.b)("p",null,"Tal ferramenta ir\xe1 encontrar erros e indicar\xe1 ao usu\xe1rio qual o tipo do erro, al\xe9m da sua localiza\xe7\xe3o, destacando-o. A figura anterior exemplifica a identifica\xe7\xe3o autom\xe1tica do erro pelo PSP-UFU, assim como sua posi\xe7\xe3o na express\xe3o inserida."),Object(b.b)("h2",{id:"exemplo-de-express\xe3o-matem\xe1tica"},"Exemplo de Express\xe3o Matem\xe1tica"),Object(b.b)("p",null,"A corrente de campo em ",Object(b.b)("span",Object(n.a)({parentName:"p"},{className:"math math-inline"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"katex"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"katex-mathml"}),Object(b.b)("math",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(b.b)("semantics",{parentName:"math"},Object(b.b)("mrow",{parentName:"semantics"},Object(b.b)("mi",{parentName:"mrow"},"p"),Object(b.b)("mi",Object(n.a)({parentName:"mrow"},{mathvariant:"normal"}),"."),Object(b.b)("mi",{parentName:"mrow"},"u"),Object(b.b)("mi",Object(n.a)({parentName:"mrow"},{mathvariant:"normal"}),".")),Object(b.b)("annotation",Object(n.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"p.u.")))),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"base"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"strut",style:{height:"0.625em",verticalAlign:"-0.19444em"}})),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault"}),"p"),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"mord"}),"."),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault"}),"u"),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"mord"}),".")))))," pode ser estimada de forma aproximada utilizando as pot\xeancias ativa (",Object(b.b)("span",Object(n.a)({parentName:"p"},{className:"math math-inline"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"katex"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"katex-mathml"}),Object(b.b)("math",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(b.b)("semantics",{parentName:"math"},Object(b.b)("mrow",{parentName:"semantics"},Object(b.b)("mi",{parentName:"mrow"},"P")),Object(b.b)("annotation",Object(n.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"P")))),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"base"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"strut",style:{height:"0.68333em",verticalAlign:"0em"}})),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.13889em"}}),"P"))))),") e reativa (",Object(b.b)("span",Object(n.a)({parentName:"p"},{className:"math math-inline"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"katex"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"katex-mathml"}),Object(b.b)("math",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(b.b)("semantics",{parentName:"math"},Object(b.b)("mrow",{parentName:"semantics"},Object(b.b)("mi",{parentName:"mrow"},"Q")),Object(b.b)("annotation",Object(n.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q")))),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"base"}),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(b.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"))))),"), assim como as reat\xe2ncias transit\xf3rias de eixo direto 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v;\n")),Object(b.b)("p",null,"Com todos os dados necess\xe1rios pode-se calcular a corrente de campo:"),Object(b.b)("pre",null,Object(b.b)("code",Object(n.a)({parentName:"pre"},{className:"language-cpp"}),"i_f := sqrt((v + yq)^2 + p^2) + (xd / xq + 1.0) *\n       ((yq * (v + yq) + yp^2) / sqrt((v + yq)^2 + p^2));\n")),Object(b.b)("p",null,"Finalmente, pode-se definir a sa\xedda do bloco (n\xe3o inserindo ",Object(b.b)("inlineCode",{parentName:"p"},";"),"):"),Object(b.b)("pre",null,Object(b.b)("code",Object(n.a)({parentName:"pre"},{className:"language-cpp"}),"i_f\n")),Object(b.b)("p",null,"Portanto, para as entradas definidas como ",Object(b.b)("inlineCode",{parentName:"p"},"p q v"),", a express\xe3o matem\xe1tica total para c\xe1lculo da corrente de campo ser\xe1:"),Object(b.b)("pre",null,Object(b.b)("code",Object(n.a)({parentName:"pre"},{className:"language-cpp"}),"xd := 0.146;\nxq := 0.0969;\nyp := xq * p / v;\nyq := xq * q / v;\ni_f := sqrt((v + yq)^2 + p^2) + (xd / xq + 1.0) *\n    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diff --git a/docs/62.617b9203.js b/docs/62.617b9203.js
deleted file mode 100644
index 92a8fac..0000000
--- a/docs/62.617b9203.js
+++ /dev/null
@@ -1,2 +0,0 @@
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Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(r.b)("a",Object(n.a)({parentName:"p"},{href:"https://pt.wikipedia.org/wiki/Unicode"}),"Unicode"),"."),Object(r.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(r.b)("h4",{id:"tens\xe3o-nominal"},"Tens\xe3o nominal"),Object(r.b)("p",null,"Utilizado para c\xe1lculo da imped\xe2ncia base de alguns elementos conectados, al\xe9m do c\xe1lculo da rela\xe7\xe3o de transforma\xe7\xe3o dos transformadores conectados entre barras. Sua unidade pode ser selecionada, sendo expressa em V ou em kV."),Object(r.b)("p",null,"A modifica\xe7\xe3o desse par\xe2metro ir\xe1 alterar toda a tens\xe3o do trecho conectado por linhas el\xe9tricas, sendo emitido um alerta ao usu\xe1rio."),Object(r.b)("h4",{id:"tens\xe3o-controlada"},"Tens\xe3o controlada"),Object(r.b)("p",null,"Caracteriza o barramento como ",Object(r.b)("strong",{parentName:"p"},"barra de tens\xe3o controlada (",Object(r.b)("a",Object(n.a)({parentName:"strong"},{href:"powerFlow"}),"Barra PV"),")"),","),Object(r.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(r.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-heading"}),Object(r.b)("h5",{parentName:"div"},Object(r.b)("span",Object(n.a)({parentName:"h5"},{className:"admonition-icon"}),Object(r.b)("svg",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(r.b)("path",Object(n.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(r.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-content"}),Object(r.b)("p",{parentName:"div"},"Essa op\xe7\xe3o \xe9 somente v\xe1lida caso alguma m\xe1quina s\xedncrona esteja conectado, caso contr\xe1rio esse valor ser\xe1 ignorado. Caso o limite de pot\xeancia reativa da m\xe1quina s\xedncrona conectada seja ultrapassado esse valor tamb\xe9m \xe9 ignorado."))),Object(r.b)("p",null,"O valor poder\xe1 ser inserido em p.u. ou em volts (ou kV caso a tens\xe3o nominal esteja nesta unidade)."),Object(r.b)("h4",{id:"barra-de-refer\xeancia"},"Barra de refer\xeancia"),Object(r.b)("p",null,"Caracteriza o barramento como ",Object(r.b)("strong",{parentName:"p"},"barra de refer\xeancia (",Object(r.b)("a",Object(n.a)({parentName:"strong"},{href:"powerFlow"}),"Barra de oscila\xe7\xe3o"),")"),". Essa op\xe7\xe3o \xe9 somente v\xe1lida caso esteja conectado um gerador s\xedncrono, caso contr\xe1rio uma mensagem de erro ser\xe1 exibida ao usu\xe1rio ao realizar algum dos c\xe1lculos do programa.")),Object(r.b)(b.a,{value:"fault",mdxType:"TabItem"},Object(r.b)("h4",{id:"inserir-falta"},"Inserir falta"),Object(r.b)("p",null,"Indica se existe um curto-circuito na barra nos estudos de ",Object(r.b)("a",Object(n.a)({parentName:"p"},{href:"fault"}),"curto-circuito"),"."),Object(r.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(r.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-heading"}),Object(r.b)("h5",{parentName:"div"},Object(r.b)("span",Object(n.a)({parentName:"h5"},{className:"admonition-icon"}),Object(r.b)("svg",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(r.b)("path",Object(n.a)({parentName:"svg"},{fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"})))),"Cuidado!")),Object(r.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-content"}),Object(r.b)("p",{parentName:"div"},"Nota-se que essa op\xe7\xe3o ir\xe1 inserir uma falta no c\xe1lculo de ",Object(r.b)("a",Object(n.a)({parentName:"p"},{href:"fault"}),"curto-circuito")," e ",Object(r.b)("strong",{parentName:"p"},"n\xe3o ser\xe1 considerado no estudo de estabilidade"),"."))),Object(r.b)("h4",{id:"tipo-de-falta"},"Tipo de falta"),Object(r.b)("p",null,"Seleciona o tipo de falha shunt da barra:"),Object(r.b)("ul",null,Object(r.b)("li",{parentName:"ul"},"trif\xe1sico para a terra"),Object(r.b)("li",{parentName:"ul"},"fase-fase"),Object(r.b)("li",{parentName:"ul"},"fase-fase-terra"),Object(r.b)("li",{parentName:"ul"},"fase-terra")),Object(r.b)("h4",{id:"local-da-falta"},"Local da falta"),Object(r.b)("p",null,"Seleciona a fase em que se situa a falta (ou combina\xe7\xe3o delas no caso de falha entre duas fases), sendo essa op\xe7\xe3o desabilitada para o tipo de falta trif\xe1sica."),Object(r.b)("h4",{id:"resist\xeancia-e-reat\xe2ncia-de-falta"},"Resist\xeancia e reat\xe2ncia de falta"),Object(r.b)("p",null,"Representam a imped\xe2ncia da falta. 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Os seguintes dados s\xe3o exibidos:"),Object(r.b)("ul",null,Object(r.b)("li",{parentName:"ul"},"m\xf3dulo da tens\xe3o do barramento"),Object(r.b)("li",{parentName:"ul"},"\xe2ngulo da tens\xe3o do barramento.")),Object(r.b)("h4",{id:"inserir-falta-1"},"Inserir falta"),Object(r.b)("p",null,"Insere uma falta trif\xe1sica na barra no c\xe1lculo de ",Object(r.b)("a",Object(n.a)({parentName:"p"},{href:"stability"}),"estabilidade"),"."),Object(r.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(r.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-heading"}),Object(r.b)("h5",{parentName:"div"},Object(r.b)("span",Object(n.a)({parentName:"h5"},{className:"admonition-icon"}),Object(r.b)("svg",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(r.b)("path",Object(n.a)({parentName:"svg"},{fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"})))),"Cuidado!")),Object(r.b)("div",Object(n.a)({parentName:"div"},{className:"admonition-content"}),Object(r.b)("p",{parentName:"div"},"Nota-se que essa op\xe7\xe3o ir\xe1 inserir uma falta no c\xe1lculo de ",Object(r.b)("a",Object(n.a)({parentName:"p"},{href:"stability"}),"estabilidade")," e ",Object(r.b)("strong",{parentName:"p"},"n\xe3o ser\xe1 considerado no estudo de ",Object(r.b)("a",Object(n.a)({parentName:"strong"},{href:"fault"}),"curto-circuito")),"."))),Object(r.b)("h4",{id:"tempo"},"Tempo"),Object(r.b)("p",null,"Instante no tempo (",Object(r.b)("span",Object(n.a)({parentName:"p"},{className:"math math-inline"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"katex"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"katex-mathml"}),Object(r.b)("math",Object(n.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(r.b)("semantics",{parentName:"math"},Object(r.b)("mrow",{parentName:"semantics"},Object(r.b)("msub",{parentName:"mrow"},Object(r.b)("mi",{parentName:"msub"},"t"),Object(r.b)("mrow",{parentName:"msub"},Object(r.b)("mi",{parentName:"mrow"},"f"),Object(r.b)("mi",{parentName:"mrow"},"a"),Object(r.b)("mi",{parentName:"mrow"},"l"),Object(r.b)("mi",{parentName:"mrow"},"t"),Object(r.b)("mi",{parentName:"mrow"},"a")))),Object(r.b)("annotation",Object(n.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"t_{falta}")))),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"base"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"strut",style:{height:"0.9011879999999999em",verticalAlign:"-0.286108em"}})),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"mord"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault"}),"t"),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"msupsub"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"vlist-r"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"vlist",style:{height:"0.3361079999999999em"}}),Object(r.b)("span",Object(n.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"mord mtight"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.10764em"}}),"f"),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"a"),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.01968em"}}),"l"),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"t"),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"a"))))),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"vlist-r"}),Object(r.b)("span",Object(n.a)({parentName:"span"},{className:"vlist",style:{height:"0.286108em"}}),Object(r.b)("span",{parentName:"span"})))))))))),") em que ocorre a falta durante os estudos de ",Object(r.b)("a",Object(n.a)({parentName:"p"},{href:"stability"}),"estabilidade"),". 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A falta ir\xe1 iniciar no instante ",Object(r.b)("span",Object(n.a)({parentName:"p"},{className:"math 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Essa barra \xe9 necessariamente geradora, uma vez que ela \xe9 respons\xe1vel pelo equil\xedbrio do balan\xe7o de pot\xeancia do sistema."))),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"O tipo de barra deve ser definido no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"bus"}),"elemento barramento"),"."),Object(n.b)("p",{parentName:"div"},"Note que ",Object(n.b)("strong",{parentName:"p"},"o sistema deve possuir somente uma barra de refer\xeancia"),"."))),Object(n.b)("p",null,"Como mencionado anteriormente, as equa\xe7\xf5es s\xe3o n\xe3o-lineares e a solu\xe7\xe3o anal\xedtica n\xe3o \xe9 pr\xe1tica. As solu\xe7\xf5es dessas equa\xe7\xf5es seguem ",Object(n.b)("strong",{parentName:"p"},Object(n.b)("a",Object(s.a)({parentName:"strong"},{href:"https://en.wikipedia.org/wiki/Iterative_method"}),"processos iterativos")),", em que s\xe3o atribu\xeddos valores estimados (ou iniciais) para as barras com tens\xf5es desconhecidas e, baseado na pot\xeancia ativa e reativa e m\xf3dulo da tens\xe3o especificados, calcula-se por meio das equa\xe7\xf5es previamente apresentadas as novas tens\xf5es complexas em cada n\xf3 do sistema."),Object(n.b)("p",null,"Na sequ\xeancia, esse conjunto de valores para as tens\xf5es em cada barra \xe9 utilizado para novamente calcular outro grupo de tens\xf5es. Cada c\xe1lculo de um novo conjunto de tens\xf5es \xe9 chamado itera\xe7\xe3o. O processo iterativo \xe9 repetido at\xe9 que as mudan\xe7as em todas as barras sejam menores do que um valor pr\xe9-estipulado, obtendo assim a converg\xeancia."),Object(n.b)("h2",{id:"execu\xe7\xe3o-do-fluxo-de-carga-no-psp-ufu"},"Execu\xe7\xe3o do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"powerEditor"}),"editor de pot\xeancia"),", a execu\xe7\xe3o do fluxo de carga \xe9 realizada no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," clicando no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Fluxo de carga"),"."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(b.a)("images/menuSimulationPF.svg"),alt:"Menu Simula\xe7\xe3o",title:"Menu Simula\xe7\xe3o"}))),Object(n.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"})))),"Dica")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"Caso o fluxo de carga tenha sido executado com sucesso, as setas de pot\xeancia ser\xe3o exibidas, a barra de status indicar\xe1 sucesso na opera\xe7\xe3o e os ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"text"}),"elementos de texto")," ser\xe3o atualizados."))),Object(n.b)("p",null,Object(n.b)("strong",{parentName:"p"},"Outra possibilidade")," \xe9 a execu\xe7\xe3o por meio do c\xe1lculo cont\xednuo, tamb\xe9m presente no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," e seu acionamento \xe9 realizado co clicar no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Habilitar solu\xe7\xe3o"),". Com essa op\xe7\xe3o, os c\xe1lculos est\xe1ticos selecionados nas ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o")," s\xe3o automaticamente realizados ao modificar quaisquer par\xe2metros da rede, como dados el\xe9tricos e acionamento dos disjuntores dos elementos (remo\xe7\xe3o ou inser\xe7\xe3o)."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"Evite construir o circuito com o c\xe1lculo cont\xednuo habilitado, uma vez que configura\xe7\xf5es tempor\xe1rias podem levar a erros de execu\xe7\xe3o da simula\xe7\xe3o."),Object(n.b)("p",{parentName:"div"},"Para desabilitar o c\xe1lculo cont\xednuo clique no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Desabilitar solu\xe7\xe3o"),"."))),Object(n.b)("p",null,"Os resultados do fluxo de carga s\xe3o exibidos nos ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"text"}),"elementos de texto vinculado"),", ao posicionar o mouse sobre os elementos e em ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"tabularReport"}),"relat\xf3rios tabulares"),"."),Object(n.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-fluxo-de-carga"},"Erros comuns na execu\xe7\xe3o do fluxo de carga"),Object(n.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao fluxo de carga."),Object(n.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-o-n\xfamero-m\xe1ximo-de-itera\xe7\xf5es-foi-alcan\xe7ado"},'A seguinte mensagem de erro \xe9 exibida: "O n\xfamero m\xe1ximo de itera\xe7\xf5es foi alcan\xe7ado"'),Object(n.b)("p",null,"Essa mensagem de erro \xe9 exibida quando o m\xe9todo de solu\xe7\xe3o num\xe9rica selecionado nas ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o")," atinge o n\xfamero m\xe1ximo de itera\xe7\xf5es inserido. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"O n\xfamero m\xe1ximo de itera\xe7\xf5es est\xe1 muito baixo"),". Alguns circuitos exigem um n\xfamero maior de itera\xe7\xf5es, portanto altere o valor do m\xe1ximo de itera\xe7\xf5es nas ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),". Tamb\xe9m tente alternar entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros de simula\xe7\xe3o est\xe3o inadequados"),". Caso um par\xe2metro do m\xe9todo de solu\xe7\xe3o esteja inadequado, como fator de acelera\xe7\xe3o ou toler\xe2ncia, o c\xe1lculo pode n\xe3o alcan\xe7ar a converg\xeancia. Altere esses par\xe2metros nas ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),".")),Object(n.b)("h4",{id:"os-dados-de-sa\xedda-s\xe3o-exibidos-como-nan-ou-nan"},'Os dados de sa\xedda s\xe3o exibidos como "NaN" ou "nan"'),Object(n.b)("p",null,'Isso ocorre devido a erros de opera\xe7\xf5es matem\xe1ticas nos c\xe1lculos de fluxo de carga. "NaN" significa ',Object(n.b)("em",{parentName:"p"},"Not a Number"),"."),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(n.b)("h2",{id:"m\xe9todos-de-solu\xe7\xe3o-num\xe9rica-do-fluxo-de-carga-no-psp-ufu"},"M\xe9todos de solu\xe7\xe3o num\xe9rica do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Os m\xe9todos implementados no programa para solu\xe7\xe3o do problema de fluxo de carga no PSP-UFU s\xe3o ",Object(n.b)("strong",{parentName:"p"},"Gauss-Seidel (GS)")," e ",Object(n.b)("strong",{parentName:"p"},"Newton-Raphson (NR)"),". Al\xe9m desses m\xe9todos cl\xe1ssicos, um m\xe9todo h\xedbrido pode ser utilizado (definido nas ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),"), em que \xe9 utilizado inicialmente o GS e na sequ\xeancia o NR, aumentando a chance de converg\xeancia do NR."),Object(n.b)("h3",{id:"gauss-seidel"},"Gauss-Seidel"),Object(n.b)("p",null,"O m\xe9todo de Gauss-Seidel tem sido bastante utilizado nas \xfaltimas d\xe9cadas para solu\xe7\xe3o do problema de fluxo de carga, uma vez que n\xe3o h\xe1 a necessidade de fatorar a matrizes, reduzindo o esfor\xe7o computacional. 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\xe9 o m\xf3dulo da tens\xe3o especificada para a barra PV. A barra de refer\xeancia tem o m\xf3dulo e \xe2ngulo de tens\xe3o fixos e n\xe3o participam no processo iterativo."),Object(n.b)("p",null,"O m\xe9todo de Gauss-Seidel mostra um ",Object(n.b)("strong",{parentName:"p"},"n\xfamero excessivo de itera\xe7\xf5es")," e, com o intuito reduzi-los, multiplicam-se as corre\xe7\xf5es de tens\xf5es por uma constante. Essa opera\xe7\xe3o amplia o valor da corre\xe7\xe3o, trazendo a tens\xe3o para mais perto do valor do valor final. Os multiplicadores que realizam essa converg\xeancia melhorada s\xe3o chamados de fatores de acelera\xe7\xe3o. Para qualquer sistema existem valores \xf3timos para os fatores de acelera\xe7\xe3o e uma escolha inadequada pode resultar em uma converg\xeancia mais lenta ou torn\xe1-la imposs\xedvel. Normalmente \xe9 utilizado um fator de acelera\xe7\xe3o igual a 1,6, valor definido como padr\xe3o no programa. 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tens\xe3o deixa de ser controlada partir de ent\xe3o."),Object(n.b)("p",null,"A verifica\xe7\xe3o de viola\xe7\xe3o e troca de tipo de barra pode ser realizada a cada itera\xe7\xe3o ou ao final da converg\xeancia do c\xe1lculo. No PSP-UFU foi implementada a \xfaltima estrat\xe9gia, uma vez que separa os conceitos de c\xe1lculo e de verifica\xe7\xe3o de limites, tornando mais f\xe1cil o desenvolvimento de novos m\xe9todos num\xe9ricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situa\xe7\xe3o n\xe3o violadora o c\xe1lculo iterativo deve ser retomado at\xe9 que obtenha novamente a converg\xeancia."),Object(n.b)("h2",{id:"refer\xeancias"},"Refer\xeancias"),Object(n.b)("ol",null,Object(n.b)("li",{parentName:"ol"},"MONTICELLI, A. J. Fluxo de Carga em Redes de Energia El\xe9trica. S\xe3o Paulo: Edgar Bl\xfccher, 1983."),Object(n.b)("li",{parentName:"ol"},"STEVENSON JR.; WILLIAN, D. Elementos de An\xe1lise de Sistemas de Pot\xeancia. 2\xaa ed. S\xe3o Paulo: McGraw-Hill, 1986."),Object(n.b)("li",{parentName:"ol"},"MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1007/978-3-642-13669-6"}),"https://doi.org/10.1007/978-3-642-13669-6")),Object(n.b)("li",{parentName:"ol"},"ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1002/9781118878286"}),"https://doi.org/10.1002/9781118878286")),Object(n.b)("li",{parentName:"ol"},"TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton\u2019s Method. IEEE Transaction on Power Apparatus and Systems, v. PAS-86, n. 11, nov. 1967. doi: ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1109/TPAS.1967.291823"}),"https://doi.org/10.1109/TPAS.1967.291823"))))}j.isMDXComponent=!0}}]);
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O vetor de pot\xeancias \xe9 ent\xe3o atualizado e os procedimentos convencionais de solu\xe7\xe3o do fluxo de pot\xeancia s\xe3o realizados.")),Object(n.b)("p",null,"Os passos de 2 a 4 s\xe3o repetidos at\xe9 que se obtenha a converg\xeancia."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"No PSP-UFU, os motores de indu\xe7\xe3o n\xe3o s\xe3o considerados no c\xe1lculo de curto-circuito."))),Object(n.b)("h3",{id:"motor-de-indu\xe7\xe3o-trif\xe1sico-no-estudo-de-estabilidade"},"Motor de indu\xe7\xe3o trif\xe1sico no estudo de estabilidade"),Object(n.b)("p",null,"Uma importante carga din\xe2mica s\xe3o os motores de indu\xe7\xe3o, uma vez que correspondem a uma parcela significativa das cargas presentes no sistema el\xe9trico. 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De forma semelhante ao modelo transit\xf3rio da m\xe1quina s\xedncrona, o motor de indu\xe7\xe3o pode ser modelado pelo circuito equivalente de Thevenin de tens\xe3o transit\xf3ria ",Object(n.b)("span",{parentName:"p",className:"math 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pot\xeancia constante")," no estudo de ",Object(n.b)("a",{parentName:"p",href:"powerFlow"},"fluxo de carga"),"."),Object(n.b)("p",{parentName:"div"},"A n\xe3o utiliza\xe7\xe3o dessa op\xe7\xe3o poder\xe1 gerar erros de regime permamente no estudo de ",Object(n.b)("a",{parentName:"p",href:"stability"},"estabilidade"),"."))),Object(n.b)("h4",{id:"utilizar-pot\xeancia-nominal-como-base"},"Utilizar pot\xeancia nominal como base"),Object(n.b)("p",null,"Caso essa op\xe7\xe3o seja marcada, o programa ir\xe1 utilizar a pot\xeancia nominal do gerador como base para a convers\xe3o das unidades, inclusive aqueles no formul\xe1rio de estabilidade, caso contr\xe1rio ser\xe1 usada a pot\xeancia base do sistema.")),Object(n.b)(c.a,{value:"stability",mdxType:"TabItem"},Object(n.b)("h4",{id:"imprimir-dados-da-m\xe1quina-de-indu\xe7\xe3o"},"Imprimir dados da m\xe1quina de indu\xe7\xe3o"),Object(n.b)("p",null,"Exibe os dados do gerador s\xedncrono nos gr\xe1ficos no tempo. 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no motor de indu\xe7\xe3o. 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das m\xe1quinas s\xedncronas s\xe3o introduzidos \u201cfatores de satura\xe7\xe3o\u201d que modificam as imped\xe2ncias do circuito equivalente, os quais dependem de uma reat\xe2ncia de dispers\xe3o efetiva, chamada de reat\xe2ncia de Potier (",Object(n.b)("span",{parentName:"p",className:"math 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magn\xe9tico que percorrem o ar na maioria de seu caminho e, portanto, \xe9 independente da satura\xe7\xe3o."),Object(n.b)("p",null,"O m\xe9todo implementado no programa permite reproduzir a satura\xe7\xe3o em ambos os eixos (direto e em quadratura), diferindo entre si devido \xe0 diverg\xeancia no tamanho do entreferro. \xc9 assumido que a soma vetorial das duas componentes do fluxo magn\xe9tico saturado est\xe1 em fase com a f.m.m. e proporcional \xe0 Tens\xe3o de Potier (",Object(n.b)("span",{parentName:"p",className:"math 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Na implementa\xe7\xe3o realizada no programa a utiliza\xe7\xe3o ou n\xe3o desse recurso \xe9 opcional e pode ser ",Object(n.b)("a",{parentName:"p",href:"simulationConfig"},"definida pelo usu\xe1rio"),"."),Object(n.b)("h2",{id:"formul\xe1rio-de-edi\xe7\xe3o-dos-geradores-s\xedncronos"},"Formul\xe1rio de edi\xe7\xe3o dos geradores s\xedncronos"),Object(n.b)("p",null,"A imagem abaixo apresenta o formul\xe1rio de inser\xe7\xe3o/altera\xe7\xe3o de dados dos geradores s\xedncronos:"),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(p.a)("images/syncGeneratorForm.png"),alt:"Formul\xe1rio dos geradores s\xedncronos no PSP-UFU",title:"Formul\xe1rio dos geradores s\xedncronos no PSP-UFU"}))),Object(n.b)("p",null,'Um segundo pelos dados de estabilidade, como mostra a figura abaixo, acessado ao clicar no bot\xe3o "Estabilidade" do formul\xe1rio principal. 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Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(n.b)("a",{parentName:"p",href:"https://pt.wikipedia.org/wiki/Unicode"},"Unicode"),"."),Object(n.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(n.b)("h4",{id:"pot\xeancia-nominal"},"Pot\xeancia nominal"),Object(n.b)("p",null,"Pot\xeancia nominal do gerador, inserida em MVA, kVA ou VA."),Object(n.b)("p",null,'Esse campo \xe9 especialmente importante caso a op\xe7\xe3o "Utilizar a pot\xeancia nominal como base" esteja marcada.'),Object(n.b)("h4",{id:"pot\xeancias-ativa-e-reativa"},"Pot\xeancias ativa e reativa"),Object(n.b)("p",null,"Pot\xeancias ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do gerador."),Object(n.b)("p",null,"Caso a barra conectada seja PV o valor de pot\xeancia reativa ser\xe1 ignorado e caso seja de refer\xeancia ambos os valores inseridos ser\xe3o desprezados."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Caso mais de um gerador esteja conectado na mesma barra, os valores de pot\xeancia reativa (nas barras de refer\xeancia e PV) e ativa (nas barras de refer\xeancia) s\xe3o igualmente distribu\xeddas, respeitando os limites individuais de pot\xeancia reativa."))),Object(n.b)("h4",{id:"pot\xeancias-reativas-m\xe1xima-e-m\xednima"},"Pot\xeancias reativas m\xe1xima e m\xednima"),Object(n.b)("p",null,"Limites de pot\xeancia reativa m\xe1xima e m\xednima do gerador para controle de tens\xe3o em barras PV. 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Essas propriedades s\xe3o atribu\xeddas e retiradas da lista gen\xe9rica por meio dos bot\xf5es "Adicionar" e "Remover", respectivamente.'))),Object(n.b)("h2",{id:"acesso-aos-controles-da-m\xe1quina-s\xedncrona"},"Acesso aos controles da m\xe1quina s\xedncrona"),Object(n.b)("p",null,"Como j\xe1 ",Object(n.b)("a",{parentName:"p",href:"syncGenerator#utilizar-avr-e-regulador-de-velocidade"},"mencionado anteriormente"),', os reguladores de velocidade e tens\xe3o da m\xe1quina s\xedncrona podem ser acionados ou inibidos por meio das caixas de sele\xe7\xe3o "',Object(n.b)("a",{parentName:"p",href:"syncGenerator#utilizar-avr-e-regulador-de-velocidade"},"Utilizar AVR e regulador de velocidade"),'". Ambas as op\xe7\xf5es ir\xe3o acessar o ',Object(n.b)("a",{parentName:"p",href:"controlEditor"},"editor de controles"),"."),Object(n.b)("p",null,"O acesso aos controles do ",Object(n.b)("strong",{parentName:"p"},"AVR"),' poder\xe3o ent\xe3o ser criados e manipulados ao clicar no bot\xe3o "Editar AVR", assim como o ',Object(n.b)("strong",{parentName:"p"},"Regulador de Velocidade"),' \xe9 acessado no bot\xe3o "Editar regulador de velocidade".'),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"No PSP-UFU a op\xe7\xe3o de editar o ",Object(n.b)("strong",{parentName:"p"},"AVR")," engloba mais que somente o controle de tens\xe3o da m\xe1quina. Nele ",Object(n.b)("strong",{parentName:"p"},"deve")," ser inserida a malha de controle da m\xe1quina assim como a ",Object(n.b)("strong",{parentName:"p"},"excitatriz da m\xe1quina s\xedncrona"),". Outras estrat\xe9gias de controle (opcionais), como PSS (",Object(n.b)("em",{parentName:"p"},"Power System Stabilizer"),") e/ou controles de sobre e sub excita\xe7\xe3o, s\xe3o tamb\xe9m implementadas em conjunto."))),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Assim como no AVR, o ",Object(n.b)("strong",{parentName:"p"},"Regulador de Velocidade")," engloba mais que a regula\xe7\xe3o prim\xe1ria da m\xe1quina. Nessa op\xe7\xe3o ",Object(n.b)("strong",{parentName:"p"},"deve")," ser inserida ao menos a malha de controle da regula\xe7\xe3o prim\xe1ria de velocidade, assim como o ",Object(n.b)("strong",{parentName:"p"},"modelo da turbina"),". Estrat\xe9gias opcionais de controle da velocidade tamb\xe9m s\xe3o inseridas nessa op\xe7\xe3o."))),Object(n.b)("h2",{id:"refer\xeancias"},"Refer\xeancias"),Object(n.b)("ol",null,Object(n.b)("li",{parentName:"ol"},"MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1007/978-3-642-13669-6"},"https://doi.org/10.1007/978-3-642-13669-6")),Object(n.b)("li",{parentName:"ol"},"ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1002/9781118878286"},"https://doi.org/10.1002/9781118878286")),Object(n.b)("li",{parentName:"ol"},"KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994."),Object(n.b)("li",{parentName:"ol"},"DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1109/TPAS.1972.293341"},"https://doi.org/10.1109/TPAS.1972.293341")),Object(n.b)("li",{parentName:"ol"},"IEEE Std 1110-2002 IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses. IEEE, New York, nov. 2003. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1109/IEEESTD.2003.94408"},"https://doi.org/10.1109/IEEESTD.2003.94408")),Object(n.b)("li",{parentName:"ol"},"KIMBARK, E. W. Power System Stability: Volume III \u2013 Synchronous Machine. New York: Wiley-IEEE Press, 1995.")))}O.isMDXComponent=!0}}]);
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Essa barra \xe9 necessariamente geradora, uma vez que ela \xe9 respons\xe1vel pelo equil\xedbrio do balan\xe7o de pot\xeancia do sistema."))),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"O tipo de barra deve ser definido no ",Object(n.b)("a",{parentName:"p",href:"bus"},"elemento barramento"),"."),Object(n.b)("p",{parentName:"div"},"Note que ",Object(n.b)("strong",{parentName:"p"},"o sistema deve possuir somente uma barra de refer\xeancia"),"."))),Object(n.b)("p",null,"Como mencionado anteriormente, as equa\xe7\xf5es s\xe3o n\xe3o-lineares e a solu\xe7\xe3o anal\xedtica n\xe3o \xe9 pr\xe1tica. As solu\xe7\xf5es dessas equa\xe7\xf5es seguem ",Object(n.b)("strong",{parentName:"p"},Object(n.b)("a",{parentName:"strong",href:"https://en.wikipedia.org/wiki/Iterative_method"},"processos iterativos")),", em que s\xe3o atribu\xeddos valores estimados (ou iniciais) para as barras com tens\xf5es desconhecidas e, baseado na pot\xeancia ativa e reativa e m\xf3dulo da tens\xe3o especificados, calcula-se por meio das equa\xe7\xf5es previamente apresentadas as novas tens\xf5es complexas em cada n\xf3 do sistema."),Object(n.b)("p",null,"Na sequ\xeancia, esse conjunto de valores para as tens\xf5es em cada barra \xe9 utilizado para novamente calcular outro grupo de tens\xf5es. Cada c\xe1lculo de um novo conjunto de tens\xf5es \xe9 chamado itera\xe7\xe3o. O processo iterativo \xe9 repetido at\xe9 que as mudan\xe7as em todas as barras sejam menores do que um valor pr\xe9-estipulado, obtendo assim a converg\xeancia."),Object(n.b)("h2",{id:"execu\xe7\xe3o-do-fluxo-de-carga-no-psp-ufu"},"Execu\xe7\xe3o do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(n.b)("a",{parentName:"p",href:"powerEditor"},"editor de pot\xeancia"),", a execu\xe7\xe3o do fluxo de carga \xe9 realizada no ",Object(n.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"menu Simula\xe7\xe3o")," clicando no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Fluxo de carga"),"."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(p.a)("images/menuSimulationPF.svg"),alt:"Menu Simula\xe7\xe3o",title:"Menu Simula\xe7\xe3o"}))),Object(n.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Dica")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Caso o fluxo de carga tenha sido executado com sucesso, as setas de pot\xeancia ser\xe3o exibidas, a barra de status indicar\xe1 sucesso na opera\xe7\xe3o e os ",Object(n.b)("a",{parentName:"p",href:"text"},"elementos de texto")," ser\xe3o atualizados."))),Object(n.b)("p",null,Object(n.b)("strong",{parentName:"p"},"Outra possibilidade")," \xe9 a execu\xe7\xe3o por meio do c\xe1lculo cont\xednuo, tamb\xe9m presente no ",Object(n.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"menu Simula\xe7\xe3o")," e seu acionamento \xe9 realizado co clicar no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Habilitar solu\xe7\xe3o"),". Com essa op\xe7\xe3o, os c\xe1lculos est\xe1ticos selecionados nas ",Object(n.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o")," s\xe3o automaticamente realizados ao modificar quaisquer par\xe2metros da rede, como dados el\xe9tricos e acionamento dos disjuntores dos elementos (remo\xe7\xe3o ou inser\xe7\xe3o)."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Evite construir o circuito com o c\xe1lculo cont\xednuo habilitado, uma vez que configura\xe7\xf5es tempor\xe1rias podem levar a erros de execu\xe7\xe3o da simula\xe7\xe3o."),Object(n.b)("p",{parentName:"div"},"Para desabilitar o c\xe1lculo cont\xednuo clique no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Desabilitar solu\xe7\xe3o"),"."))),Object(n.b)("p",null,"Os resultados do fluxo de carga s\xe3o exibidos nos ",Object(n.b)("a",{parentName:"p",href:"text"},"elementos de texto vinculado"),", ao posicionar o mouse sobre os elementos e em ",Object(n.b)("a",{parentName:"p",href:"tabularReport"},"relat\xf3rios tabulares"),"."),Object(n.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-fluxo-de-carga"},"Erros comuns na execu\xe7\xe3o do fluxo de carga"),Object(n.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao fluxo de carga."),Object(n.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-o-n\xfamero-m\xe1ximo-de-itera\xe7\xf5es-foi-alcan\xe7ado"},'A seguinte mensagem de erro \xe9 exibida: "O n\xfamero m\xe1ximo de itera\xe7\xf5es foi alcan\xe7ado"'),Object(n.b)("p",null,"Essa mensagem de erro \xe9 exibida quando o m\xe9todo de solu\xe7\xe3o num\xe9rica selecionado nas ",Object(n.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o")," atinge o n\xfamero m\xe1ximo de itera\xe7\xf5es inserido. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"O n\xfamero m\xe1ximo de itera\xe7\xf5es est\xe1 muito baixo"),". Alguns circuitos exigem um n\xfamero maior de itera\xe7\xf5es, portanto altere o valor do m\xe1ximo de itera\xe7\xf5es nas ",Object(n.b)("a",{parentName:"li",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),". Tamb\xe9m tente alternar entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros de simula\xe7\xe3o est\xe3o inadequados"),". Caso um par\xe2metro do m\xe9todo de solu\xe7\xe3o esteja inadequado, como fator de acelera\xe7\xe3o ou toler\xe2ncia, o c\xe1lculo pode n\xe3o alcan\xe7ar a converg\xeancia. Altere esses par\xe2metros nas ",Object(n.b)("a",{parentName:"li",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),".")),Object(n.b)("h4",{id:"os-dados-de-sa\xedda-s\xe3o-exibidos-como-nan-ou-nan"},'Os dados de sa\xedda s\xe3o exibidos como "NaN" ou "nan"'),Object(n.b)("p",null,'Isso ocorre devido a erros de opera\xe7\xf5es matem\xe1ticas nos c\xe1lculos de fluxo de carga. "NaN" significa ',Object(n.b)("em",{parentName:"p"},"Not a Number"),"."),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(n.b)("h2",{id:"m\xe9todos-de-solu\xe7\xe3o-num\xe9rica-do-fluxo-de-carga-no-psp-ufu"},"M\xe9todos de solu\xe7\xe3o num\xe9rica do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Os m\xe9todos implementados no programa para solu\xe7\xe3o do problema de fluxo de carga no PSP-UFU s\xe3o ",Object(n.b)("strong",{parentName:"p"},"Gauss-Seidel (GS)")," e ",Object(n.b)("strong",{parentName:"p"},"Newton-Raphson (NR)"),". Al\xe9m desses m\xe9todos cl\xe1ssicos, um m\xe9todo h\xedbrido pode ser utilizado (definido nas ",Object(n.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),"), em que \xe9 utilizado inicialmente o GS e na sequ\xeancia o NR, aumentando a chance de converg\xeancia do NR."),Object(n.b)("h3",{id:"gauss-seidel"},"Gauss-Seidel"),Object(n.b)("p",null,"O m\xe9todo de Gauss-Seidel tem sido bastante utilizado nas \xfaltimas d\xe9cadas para solu\xe7\xe3o do problema de fluxo de carga, uma vez que n\xe3o h\xe1 a necessidade de fatorar a matrizes, reduzindo o esfor\xe7o computacional. Atualmente, restri\xe7\xf5es computacionais s\xe3o menos problem\xe1ticas e outros m\xe9todos s\xe3o normalmente escolhidos, por\xe9m o Gauss-Seidel ainda possui valor did\xe1tico e, visto que o PSP-UFU tamb\xe9m possui fins educacionais, optou-se pela implementa\xe7\xe3o desse m\xe9todo."),Object(n.b)("p",null,"Para iniciar as itera\xe7\xf5es do m\xe9todo s\xe3o necess\xe1rios valores iniciais para as tens\xf5es (",Object(n.b)("span",{parentName:"p",className:"math math-inline"},Object(n.b)("span",{parentName:"span",className:"katex"},Object(n.b)("span",{parentName:"span",className:"katex-mathml"},Object(n.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mover",{parentName:"msubsup",accent:"true"},Object(n.b)("mi",{parentName:"mover"},"V"),Object(n.b)("mo",{parentName:"mover"},"\u02d9")),Object(n.b)("mi",{parentName:"msubsup"},"i"),Object(n.b)("mn",{parentName:"msubsup"},"0"))),Object(n.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\dot{V}_i^0")))),Object(n.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(n.b)("span",{parentName:"span",className:"base"},Object(n.b)("span",{parentName:"span",className:"strut",style:{height:"1.178854em",verticalAlign:"-0.258664em"}}),Object(n.b)("span",{parentName:"span",className:"mord"},Object(n.b)("span",{parentName:"span",className:"mord accent"},Object(n.b)("span",{parentName:"span",className:"vlist-t"},Object(n.b)("span",{parentName:"span",className:"vlist-r"},Object(n.b)("span",{parentName:"span",className:"vlist",style:{height:"0.9201900000000001em"}},Object(n.b)("span",{parentName:"span",style:{top:"-3em"}},Object(n.b)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),Object(n.b)("span",{parentName:"span",className:"mord"},Object(n.b)("span",{parentName:"span",className:"mord mathdefault",style:{marginRight:"0.22222em"}},"V"))),Object(n.b)("span",{parentName:"span",style:{top:"-3.25233em"}},Object(n.b)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),Object(n.b)("span",{parentName:"span",className:"accent-body",style:{left:"-0.13889em"}},Object(n.b)("span",{parentName:"span",className:"mord"},"\u02d9"))))))),Object(n.b)("span",{parentName:"span",className:"msupsub"},Object(n.b)("span",{parentName:"span",className:"vlist-t vlist-t2"},Object(n.b)("span",{parentName:"span",className:"vlist-r"},Object(n.b)("span",{parentName:"span",className:"vlist",style:{height:"0.8141079999999999em"}},Object(n.b)("span",{parentName:"span",style:{top:"-2.441336em",marginRight:"0.05em"}},Object(n.b)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),Object(n.b)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},Object(n.b)("span",{parentName:"span",className:"mord mathdefault mtight"},"i"))),Object(n.b)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},Object(n.b)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),Object(n.b)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},Object(n.b)("span",{parentName:"span",className:"mord mtight"},"0")))),Object(n.b)("span",{parentName:"span",className:"vlist-s"},"\u200b")),Object(n.b)("span",{parentName:"span",className:"vlist-r"},Object(n.b)("span",{parentName:"span",className:"vlist",style:{height:"0.258664em"}},Object(n.b)("span",{parentName:"span"})))))))))),") que devem ser calculadas. 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mtight"},Object(n.b)("span",{parentName:"span",className:"mord mathdefault mtight"},"c"),Object(n.b)("span",{parentName:"span",className:"mord mathdefault mtight"},"a"),Object(n.b)("span",{parentName:"span",className:"mord mathdefault mtight",style:{marginRight:"0.01968em"}},"l"),Object(n.b)("span",{parentName:"span",className:"mord mathdefault mtight"},"c"))))),Object(n.b)("span",{parentName:"span",className:"vlist-s"},"\u200b")),Object(n.b)("span",{parentName:"span",className:"vlist-r"},Object(n.b)("span",{parentName:"span",className:"vlist",style:{height:"0.247em"}},Object(n.b)("span",{parentName:"span"}))))))))))),Object(n.b)("p",null,"O passo seguinte \xe9 a forma\xe7\xe3o da matriz jacobiana. Com isso \xe9 poss\xedvel calcular as corre\xe7\xf5es de m\xf3dulo e \xe2ngulo das tens\xf5es de todas as barras (com exce\xe7\xe3o da barra de refer\xeancia). Para tanto, no PSP-UFU utiliza-se o m\xe9todo de ",Object(n.b)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/Gaussian_elimination"},"Elimina\xe7\xe3o Gaussiana")," e em sequ\xeancia a substitui\xe7\xe3o regressiva. 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London: Springer, 2010. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1007/978-3-642-13669-6"},"https://doi.org/10.1007/978-3-642-13669-6")),Object(n.b)("li",{parentName:"ol"},"ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1002/9781118878286"},"https://doi.org/10.1002/9781118878286")),Object(n.b)("li",{parentName:"ol"},"TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton\u2019s Method. IEEE Transaction on Power Apparatus and Systems, v. PAS-86, n. 11, nov. 1967. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1109/TPAS.1967.291823"},"https://doi.org/10.1109/TPAS.1967.291823"))))}N.isMDXComponent=!0}}]);
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A edi\xe7\xe3o dos dados el\xe9tricos \xe9 feita por meio de formul\xe1rios de dados acessados clicando duas vezes sobre os elementos."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(i.a)("images/workspace.png"),alt:"\xc1rea de Trabalho",title:"\xc1rea de Trabalho"}))),Object(n.b)("h3",{id:"barra-de-status"},"Barra de Status"),Object(n.b)("p",null,"A barra de status \xe9 respons\xe1vel por informa\xe7\xf5es interessantes sobre as circunst\xe2ncias atuais do programa: modo de opera\xe7\xe3o do mouse (editar, mover, arrastar), zoom aplicado e posi\xe7\xe3o do mouse na \xe1rea de trabalho, al\xe9m de dados sobre as a\xe7\xf5es dos usu\xe1rios, como: inser\xe7\xe3o de elementos, informa\xe7\xe3o sobre copiar e colar, etc."),Object(n.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Dica")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},Object(n.b)("strong",{parentName:"p"},"Sempre fique atendo \xe0 barra de status"),", pois s\xe3o fornecidas informa\xe7\xf5es importantes acerca da opera\xe7\xe3o do programa. Caso tenha alguma d\xfavida siga as instru\xe7\xf5es apresentadas nesse componente."))),Object(n.b)("h2",{id:"configura\xe7\xf5es-gerais"},"Configura\xe7\xf5es gerais"),Object(n.b)("p",null,"As configura\xe7\xf5es gerais do programa s\xe3o acessadas no submenu Ribbon ",Object(n.b)("strong",{parentName:"p"},"Arquivo"),". Essas configura\xe7\xf5es s\xe3o aplicadas para todos os projetos e permanecem gravadas no disco."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(i.a)("images/generalSettings.png"),alt:"Configura\xe7\xf5es gerais",title:"Configura\xe7\xf5es gerais"}))),Object(n.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Cuidado")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Algumas configura\xe7\xf5es de seguran\xe7a do seu computador (principalmente em sistemas Windows) podem gerar uma mensagem de erro ao confirmar as altera\xe7\xf5es das configura\xe7\xf5es gerais."),Object(n.b)("p",{parentName:"div"},"Para resolver esse problema basta ",Object(n.b)("strong",{parentName:"p"},"executar o PSP-UFU como administrador")," (clicar com bot\xe3o direito no atalho do programa e posteriormente em executar como administrador). Altere novamente as configura\xe7\xf5es e reinicie o programa."))),Object(n.b)("h3",{id:"idioma"},"Idioma"),Object(n.b)("p",null,"Atualmente os seguintes idiomas est\xe3o dispon\xedveis no PSP-UFU:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},"Ingl\xeas"),Object(n.b)("li",{parentName:"ul"},"Portugu\xeas")),Object(n.b)("div",{className:"admonition admonition-info alert alert--info"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"}))),"Informa\xe7\xe3o")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"O programa deve ser reiniciado para surtir efeito da altera\xe7\xe3o do idioma."))),Object(n.b)("h3",{id:"renderiza\xe7\xe3o"},"Renderiza\xe7\xe3o"),Object(n.b)("p",null,"Define como os elementos gr\xe1ficos do ",Object(n.b)("a",{parentName:"p",href:"powerEditor"},"editor de pot\xeancia")," e do ",Object(n.b)("a",{parentName:"p",href:"controlEditor"},"editor de controle")," s\xe3o desenhados na tela. Atualmente duas op\xe7\xf5es est\xe3o dispon\xedveis:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("a",{parentName:"li",href:"https://www.opengl.org/about/"},"OpenGL")),Object(n.b)("li",{parentName:"ul"},Object(n.b)("a",{parentName:"li",href:"https://docs.wxwidgets.org/3.0/classwx_graphics_context.html"},"Device Context"))),Object(n.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Cuidado!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},'o OpenGL pode n\xe3o ser suportado pelo seu computador. Nesse caso, ao criar um novo projeto a tela de trabalho n\xe3o \xe9 exibida ou \xe9 exibida uma tela preta. A solu\xe7\xe3o \xe9 alterar o renderizador para "Device Context". '),Object(n.b)("p",{parentName:"div"},"Caso apare\xe7a uma mensagem de erro, execute o programa como administrador e repita o processo."))),Object(n.b)("div",{className:"admonition admonition-info alert alert--info"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"}))),"Informa\xe7\xe3o")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"O programa deve ser reiniciado para surtir efeito de altera\xe7\xe3o do renderizador."))),Object(n.b)("h2",{id:"arquivos-de-projeto-do-psp-ufu"},"Arquivos de projeto do PSP-UFU"),Object(n.b)("p",null,"As op\xe7\xf5es de cria\xe7\xe3o, grava\xe7\xe3o e abertura de projetos no disco, assim como importa\xe7\xe3o de arquivos de outros programas est\xe3o presentes no submenu Ribbon ",Object(n.b)("strong",{parentName:"p"},"Arquivo"),"."),Object(n.b)("h3",{id:"novo-projeto"},"Novo projeto"),Object(n.b)("p",null,"A cria\xe7\xe3o de um novo projeto \xe9 realizada clicando no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Novo projeto")," no ",Object(n.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"submenu ",Object(n.b)("strong",{parentName:"a"},"Arquivo")),". Essa a\xe7\xe3o cria um sistema em branco na \xe1rea de trabalho, local onde \xe9 poss\xedvel inserir os elementos el\xe9tricos por meio do ",Object(n.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"submenu ",Object(n.b)("strong",{parentName:"a"},"Ferramentas"))," ou pelas ",Object(n.b)("a",{parentName:"p",href:"powerEditor#teclas-de-atalho"},"teclas de atalho"),"."),Object(n.b)("p",null,"Esse sistema em branco pode ser ciado utilizando o ",Object(n.b)("a",{parentName:"p",href:"powerEditor"},"Editor de Pot\xeancia"),"."),Object(n.b)("h3",{id:"salvar-e-salvar-como"},"Salvar e Salvar como..."),Object(n.b)("p",null,'A op\xe7\xe3o "Salvar" sobrep\xf5e as altera\xe7\xf5es realizadas no projeto aberto e grava no disco. A op\xe7\xe3o "Salvar como..." cria um novo arquivo e grava o projeto no disco com o aux\xedlio de uma janela de sele\xe7\xe3o de pasta (e defini\xe7\xe3o do nome do arquivo).'),Object(n.b)("div",{className:"admonition admonition-info alert alert--info"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"}))),"Informa\xe7\xe3o")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},'Para projetos que est\xe3o sendo gravados pela primeira vez a op\xe7\xe3o "Salvar" se comporta de forma id\xeantica \xe0 op\xe7\xe3o "Salvar como...".'))),Object(n.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Dica")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"O PSP-UFU grava os arquivos com a extens\xe3o ",Object(n.b)("em",{parentName:"p"},".psp"),". Esses arquivos nada mais s\xe3o que arquivos de texto utilizando a ",Object(n.b)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/Markup_language"},"linguagem de marca\xe7\xe3o")," XML (",Object(n.b)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/XML"},"eXtensible Markup Language"),"), que define uma s\xe9rie de regras de formata\xe7\xe3o dos dados de forma que eles s\xe3o tanto leg\xedveis por humanos quanto por m\xe1quinas."),Object(n.b)("p",{parentName:"div"},"Portanto, os dados el\xe9tricos contidos neles podem ser facilmente identificados e alterados, caso necess\xe1rio."))),Object(n.b)("h3",{id:"abrir-projeto"},"Abrir projeto"),Object(n.b)("p",null,"Essa op\xe7\xe3o abre os projetos gravados no disco por meio com o aux\xedlio de uma janela de sele\xe7\xe3o de arquivos. "),Object(n.b)("h3",{id:"importar-projeto"},"Importar projeto"),Object(n.b)("p",null,"O PSP-UFU permite a importa\xe7\xe3o de arquivos dos seguintes programas:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("a",{parentName:"li",href:"http://www.cepel.br/pt_br/produtos/programas-computacionais-por-categoria/menu/anarede-analise-de-redes-eletricas.htm"},"ANAREDE")),Object(n.b)("li",{parentName:"ul"},Object(n.b)("a",{parentName:"li",href:"https://matpower.org/"},"Matpower"))),Object(n.b)("p",null,"Para importa\xe7\xe3o dos arquivos do ",Object(n.b)("strong",{parentName:"p"},"ANAREDE")," s\xe3o utilizados tanto o arquivo de dados el\xe9tricos (",Object(n.b)("em",{parentName:"p"},".pwf"),") quanto de dados gr\xe1ficos dos elementos (",Object(n.b)("em",{parentName:"p"},".lst"),")."),Object(n.b)("p",null,"Uma vez que o arquivo do ",Object(n.b)("strong",{parentName:"p"},"Matpower")," (",Object(n.b)("em",{parentName:"p"},".m"),") n\xe3o possui dados gr\xe1ficos dos elementos el\xe9tricos, o diagrama unifilar \xe9 automaticamente gerado pelo PSP-UFU ao import\xe1-lo."),Object(n.b)("h4",{id:"gera\xe7\xe3o-autom\xe1tica-do-layout-de-diagramas-unifilares"},"Gera\xe7\xe3o autom\xe1tica do ",Object(n.b)("em",{parentName:"h4"},"layout")," de diagramas unifilares"),Object(n.b)("p",null,"Para criar o ",Object(n.b)("em",{parentName:"p"},"layout")," autom\xe1tico, utilizou-se a teoria dos grafos baseada no posicionamento direcionado \xe0 for\xe7a e aplica\xe7\xe3o de grafos com arestas ponderadas. De acordo com a teoria dos grafos, o sistema \xe9 modelado como v\xe9rtices e arestas. Nesse contexto, os barramentos s\xe3o os v\xe9rtices e os ramos do sistema (linhas e transformadores) s\xe3o as arestas. As localiza\xe7\xf5es de elementos de deriva\xe7\xe3o, assim como os n\xf3s dos elementos, s\xe3o automaticamente controladas pelo PSP-UFU."),Object(n.b)("p",null,"A ferramenta de ",Object(n.b)("em",{parentName:"p"},"layout")," autom\xe1tico \xe9 composta por um processo iterativo e os resultados s\xe3o mais refinados com um n\xfamero maior de itera\xe7\xf5es. Os melhores resultados s\xe3o obtidos utilizando o n\xfamero de itera\xe7\xf5es igual ou superior a cinco vezes o n\xfamero de barramentos do sistema."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"O ",Object(n.b)("em",{parentName:"p"},"layout")," autom\xe1tico ainda est\xe1 em desenvolvimento e algumas melhorias podem ser feitas para evitar cruzamentos e sobreposi\xe7\xf5es, por\xe9m, em seu estado atual, \xe9 totalmente funcional e satisfat\xf3rio."))),Object(n.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Dica")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Uma vez que o Matpower pode converter os formatos CDF (",Object(n.b)("em",{parentName:"p"},"Common Data Format"),") e PSS/E RAW para arquivos ",Object(n.b)("em",{parentName:"p"},".m"),", estes podem ser importados de forma indireta para o PSP-UFU."))))}p.isMDXComponent=!0}}]);
\ No newline at end of file
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Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(m.b)("a",{parentName:"p",href:"https://pt.wikipedia.org/wiki/Unicode"},"Unicode"),"."),Object(m.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(m.b)("h4",{id:"pot\xeancia-ativa"},"Pot\xeancia ativa"),Object(m.b)("p",null,"Parcela de pot\xeancia ativa da carga. Pode ser inserido em MW, kW W ou ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"p"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},"."),Object(m.b)("mi",{parentName:"mrow"},"u"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},".")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p.u.")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.19444em"}}),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"p"),Object(m.b)("span",{parentName:"span",className:"mord"},"."),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"u"),Object(m.b)("span",{parentName:"span",className:"mord"},".")))))," (na base do sistema)."),Object(m.b)("h4",{id:"pot\xeancia-reativa"},"Pot\xeancia reativa"),Object(m.b)("p",null,"Parcela de pot\xeancia reativa da carga. 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Normalmente utilizada no c\xe1lculo do erro de velocidade nos reguladores de velocidade, al\xe9m de entrada do PSS em AVRs;")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Pot\xeancia ativa e reativa")," (",Object(m.b)("em",{parentName:"p"},"entrada: AVR"),"): Pot\xeancia ativa fornecida pela m\xe1quina s\xedncrona, em ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"p"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},"."),Object(m.b)("mi",{parentName:"mrow"},"u"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},".")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p.u.")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.19444em"}}),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"p"),Object(m.b)("span",{parentName:"span",className:"mord"},"."),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"u"),Object(m.b)("span",{parentName:"span",className:"mord"},"."))))),", vari\xe1vel no tempo. Normalmente utilizada como entrada do PSS (pot\xeancia ativa) e controle de sub e sobrecorrente de excita\xe7\xe3o nos AVRs;")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Tens\xe3o terminal inicial")," (",Object(m.b)("em",{parentName:"p"},"entrada: AVR"),"): M\xf3dulo da tens\xe3o no barramento da m\xe1quina s\xedncrona pr\xe9via ao estudo din\xe2mico originado do fluxo de carga, em ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"p"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},"."),Object(m.b)("mi",{parentName:"mrow"},"u"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},".")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p.u.")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.19444em"}}),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"p"),Object(m.b)("span",{parentName:"span",className:"mord"},"."),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"u"),Object(m.b)("span",{parentName:"span",className:"mord"},"."))))),", fixo no tempo. Essa vari\xe1vel est\xe1 normalmente associada \xe0 refer\xeancia de tens\xe3o do AVR;")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Velocidade inicial")," (",Object(m.b)("em",{parentName:"p"},"entrada: AVR e RV"),"): Velocidade do sistema (",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"2"),Object(m.b)("mi",{parentName:"mrow"},"\u03c0"),Object(m.b)("msub",{parentName:"mrow"},Object(m.b)("mi",{parentName:"msub"},"f"),Object(m.b)("mrow",{parentName:"msub"},Object(m.b)("mi",{parentName:"mrow"},"r"),Object(m.b)("mi",{parentName:"mrow"},"e"),Object(m.b)("mi",{parentName:"mrow"},"f")))),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2 \\pi f_{ref}")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.980548em",verticalAlign:"-0.286108em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"2"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault",style:{marginRight:"0.03588em"}},"\u03c0"),Object(m.b)("span",{parentName:"span",className:"mord"},Object(m.b)("span",{parentName:"span",className:"mord mathdefault",style:{marginRight:"0.10764em"}},"f"),Object(m.b)("span",{parentName:"span",className:"msupsub"},Object(m.b)("span",{parentName:"span",className:"vlist-t vlist-t2"},Object(m.b)("span",{parentName:"span",className:"vlist-r"},Object(m.b)("span",{parentName:"span",className:"vlist",style:{height:"0.3361079999999999em"}},Object(m.b)("span",{parentName:"span",style:{top:"-2.5500000000000003em",marginLeft:"-0.10764em",marginRight:"0.05em"}},Object(m.b)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),Object(m.b)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},Object(m.b)("span",{parentName:"span",className:"mord mtight"},Object(m.b)("span",{parentName:"span",className:"mord mathdefault mtight",style:{marginRight:"0.02778em"}},"r"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault mtight"},"e"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault mtight",style:{marginRight:"0.10764em"}},"f"))))),Object(m.b)("span",{parentName:"span",className:"vlist-s"},"\u200b")),Object(m.b)("span",{parentName:"span",className:"vlist-r"},Object(m.b)("span",{parentName:"span",className:"vlist",style:{height:"0.286108em"}},Object(m.b)("span",{parentName:"span"})))))))))),"), definida nas ",Object(m.b)("a",{parentName:"p",href:"simulationConfig"},"op\xe7\xf5es de simula\xe7\xe3o"),", em rad/s, fixa no tempo. Normalmente utiliza-se essa vari\xe1vel como refer\xeancia de velocidade em RVs e normaliza\xe7\xe3o da velocidade;")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Pot\xeancia mec\xe2nica inicial")," (",Object(m.b)("em",{parentName:"p"},"entrada: RV"),"): Pot\xeancia mec\xe2nica inicial, calculada ap\xf3s a inicializa\xe7\xe3o das m\xe1quinas s\xedncronas com os dados originados do fluxo de carga, em ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"p"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},"."),Object(m.b)("mi",{parentName:"mrow"},"u"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},".")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p.u.")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.19444em"}}),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"p"),Object(m.b)("span",{parentName:"span",className:"mord"},"."),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"u"),Object(m.b)("span",{parentName:"span",className:"mord"},"."))))),", fixa no tempo. Normalmente \xe9 utilizada como refer\xeancia de pot\xeancia mec\xe2nica nos reguladores de velocidade;")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Varia\xe7\xe3o de velocidade e pot\xeancia ativa")," (",Object(m.b)("em",{parentName:"p"},"entrada: AVR"),"): C\xe1lculo da varia\xe7\xe3o dessas entradas entre os passos de integra\xe7\xe3o normalizada pelo passo de integra\xe7\xe3o, conforme a equa\xe7\xe3o:"),Object(m.b)("div",{parentName:"li",className:"math 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Para atingir a angula\xe7\xe3o pretendida basta repetir o processo de rota\xe7\xe3o."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/rotate.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/rotate.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"arrastar"},"Arrastar"),Object(n.b)("p",null,"Tr\xeas ferramentas muito \xfateis na navega\xe7\xe3o do circuito, principalmente em grandes redes, s\xe3o: arrastar, zoom e encaixar."),Object(n.b)("p",null,"A ferramenta arrastar possibilita mover todo o circuito ",Object(n.b)("strong",{parentName:"p"},"preservando as posi\xe7\xf5es relativas entre os elementos"),". Esta ferramenta pode ser acessada pelo submenu \u201cFerramentas\u201d ou ",Object(n.b)("strong",{parentName:"p"},"mantendo pressionada o ",Object(n.b)("em",{parentName:"strong"},"scroll")," do mouse"),", ativando assim o \u201cmodo arrastar\u201d. Ao ativ\xe1-lo todo o circuito ser\xe1 movido clicando e arrastando na dire\xe7\xe3o desejada."),Object(n.b)("p",null,"Para sair do \u201cmodo arrastar\u201d basta pressionar a tecla Escape do teclado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/drag.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/drag.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"zoom"},"Zoom"),Object(n.b)("p",null,"O zoom ser\xe1 aplicado ao utilizar o ",Object(n.b)("strong",{parentName:"p"},"scroll do mouse")," para mais ou menos zoom. A ferramenta zoom ir\xe1 aproximar ou distanciar o circuito a partir da posi\xe7\xe3o do ponteiro do mouse na tela para encontrar objetos ou destaca-los."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/zoom.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/zoom.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"encaixar"},"Encaixar"),Object(n.b)("p",null,"A ferramenta encaixar ir\xe1 mover o circuito e aplicar o zoom necess\xe1rio para que todos os componentes da rede sejam exibidos na tela. Esta ferramenta pode ser acessada atrav\xe9s do submenu \u201cFerramentas\u201d ou pela combina\xe7\xe3o de teclas \u201cShift + F\u201d."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/fit.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/fit.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"copiar-e-colar"},"Copiar e Colar"),Object(n.b)("p",null,"Duas ferramentas importantes durante o processo de cria\xe7\xe3o da rede el\xe9trica de pot\xeancia no ",Object(n.b)("em",{parentName:"p"},"software")," s\xe3o copiar e colar. Qualquer circuito criado no PSP-UFU poder\xe1 ser duplicado completamente ou parcialmente no mesmo projeto ou em m\xfaltiplos atrav\xe9s dessas ferramentas."),Object(n.b)("p",null,"Para copiar basta selecionar os elementos que se deseja duplicar e acessar a ferramenta no submenu \u201cFerramentas\u201d ou pressionando a combina\xe7\xe3o das teclas \u201cCtrl + C\u201d, ent\xe3o os dados do circuito ser\xe3o copiados para a \xe1rea de transfer\xeancia. Ao copiar um circuito ele poder\xe1 ser colado no mesmo projeto ou em um projeto distinto, utilizando o mesmo submenu ou pressionando a combina\xe7\xe3o das teclas \u201cCtrl + V\u201d."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/copyPaste.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/copyPaste.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"excluir"},"Excluir"),Object(n.b)("p",null,"Qualquer objeto poder\xe1 ser exclu\xeddo do projeto por meio da ferramenta deletar, presente no menu Ribbon, pela tecla de atalho Delete ou pelo acesso ao menu de contexto clicando com o bot\xe3o direito sobre o elemento selecionado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/delete.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/delete.webm"),type:"video/webm"})),Object(n.b)("h2",{id:"personaliza\xe7\xe3o-gr\xe1fica-os-elementos"},"Personaliza\xe7\xe3o gr\xe1fica os elementos"),Object(n.b)("p",null,"Alguns elementos permitem sua personaliza\xe7\xe3o gr\xe1fica, como as barras e linhas."),Object(n.b)("h3",{id:"barra"},"Barra"),Object(n.b)("p",null,"No caso dos barramentos, pode-se alterar o ",Object(n.b)("strong",{parentName:"p"},"comprimento da barra")," clicando e arrastando no ",Object(n.b)("em",{parentName:"p"},"pickbox")," exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado."),Object(n.b)("h3",{id:"linha"},"Linha"),Object(n.b)("p",null,"As linhas podem ser inseridas com ",Object(n.b)("strong",{parentName:"p"},'"n\xf3s" (pontos de ancoragem)'),", personalizando a disposi\xe7\xe3o das linhas no diagrama unifilar. Esses n\xf3s s\xe3o inseridos durante o processo de cria\xe7\xe3o do elemento, anteriormente \xe0 sele\xe7\xe3o da segunda barra."),Object(n.b)("p",null,"Ap\xf3s a sua inser\xe7\xe3o, novos n\xf3s podem ser anexados pelo menu de contexto acessado ao clicar com o bot\xe3o direito sobre a linha selecionada. Sua remo\xe7\xe3o tamb\xe9m \xe9 alcan\xe7ada por uma op\xe7\xe3o do menu de contexto."),Object(n.b)("p",null,"Assim como nos barramentos, a altera\xe7\xe3o da posi\xe7\xe3o dos n\xf3s \xe9 obtida clicando e arrastando o ",Object(n.b)("em",{parentName:"p"},"pickbox")," exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/graphicPropsChange.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/graphicPropsChange.webm"),type:"video/webm"})),Object(n.b)("h2",{id:"conex\xe3o-e-desconex\xe3o-de-elemetos-na-barra"},"Conex\xe3o e desconex\xe3o de elemetos na barra"),Object(n.b)("p",null,"Os elementos podem ser desconectados e reconectados \xe0s barras ap\xf3s sua inser\xe7\xe3o somente pela ferramenta ",Object(n.b)("em",{parentName:"p"},"drag-and-drop"),". Para isso, o elemento deve ser selecionado e o seu n\xf3 de conex\xe3o deve ser arrastado para o local requerido."),Object(n.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Cuidado!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"O elemento a ser desconectado/reconectado ",Object(n.b)("strong",{parentName:"p"},"deve ser previamente selecionado")," para efetuar a opera\xe7\xe3o. Caso contr\xe1rio a barra ser\xe1 movida ou nenhuma opera\xe7\xe3o ser\xe1 realizada."))),Object(n.b)("p",null,"No caso de desconex\xe3o, o elemento ser\xe1 automaticamente removido da simula\xe7\xe3o, sendo indicado pela sua cor (o elemento passa a ter cor ",Object(n.b)(d,{color:"#7f7f7f",mdxType:"Highlight"},"cinza"),")."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Ao reconectar o elemento ao circuito, deve-se ",Object(n.b)("strong",{parentName:"p"},'atentar para estado do seu "disjuntor"'),", indicado por um quadrado pr\xf3ximo ao seu n\xf3 de conex\xe3o. Um elemento reconectado volta ao circuito com seu disjuntor aberto (",Object(n.b)(d,{color:"#ff1919",mdxType:"Highlight"},"vermelho"),"), sendo necess\xe1rio clicar sobre o mesmo a fim de inserir o elemento no circuito."))),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/disconnect.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/disconnect.webm"),type:"video/webm"})))}m.isMDXComponent=!0},140:function(e,o,a){"use strict";a.d(o,"a",(function(){return b})),a.d(o,"b",(function(){return u}));var r=a(0),t=a.n(r);function n(e,o,a){return o in e?Object.defineProperty(e,o,{value:a,enumerable:!0,configurable:!0,writable:!0}):e[o]=a,e}function s(e,o){var a=Object.keys(e);if(Object.getOwnPropertySymbols){var r=Object.getOwnPropertySymbols(e);o&&(r=r.filter((function(o){return Object.getOwnPropertyDescriptor(e,o).enumerable}))),a.push.apply(a,r)}return a}function c(e){for(var o=1;o<arguments.length;o++){var a=null!=arguments[o]?arguments[o]:{};o%2?s(Object(a),!0).forEach((function(o){n(e,o,a[o])})):Object.getOwnPropertyDescriptors?Object.defineProperties(e,Object.getOwnPropertyDescriptors(a)):s(Object(a)).forEach((function(o){Object.defineProperty(e,o,Object.getOwnPropertyDescriptor(a,o))}))}return e}function i(e,o){if(null==e)return{};var a,r,t=function(e,o){if(null==e)return{};var a,r,t={},n=Object.keys(e);for(r=0;r<n.length;r++)a=n[r],o.indexOf(a)>=0||(t[a]=e[a]);return t}(e,o);if(Object.getOwnPropertySymbols){var n=Object.getOwnPropertySymbols(e);for(r=0;r<n.length;r++)a=n[r],o.indexOf(a)>=0||Object.prototype.propertyIsEnumerable.call(e,a)&&(t[a]=e[a])}return t}var d=t.a.createContext({}),l=function(e){var o=t.a.useContext(d),a=o;return e&&(a="function"==typeof e?e(o):c(c({},o),e)),a},b=function(e){var o=l(e.components);return t.a.createElement(d.Provider,{value:o},e.children)},m={inlineCode:"code",wrapper:function(e){var o=e.children;return t.a.createElement(t.a.Fragment,{},o)}},p=t.a.forwardRef((function(e,o){var a=e.components,r=e.mdxType,n=e.originalType,s=e.parentName,d=i(e,["components","mdxType","originalType","parentName"]),b=l(a),p=r,u=b["".concat(s,".").concat(p)]||b[p]||m[p]||n;return a?t.a.createElement(u,c(c({ref:o},d),{},{components:a})):t.a.createElement(u,c({ref:o},d))}));function u(e,o){var a=arguments,r=o&&o.mdxType;if("string"==typeof e||r){var n=a.length,s=new Array(n);s[0]=p;var c={};for(var i in o)hasOwnProperty.call(o,i)&&(c[i]=o[i]);c.originalType=e,c.mdxType="string"==typeof e?e:r,s[1]=c;for(var d=2;d<n;d++)s[d]=a[d];return t.a.createElement.apply(null,s)}return t.a.createElement.apply(null,a)}p.displayName="MDXCreateElement"},141:function(e,o,a){"use strict";a.d(o,"b",(function(){return n})),a.d(o,"a",(function(){return s}));var r=a(16),t=a(142);function n(){const{siteConfig:{baseUrl:e="/",url:o}={}}=Object(r.default)();return{withBaseUrl:(a,r)=>function(e,o,a,{forcePrependBaseUrl:r=!1,absolute:n=!1}={}){if(!a)return a;if(a.startsWith("#"))return a;if(Object(t.b)(a))return a;if(r)return o+a;const s=a.startsWith(o)?a:o+a.replace(/^\//,"");return n?e+s:s}(o,e,a,r)}}function s(e,o={}){const{withBaseUrl:a}=n();return a(e,o)}},142:function(e,o,a){"use strict";function r(e){return!0===/^(\w*:|\/\/)/.test(e)}function t(e){return void 0!==e&&!r(e)}a.d(o,"b",(function(){return r})),a.d(o,"a",(function(){return t}))}}]);
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diff --git a/docs/assets/js/8ed8a9e1.4a0217a1.js b/docs/assets/js/8ed8a9e1.4a0217a1.js
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A solu\xe7\xe3o das equa\xe7\xf5es se d\xe1 por meio de um ",Object(m.b)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/Numerical_integration"},"m\xe9todo de integra\xe7\xe3o num\xe9rica")," e, ent\xe3o, \xe9 observada a resposta do sistema de pot\xeancia."),Object(m.b)("p",null,"Caso a solu\xe7\xe3o no dom\xednio do tempo, ap\xf3s alguma perturba\xe7\xe3o, leve um ou mais par\xe2metros do sistema a valores fisicamente imposs\xedveis ou, at\xe9 mesmo, a solu\xe7\xe3o das equa\xe7\xf5es diferenciais n\xe3o obtenha converg\xeancia em processos iterativos, o sistema \xe9 dito ",Object(m.b)("strong",{parentName:"p"},"inst\xe1vel"),". Em contrapartida, caso as vari\xe1veis de estado dos elementos do sistema oscilem ap\xf3s uma perturba\xe7\xe3o e se estabele\xe7am em um novo ponto de opera\xe7\xe3o, mesmo que distinto dos valores iniciais, o sistema \xe9 considerado ",Object(m.b)("strong",{parentName:"p"},"est\xe1vel"),"."),Object(m.b)("h2",{id:"execu\xe7\xe3o-do-estudo-de-estabilidade-no-psp-ufu"},"Execu\xe7\xe3o do estudo de estabilidade no PSP-UFU"),Object(m.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(m.b)("a",{parentName:"p",href:"powerEditor"},"editor de pot\xeancia"),", assim como os ",Object(m.b)("a",{parentName:"p",href:"controlEditor"},"controles")," das ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quinas s\xedncronas"),", a execu\xe7\xe3o da estabilidade \xe9 realizada no ",Object(m.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"menu Simula\xe7\xe3o")," clicando no \xedcone do bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"Executar Estabilidade"),'. Ao clicar na parte inferior do bot\xe3o "Executar Estabilidade" ser\xe1 exibido um menu suspenso com a op\xe7\xe3o ',Object(m.b)("strong",{parentName:"p"},"Lista de eventos de estabilidade"),"."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/menuSimulationStability.svg"),alt:"Menu Simula\xe7\xe3o",title:"Menu Simula\xe7\xe3o"}))),Object(m.b)("p",null,"Ao executar o c\xe1lculo de estabilidade, uma caixa com o status do processo de simula\xe7\xe3o ser\xe1 exibida, indicando primeiramente a inicializa\xe7\xe3o do estudo e posteriormente o tempo de simula\xe7\xe3o calculado."),Object(m.b)("p",null,"Ao clicar na lista de eventos de estabilidade, ser\xe1 exibida uma janela com a descri\xe7\xe3o dos eventos de estabilidade inseridos."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabList.png"),alt:"Lista de eventos de estabilidade",title:"Lista de eventos de estabilidade"}))),Object(m.b)("h3",{id:"inserindo-um-evento-de-estabilidade"},"Inserindo um evento de estabilidade"),Object(m.b)("p",null,"Os dist\xfarbios mais comuns aplicados em estudos de estabilidade transit\xf3ria s\xe3o faltas e chaveamentos. Tais opera\xe7\xf5es s\xe3o facilmente realizadas no PSP-UFU:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Faltas"),": Curtos-circuitos trif\xe1sicos podem ser inseridos nos ",Object(m.b)("a",{parentName:"p",href:"bus"},"barramentos")," do sistema por meio da inclus\xe3o da imped\xe2ncia de falta na matriz admit\xe2ncia de barras da mesma maneira realizada em um elemento ",Object(m.b)("em",{parentName:"p"},"shunt"),". Tal valor \xe9 definido pelo usu\xe1rio e caso seja um curto-circuito franco, um valor n\xe3o nulo, mas suficientemente pr\xf3ximo de zero, \xe9 aplicado, de forma que a tens\xe3o no barramento \xe9 levada a zero durante o dist\xfarbio.")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Chaveamento de ramo"),": Da mesma forma que as faltas, o chaveamento de ramos \xe9 realizado por meio da altera\xe7\xe3o na matriz admit\xe2ncia, removendo ou inserindo os par\xe2metros do elemento a ser chaveado.\nCada ",Object(m.b)("a",{parentName:"p",href:"powerEditor"},"elemento de pot\xeancia"),' possui um bot\xe3o de "Chaveamento" ou "Estabilidade", em que pode ser inserido os tempos de remo\xe7\xe3o e/ou inser\xe7\xe3o do componente.')),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Chaveamento de m\xe1quinas"),": A remo\xe7\xe3o de uma m\xe1quina s\xedncrona \xe9 efetivada com a retirada de sua participa\xe7\xe3o no vetor de correntes, al\xe9m da remo\xe7\xe3o de sua admit\xe2ncia fict\xedcia.\nAssim como os ",Object(m.b)("a",{parentName:"p",href:"powerEditor"},"elementos de pot\xeancia"),", as ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quinas s\xedncronas"),' possui um bot\xe3o "Chaveamento", em que pode ser inserido os tempos de remo\xe7\xe3o e/ou inser\xe7\xe3o do componente.'),Object(m.b)("div",{parentName:"li",className:"admonition admonition-caution alert alert--warning"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Mesmo que removida da barra, os par\xe2metros das ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quinas s\xedncronas")," continuam a ser calculados com a corrente do estator nula, podendo fornecer resultados em uma eventual reconex\xe3o."))))),Object(m.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Dica")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Eventos nos ",Object(m.b)("strong",{parentName:"p"},"sistemas de controle")," podem ser facilmente introduzidos com o bloco de ",Object(m.b)("a",{parentName:"p",href:"mathExpression"},"express\xe3o matem\xe1tica"),". Nesse caso, tais eventos ",Object(m.b)("strong",{parentName:"p"},"n\xe3o")," ser\xe3o exibidos na lista de eventos de estabilidade."))),Object(m.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-estudo-de-estabilidade"},"Erros comuns na execu\xe7\xe3o do estudo de estabilidade"),Object(m.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao estudo de estabilidade."),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-n\xe3o-foi-poss\xedvel-construir-a-matriz-admit\xe2ncia"},'A seguinte mensagem de erro \xe9 exibida: "N\xe3o foi poss\xedvel construir a matriz admit\xe2ncia"'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel construir a matriz admit\xe2ncia de barras. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-erro-ao-inicializar-os-valores-de-satura\xe7\xe3o-do"},'A seguinte mensagem de erro \xe9 exibida: "Erro ao inicializar os valores de satura\xe7\xe3o do..."'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel calcular os fatores de satura\xe7\xe3o da ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quina s\xedncrona"),". As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O fator de satura\xe7\xe3o \xe9 menor que 1,2"),". Esse valor deve ser maior que 1,2, ou ir\xe1 gerar erros na simula\xe7\xe3o. Caso n\xe3o seja informado, a satura\xe7\xe3o da m\xe1quina n\xe3o \xe9 considerada nos c\xe1lculos."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros da m\xe1quina s\xedncrona est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o das m\xe1quinas s\xedncronas sejam inseridos, o c\xe1lculo dos fatores de satura\xe7\xe3o pode divergir. Verifique se os dados foram inseridos corretamente.")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-erro-ao-inicializar-o-avr--regulador-de-velocidade"},'A seguinte mensagem de erro \xe9 exibida: "Erro ao inicializar o AVR / regulador de velocidade..."'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel ",Object(m.b)("a",{parentName:"p",href:"controlEditor#inicializa%C3%A7%C3%A3o-do-sistema-de-controle"},"inicializar o sistema de controle")," de uma ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quina s\xedncrona"),". As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O sistema de controle est\xe1 vazio"),". Caso esteja habilitado o AVR e/ou o regulador de velocidade e o controle n\xe3o foi inserido, esse erro pode ser acionado. Insira o controle da m\xe1quina ou desmarque a op\xe7\xe3o de utiliza\xe7\xe3o do AVR e/ou regulador de velocidade."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O sistema de controle n\xe3o possui ao menos uma entrada e uma sa\xedda"),". O sistema de controle deve ter ao menos uma ",Object(m.b)("a",{parentName:"li",href:"io"},"entrada e uma sa\xedda"),", caso contr\xe1rio apresentar\xe1 erro de execu\xe7\xe3o."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O passo de integra\xe7\xe3o est\xe1 muito pequeno"),". Caso o passo de integra\xe7\xe3o esteja muito pequeno, os c\xe1lculos gerar\xe3o erros e ir\xe3o divergir. Reduza o passo de integra\xe7\xe3o nas ",Object(m.b)("a",{parentName:"li",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),".")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-imposs\xedvel-resolver-as-m\xe1quinas-do-sistema"},'A seguinte mensagem de erro \xe9 exibida: "Imposs\xedvel resolver as m\xe1quinas do sistema"'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel resolver as EADs das m\xe1quinas s\xedncronas inseridas no sistemas. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros da m\xe1quina s\xedncrona est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o das m\xe1quinas s\xedncronas sejam inseridos, o c\xe1lculo das EADs pode se tornar imposs\xedvel. Verifique se os dados foram inseridos corretamente."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O passo de integra\xe7\xe3o est\xe1 muito pequeno"),". Caso o passo de integra\xe7\xe3o esteja muito pequeno, os c\xe1lculos gerar\xe3o erros e ir\xe3o divergir. Reduza o passo de integra\xe7\xe3o nas ",Object(m.b)("a",{parentName:"li",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),"."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-erro-ao-inicializar-o-escorregamento-do-motor"},'A seguinte mensagem de erro \xe9 exibida: "Erro ao inicializar o escorregamento do motor..."'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel calcular o valor de escorregamento inicial do motor de indu\xe7\xe3o. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros de estabilidade do motor est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o dos motores de indu\xe7\xe3o sejam inseridos, o c\xe1lculo do escorregamento pode se tornar imposs\xedvel. Verifique se os dados foram inseridos corretamente.")),Object(m.b)("h2",{id:"estrutura-da-ferramenta-de-estabilidade"},"Estrutura da ferramenta de estabilidade"),Object(m.b)("p",null,"A estabilidade de um SEP \xe9 um problema din\xe2mico e necessita de modelos mais elaborados de elementos de pot\xeancia comparados \xe0queles apresentados nos outros estudos. Esses modelos s\xe3o descritos individualmente, com destaque \xe0s ",Object(m.b)("a",{parentName:"p",href:"syncGenerator#gerador-s%C3%ADncrono-no-estudo-de-estabilidade"},"m\xe1quinas s\xedncronas"),", ",Object(m.b)("a",{parentName:"p",href:"load#carga-no-estudo-de-estabilidade"},"cargas ZIP")," e ",Object(m.b)("a",{parentName:"p",href:"indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-estudo-de-estabilidade"},"motores de indu\xe7\xe3o"),"."),Object(m.b)("p",null,"A representa\xe7\xe3o dos demais componentes do sistema el\xe9trico: ",Object(m.b)("a",{parentName:"p",href:"line"},"linhas de transmiss\xe3o"),", ",Object(m.b)("a",{parentName:"p",href:"transformer"},"transformadores")," e elementos ",Object(m.b)("em",{parentName:"p"},"shunt")," (com exce\xe7\xe3o de cargas ZIP), que formam a rede de transmiss\xe3o ou distribui\xe7\xe3o balanceada, \xe9 realizada utilizando os mesmos modelos do ",Object(m.b)("a",{parentName:"p",href:"powerFlow"},"fluxo de carga"),"."),Object(m.b)("p",null,"Os modelos din\xe2micos para a an\xe1lise no dom\xednio do tempo s\xe3o na forma de um sistema de equa\xe7\xf5es alg\xe9brico-diferenciais (EADs), descritas a seguir:"),Object(m.b)("div",{className:"math 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alg\xe9bricas"),Object(m.b)("li",{parentName:"ul"},Object(m.b)("span",{parentName:"li",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"u")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"u")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.43056em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"u"))))),"\ts\xe3o as vari\xe1veis de entrada")),Object(m.b)("p",null,"As vari\xe1veis de entrada s\xe3o inseridas pelo usu\xe1rio e permanecem constantes durante todo o processo de c\xe1lculo. Para o problema de estabilidade, as vari\xe1veis alg\xe9bricas iniciais correspondem \xe0s tens\xf5es e \xe2ngulos das barras calculadas no ",Object(m.b)("a",{parentName:"p",href:"powerFlow"},"fluxo de carga"),". As vari\xe1veis de estado iniciais s\xe3o estimadas ap\xf3s a converg\xeancia do fluxo de pot\xeancia (inicializa\xe7\xe3o dos elementos din\xe2micos)."),Object(m.b)("p",null,"Na sequ\xeancia \xe9 apresentado, na forma de fluxogramas, a estrutura do m\xf3dulo de estabilidade implementado no PSP-UFU. O fluxograma abaixo mostra a estrutura geral da ferramenta de estabilidade."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabGeneral.svg"),alt:"Estrutura geral da ferramenta de estabilidade",title:"Estrutura geral da ferramenta de estabilidade"}))),Object(m.b)("p",null,"A inicializa\xe7\xe3o da rede el\xe9trica e dos elementos din\xe2micos, assim como o precesso iterativo para solu\xe7\xe3o das EADs s\xe3o apresentados nos dois fluxogramas conseguintes."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabInit.svg"),alt:"Inicializa\xe7\xe3o da ferramenta de estabilidade",title:"Inicializa\xe7\xe3o da ferramenta de estabilidade"}))),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabIterative.svg"),alt:"Processo iterativo para c\xe1lculo dos elementos din\xe2micos",title:"Estrutura geral da ferramenta de estabilidade"}))),Object(m.b)("h3",{id:"integra\xe7\xe3o-num\xe9rica"},"Integra\xe7\xe3o num\xe9rica"),Object(m.b)("p",null,"Para resolver as equa\xe7\xf5es diferenciais da m\xe1quina \xe9 necess\xe1rio um m\xe9todo de integra\xe7\xe3o num\xe9rica. Tais m\xe9todos s\xe3o classificados em dois grupos: m\xe9todos expl\xedcitos e m\xe9todos impl\xedcitos. Os m\xe9todos expl\xedcitos, devido \xe0 sua formula\xe7\xe3o, calculam diretamente o estado do sistema em um instante de tempo posterior, enquanto m\xe9todos impl\xedcitos envolvem estados atuais e posteriores em suas equa\xe7\xf5es, exigindo, portanto, um processo iterativo."),Object(m.b)("p",null,"As constantes de tempo presentes no estudo de estabilidade t\xeam uma grande varia\xe7\xe3o em seu valor (podem variar de ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"1"),Object(m.b)("msup",{parentName:"mrow"},Object(m.b)("mn",{parentName:"msup"},"0"),Object(m.b)("mrow",{parentName:"msup"},Object(m.b)("mo",{parentName:"mrow"},"\u2212"),Object(m.b)("mn",{parentName:"mrow"},"3"))),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"s")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"10^{-3}~s")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.8141079999999999em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"1"),Object(m.b)("span",{parentName:"span",className:"mord"},Object(m.b)("span",{parentName:"span",className:"mord"},"0"),Object(m.b)("span",{parentName:"span",className:"msupsub"},Object(m.b)("span",{parentName:"span",className:"vlist-t"},Object(m.b)("span",{parentName:"span",className:"vlist-r"},Object(m.b)("span",{parentName:"span",className:"vlist",style:{height:"0.8141079999999999em"}},Object(m.b)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},Object(m.b)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),Object(m.b)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},Object(m.b)("span",{parentName:"span",className:"mord mtight"},Object(m.b)("span",{parentName:"span",className:"mord mtight"},"\u2212"),Object(m.b)("span",{parentName:"span",className:"mord mtight"},"3"))))))))),Object(m.b)("span",{parentName:"span",className:"mspace nobreak"},"\xa0"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"s")))))," a ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"10"),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"s")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"10~s")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"1"),Object(m.b)("span",{parentName:"span",className:"mord"},"0"),Object(m.b)("span",{parentName:"span",className:"mspace nobreak"},"\xa0"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"s"))))),"). Isso torna o sistema de equa\xe7\xf5es diferenciais da m\xe1quina s\xedncrona como um sistema r\xedgido (",Object(m.b)("em",{parentName:"p"},Object(m.b)("a",{parentName:"em",href:"https://en.wikipedia.org/wiki/Stiff_equation"},"stiff equation")),"). Caso a an\xe1lise da estabilidade num\xe9rica tanto das equa\xe7\xf5es diferenciais r\xedgidas quanto do m\xe9todo de integra\xe7\xe3o obtenham o mesmo comportamento, o m\xe9todo \xe9 chamado de absolutamente est\xe1vel, ou A-est\xe1vel."),Object(m.b)("p",null,"M\xe9todos de integra\xe7\xe3o num\xe9rica expl\xedcitos, como por exemplo o ",Object(m.b)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods"},"Runge-Kutta")," de quarta ordem, n\xe3o podem ser A-est\xe1veis e, portanto, normalmente possuem comportamento ruim em problemas com equa\xe7\xf5es diferenciais r\xedgidas. Por outro lado, m\xe9todos impl\xedcitos podem ser A-est\xe1veis. Um m\xe9todo impl\xedcito adequado para solu\xe7\xe3o do comportamento din\xe2mico de sistemas el\xe9tricos \xe9 o ",Object(m.b)("strong",{parentName:"p"},"Trapezoidal Impl\xedcito"),", por possuir as seguintes vantagens:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},"\xc9 numericamente est\xe1vel (A-est\xe1vel);"),Object(m.b)("li",{parentName:"ul"},"\xc9 bastante r\xe1pida;"),Object(m.b)("li",{parentName:"ul"},"Possui boa precis\xe3o (dependendo somente do passo de integra\xe7\xe3o utilizado).")),Object(m.b)("p",null,"Tal m\xe9todo foi implementado no PSP-UFU tanto para solu\xe7\xe3o das equa\xe7\xf5es diferenciais da m\xe1quina s\xedncrona quanto nas ",Object(m.b)("a",{parentName:"p",href:"transferFunction"},"fun\xe7\xf5es transfer\xeancia")," do ",Object(m.b)("a",{parentName:"p",href:"controlEditor"},"sistema de controle"),". 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Power System Modelling and Scripting. London: Springer, 2010. doi: ",Object(m.b)("a",{parentName:"li",href:"https://doi.org/10.1007/978-3-642-13669-6"},"https://doi.org/10.1007/978-3-642-13669-6")),Object(m.b)("li",{parentName:"ol"},"KUNDUR, P.; et al. Definition and classification of power system stability. IEEE Transactions Power Systems, v. 19, n. 2, mai 2004, p. 1387-1401. doi: ",Object(m.b)("a",{parentName:"li",href:"https://doi.org/10.1109/TPWRS.2004.825981"},"https://doi.org/10.1109/TPWRS.2004.825981")),Object(m.b)("li",{parentName:"ol"},"ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: ",Object(m.b)("a",{parentName:"li",href:"https://doi.org/10.1002/9781118878286"},"https://doi.org/10.1002/9781118878286")),Object(m.b)("li",{parentName:"ol"},"PARK, R. H. Two-reactions Theory of Synchronous Machine. Generalized Method of Analysis \u2013 Part I. AIEE Transactions, v. 48, n. 3, jul 1929. doi: ",Object(m.b)("a",{parentName:"li",href:"https://doi.org/10.1109/T-AIEE.1929.5055275"},"https://doi.org/10.1109/T-AIEE.1929.5055275")),Object(m.b)("li",{parentName:"ol"},"ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: ",Object(m.b)("a",{parentName:"li",href:"https://doi.org/10.1109/9780470545577"},"https://doi.org/10.1109/9780470545577")),Object(m.b)("li",{parentName:"ol"},"SAUER, P. W.; PAI, M. A. Power System Dynamics and Stability. Pretience Hall, Upper Saddle River, 1998."),Object(m.b)("li",{parentName:"ol"},"KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994."),Object(m.b)("li",{parentName:"ol"},"DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. 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+(window.webpackJsonp=window.webpackJsonp||[]).push([[47],{121:function(e,a,t){"use strict";t.r(a),t.d(a,"frontMatter",(function(){return r})),t.d(a,"metadata",(function(){return c})),t.d(a,"toc",(function(){return b})),t.d(a,"default",(function(){return l}));var s=t(3),n=t(7),m=(t(0),t(140)),p=t(141),r={id:"controlEditor",title:"Editor de Controle",sidebar_label:"Editor de Controle"},c={unversionedId:"controlEditor",id:"controlEditor",isDocsHomePage:!1,title:"Editor de Controle",description:"Os controles das m\xe1quinas s\xedncronas s\xe3o criados, modificados, importados e exportados em um editor distinto dos elementos de pot\xeancia, apresentado na figura abaixo.",source:"@site/docs/controlEditor.md",slug:"/controlEditor",permalink:"/PSP/docs/controlEditor",editUrl:"https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/controlEditor.md",version:"current",sidebar_label:"Editor de Controle",sidebar:"docs",previous:{title:"Motor de Indu\xe7\xe3o",permalink:"/PSP/docs/indMotor"},next:{title:"Entrada / Sa\xedda",permalink:"/PSP/docs/io"}},b=[{value:"Elementos de Controle",id:"elementos-de-controle",children:[]},{value:"Inicializa\xe7\xe3o do sistema de controle",id:"inicializa\xe7\xe3o-do-sistema-de-controle",children:[]},{value:"Teste do diagrama de controle",id:"teste-do-diagrama-de-controle",children:[]}],i={toc:b};function l(e){var a=e.components,t=Object(n.a)(e,["components"]);return Object(m.b)("wrapper",Object(s.a)({},i,t,{components:a,mdxType:"MDXLayout"}),Object(m.b)("link",{rel:"stylesheet",href:Object(p.a)("katex/katex.min.css")}),Object(m.b)("p",null,"Os controles das ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quinas s\xedncronas")," s\xe3o criados, modificados, importados e exportados em um editor distinto dos elementos de pot\xeancia, apresentado na figura abaixo."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(p.a)("images/controlEditor.png"),alt:"Editor de elementos de controle do PSP-UFU",title:"Editor de elementos de controle do PSP-UFU"}))),Object(m.b)("p",null,"O acesso ao editor \xe9 realizado por meio de dois bot\xf5es presentes no ",Object(m.b)("a",{parentName:"p",href:"syncGenerator#formul%C3%A1rio-de-edi%C3%A7%C3%A3o-dos-geradores-s%C3%ADncronos"},"formul\xe1rio de edi\xe7\xe3o dos dados de estabilidade dos geradores s\xedncronos"),":"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"syncGenerator#utilizar-avr-e-regulador-de-velocidade"},"Editar AVR")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"syncGenerator#utilizar-avr-e-regulador-de-velocidade"},"Editar regulador de velocidade"))),Object(m.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"No PSP-UFU a op\xe7\xe3o de editar o ",Object(m.b)("strong",{parentName:"p"},"AVR")," engloba mais que somente o controle de tens\xe3o da m\xe1quina. 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A ",Object(m.b)("strong",{parentName:"p"},"edi\xe7\xe3o dos dados")," dos componentes inseridos tamb\xe9m \xe9 realizada com ",Object(m.b)("strong",{parentName:"p"},"duplo clique")," sobre o elemento inserido, exibindo um formul\xe1rio de edi\xe7\xe3o de dados."),Object(m.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(m.b)("source",{src:"/PSP/videos/timelapseControlEditor.mp4",type:"video/mp4"}),Object(m.b)("source",{src:"/PSP/videos/timelapseControlEditor.webm",type:"video/webm"})),Object(m.b)("p",null,"Os seguintes blocos de controle est\xe3o presentes no PSP-UFU:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"io"},"Entrada e Sa\xedda")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"transferFunction"},"Fun\xe7\xe3o Transfer\xeancia")),Object(m.b)("li",{parentName:"ul"},"Matem\xe1tica",Object(m.b)("ul",{parentName:"li"},Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"sum"},"Somador")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"multiplier"},"Multiplicador")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"divider"},"Divisor")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"exponential"},"Exponencial")))),Object(m.b)("li",{parentName:"ul"},"Limitadores",Object(m.b)("ul",{parentName:"li"},Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"limiter"},"Limitador absoluto")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"rateLimiter"},"Limitador de taxa")))),Object(m.b)("li",{parentName:"ul"},"Constantes",Object(m.b)("ul",{parentName:"li"},Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"const"},"Constante")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"gain"},"Ganho")))),Object(m.b)("li",{parentName:"ul"},Object(m.b)("a",{parentName:"li",href:"mathExpression"},"Express\xe3o Matem\xe1tica"))),Object(m.b)("h2",{id:"inicializa\xe7\xe3o-do-sistema-de-controle"},"Inicializa\xe7\xe3o do sistema de controle"),Object(m.b)("p",null,"Caso as entradas do sistema de controle apresentem valores diferentes de zero ou a primeira opera\xe7\xe3o entre elas n\xe3o resulte em uma sa\xedda nula, o sistema necessita de inicializa\xe7\xe3o, de forma a adequar os valores de entradas e sa\xedda dos blocos elementares e dos vetores de estado das ",Object(m.b)("a",{parentName:"p",href:"transferFunction"},"fun\xe7\xf5es transfer\xeancias")," presentes. 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Nesse tipo de c\xe1lculo \xe9 poss\xedvel calcular faltas ",Object(m.b)("em",{parentName:"li"},"shunt")," nos ",Object(m.b)("a",{parentName:"li",href:"bus"},"barramentos")," balanceadas e desbalanceadas."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"N\xedvel de curto-circuito"),": Calcula o n\xedvel de curto-circuito (falta trif\xe1sica) em todos ",Object(m.b)("a",{parentName:"li",href:"bus"},"barramentos")," do sistema.")),Object(m.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(m.b)("a",{parentName:"p",href:"powerEditor"},"editor de pot\xeancia"),", a execu\xe7\xe3o do c\xe1lculo de curto-circuito \xe9 realizada no ",Object(m.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"menu Simula\xe7\xe3o")," clicando no bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"Falta"),". 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Com essa op\xe7\xe3o, os c\xe1lculos est\xe1ticos selecionados nas ",Object(m.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o")," s\xe3o automaticamente realizados ao modificar quaisquer par\xe2metros da rede, como dados el\xe9tricos e acionamento dos disjuntores dos elementos (remo\xe7\xe3o ou inser\xe7\xe3o)."),Object(m.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Cuidado!")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Os c\xe1lculos de curtos-circuitos n\xe3o s\xe3o habilitados por padr\xe3o no c\xe1lculo cont\xednuo e devem ser inseridos nas ",Object(m.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),"."))),Object(m.b)("p",null,"Os resultados do c\xe1lculo de curto-circuito s\xe3o exibidos nos ",Object(m.b)("a",{parentName:"p",href:"text"},"elementos de texto vinculado"),", ao posicionar o mouse sobre os barramentos e em ",Object(m.b)("a",{parentName:"p",href:"tabularReport"},"relat\xf3rios tabulares"),"."),Object(m.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-c\xe1lculo-de-curto-circuito"},"Erros comuns na execu\xe7\xe3o do c\xe1lculo de curto-circuito"),Object(m.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao calculo de curto-circuito."),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-falha-ao-inverter-a-matriz-admit\xe2ncia-de-sequ\xeancia-zero"},'A seguinte mensagem de erro \xe9 exibida: "Falha ao inverter a matriz admit\xe2ncia de sequ\xeancia zero"'),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Impossibilidade de circula\xe7\xe3o da corrente de sequ\xeancia zero"),". Caso o gerador n\xe3o seja aterrado, n\xe3o circular\xe1 corrente de sequ\xeancia zero por ele. Nesse caso, dependendo da conex\xe3o do transformador pr\xf3ximo ao gerador sem aterramento, a matriz admit\xe2ncia de sequ\xeancia zero \xe9 singular. Para contornar esse problema escolha uma das duas solu\xe7\xf5es abaixo:",Object(m.b)("ul",{parentName:"li"},Object(m.b)("li",{parentName:"ul"},'Marque a op\xe7\xe3o "Neutro aterrado" e insira um alto valor de reat\xe2ncia de aterramento (',Object(m.b)("span",{parentName:"li",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"j"),Object(m.b)("mn",{parentName:"mrow"},"9999"),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"p"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},"."),Object(m.b)("mi",{parentName:"mrow"},"u"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},".")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"j9999~p.u.")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.85396em",verticalAlign:"-0.19444em"}}),Object(m.b)("span",{parentName:"span",className:"mord mathdefault",style:{marginRight:"0.05724em"}},"j"),Object(m.b)("span",{parentName:"span",className:"mord"},"9"),Object(m.b)("span",{parentName:"span",className:"mord"},"9"),Object(m.b)("span",{parentName:"span",className:"mord"},"9"),Object(m.b)("span",{parentName:"span",className:"mord"},"9"),Object(m.b)("span",{parentName:"span",className:"mspace nobreak"},"\xa0"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"p"),Object(m.b)("span",{parentName:"span",className:"mord"},"."),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"u"),Object(m.b)("span",{parentName:"span",className:"mord"},"."))))),", por exemplo);"),Object(m.b)("li",{parentName:"ul"},"Ou, na barra do gerador, insira um reator de baixo valor de pot\xeancia reativa (",Object(m.b)("span",{parentName:"li",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"1"),Object(m.b)("mo",{parentName:"mrow",separator:"true",lspace:"0em",rspace:"0em"},","),Object(m.b)("mn",{parentName:"mrow"},"0"),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"v"),Object(m.b)("mi",{parentName:"mrow"},"a"),Object(m.b)("mi",{parentName:"mrow"},"r")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"1{,}0~var")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.8388800000000001em",verticalAlign:"-0.19444em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"1"),Object(m.b)("span",{parentName:"span",className:"mord"},Object(m.b)("span",{parentName:"span",className:"mpunct"},",")),Object(m.b)("span",{parentName:"span",className:"mord"},"0"),Object(m.b)("span",{parentName:"span",className:"mspace nobreak"},"\xa0"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault",style:{marginRight:"0.03588em"}},"v"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"a"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault",style:{marginRight:"0.02778em"}},"r"))))),", por exemplo).")))),Object(m.b)("h2",{id:"o-c\xe1lculo-de-curto-circuito"},"O c\xe1lculo de curto-circuito"),Object(m.b)("p",null,"Como j\xe1 foi apresentado anteriormente, as faltas que ocorrem com maior frequ\xeancia em sistemas de pot\xeancia s\xe3o assim\xe9tricas. Como qualquer falta assim\xe9trica provoca fluxo de corrente desequilibrada \xe9 necess\xe1rio empregar o m\xe9todo das componentes sim\xe9tricas. Esse m\xe9todo permite o estudo de sistemas balanceados com cargas desbalanceadas."),Object(m.b)("h3",{id:"m\xe9todo-das-componentes-sim\xe9tricas"},"M\xe9todo das componentes sim\xe9tricas"),Object(m.b)("p",null,"Esse m\xe9todo proposto por C. L. Fortescue, permite definir um sistema de n fasores desbalanceados em n \u2013 1 sistemas de n fases balanceados e um sistema de fase zero. O sistema de fase zero \xe9 definido por todas as fases de mesmo m\xf3dulo e \xe2ngulo.\nPara um sistema trif\xe1sico pode-se definir tr\xeas componentes de sequ\xeancia:"),Object(m.b)("ol",null,Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia positiva, constituindo em tr\xeas fasores iguais em m\xf3dulo, 120\xba defasados entre si, e tendo a mesma sequ\xeancia de fase que os fasores originais;"),Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia negativa, constituindo em tr\xeas fasores iguais em m\xf3dulo, 120\xba defasados entre si, e tendo a sequ\xeancia de fase oposta \xe0 dos fasores originais."),Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia zero, constituindo em tr\xeas fasores iguais em m\xf3dulo e com defasagem nula entre si.")),Object(m.b)("p",null,"Com isso pode-se decompor as tens\xf5es de fase em componentes sim\xe9tricas pelas seguintes equa\xe7\xf5es:"),Object(m.b)("div",{className:"math 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PSP-UFU ",Object(m.b)("strong",{parentName:"p"},"n\xe3o considera")," o ramo magnetizante ou imped\xe2ncias m\xfatuas entre as fases."))),Object(m.b)("p",null,"A figura abaixo mostra o modelo do transformador implementado no PSP-UFU:"),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(p.a)("images/transformerModel.svg"),alt:"Modelo PI da linha",title:"Modelo PI da linha"}))),Object(m.b)("p",null,"Em que:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("span",{parentName:"li",className:"math 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Esse formul\xe1rio, comum a v\xe1rios outros elementos, permite a inser\xe7\xe3o e/ou remo\xe7\xe3o do transformador durante o estudo de ',Object(m.b)("a",{parentName:"p",href:"stability"},"estabilidade"),"."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(p.a)("images/trafoSw.png"),alt:"Formul\xe1rio de chaveamento do transformador",title:"Formul\xe1rio de chaveamento do transformador"}))),Object(m.b)(b.a,{groupId:"transformer-tabs",defaultValue:"general",values:[{label:"Geral",value:"general"},{label:"Falta",value:"fault"},{label:"Bot\xe3o Estabilidade",value:"stability"}],mdxType:"Tabs"},Object(m.b)(r.a,{value:"general",mdxType:"TabItem"},Object(m.b)("h4",{id:"nome"},"Nome"),Object(m.b)("p",null,"Identifica\xe7\xe3o do elemento el\xe9trico. Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(m.b)("a",{parentName:"p",href:"https://pt.wikipedia.org/wiki/Unicode"},"Unicode"),"."),Object(m.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(m.b)("h4",{id:"tens\xe3o-nominal"},"Tens\xe3o nominal"),Object(m.b)("p",null,"Campo de informa\xe7\xe3o ",Object(m.b)("em",{parentName:"p"},"n\xe3o edit\xe1vel")," que apresenta a tens\xe3o prim\xe1ria e secund\xe1ria do transformador. 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Retorna o valor de e elevado a pot\xeancia A.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"exp2(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Exponencial de ",Object(m.b)("inlineCode",{parentName:"td"},"A")," na base ",Object(m.b)("span",{parentName:"td",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"2")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"2"))))),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"floor(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Piso de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),". Arredonda para o pr\xf3ximo menor inteiro.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"hypot(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Fun\xe7\xe3o de dist\xe2ncia Euclidiana.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"if(A,B,C)")),Object(m.b)("td",{parentName:"tr",align:null},"Se ",Object(m.b)("inlineCode",{parentName:"td"},"int(A)")," \xe9 diferente de ",Object(m.b)("span",{parentName:"td",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"0")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"0")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"0"))))),", retorna o valor de ",Object(m.b)("inlineCode",{parentName:"td"},"B"),", sen\xe3o retorna ",Object(m.b)("inlineCode",{parentName:"td"},"C"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"imag(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Retorna a parte imagin\xe1ria do n\xfamero complexo ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"int(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Arredonda ",Object(m.b)("inlineCode",{parentName:"td"},"A")," para o inteiro mais pr\xf3ximo.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"log(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Logaritmo natural (base ",Object(m.b)("span",{parentName:"td",className:"math 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",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"log2(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Logaritmo na base ",Object(m.b)("span",{parentName:"td",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"2")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"2")))))," de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"log10(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Logaritmo na base ",Object(m.b)("span",{parentName:"td",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"10")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"10")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"1"),Object(m.b)("span",{parentName:"span",className:"mord"},"0")))))," de A.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"max(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Se ",Object(m.b)("inlineCode",{parentName:"td"},"A>B"),", o resultado \xe9 ",Object(m.b)("inlineCode",{parentName:"td"},"A"),", sen\xe3o \xe9 ",Object(m.b)("inlineCode",{parentName:"td"},"B"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"min(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Se ",Object(m.b)("inlineCode",{parentName:"td"},"A<B"),", o resultado \xe9 ",Object(m.b)("inlineCode",{parentName:"td"},"A"),", sen\xe3o \xe9 ",Object(m.b)("inlineCode",{parentName:"td"},"B"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"polar(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Retorna o n\xfamero complexo de magnitude ",Object(m.b)("inlineCode",{parentName:"td"},"A")," e \xe2ngulo de fase ",Object(m.b)("inlineCode",{parentName:"td"},"B")," (em radianos).")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"pow(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Exponencia\xe7\xe3o (",Object(m.b)("inlineCode",{parentName:"td"},"A")," elevado \xe0 pot\xeancia ",Object(m.b)("inlineCode",{parentName:"td"},"B"),")")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"real(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Retorna a parte real do n\xfamero complexo ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"sec(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Secante de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"sin(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Seno de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),". Retorna o \xe2ngulo em radianos.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"sinh(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Mesmo que ",Object(m.b)("inlineCode",{parentName:"td"},"sin()"),", mas para seno hiperb\xf3lico.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"sqrt(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Raiz quadrada de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"tan(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Tangente de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),". Retorna o \xe2ngulo em radianos.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"tanh(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Mesmo que ",Object(m.b)("inlineCode",{parentName:"td"},"tan()"),", mas para tangente hiperb\xf3lica.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"trunc(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Valor truncado de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),". Retorna o n\xfamero inteiro de ",Object(m.b)("inlineCode",{parentName:"td"},"A")," sem a parcela fracionada.")))),Object(m.b)("p",null,"\xc9 poss\xedvel atribuir novas vari\xe1veis utilizando a seguinte sintaxe: "),Object(m.b)("pre",null,Object(m.b)("code",{parentName:"pre",className:"language-cpp"},"<nome da vari\xe1vel> := <express\xe3o>; <fun\xe7\xe3o>\n")),Object(m.b)("p",null,"Por exemplo:"),Object(m.b)("pre",null,Object(m.b)("code",{parentName:"pre",className:"language-cpp"},"comprimento := sqrt(x*x+y*y); 2*comprimento*sin(comprimento)\n")),Object(m.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Dica")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"O caractere de espa\xe7o e de nova linha s\xe3o ignorados na interpreta\xe7\xe3o da express\xe3o, portanto para maior organiza\xe7\xe3o, o c\xf3digo anterior pode ser escrito da seguinte forma:"),Object(m.b)("pre",{parentName:"div"},Object(m.b)("code",{parentName:"pre",className:"language-cpp"},"comprimento := sqrt(x*x+y*y);\n2*comprimento*sin(comprimento)\n")))),Object(m.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Cuidado!")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Note que a express\xe3o que determina o valor de sa\xedda do bloco ",Object(m.b)("strong",{parentName:"p"},"n\xe3o possui")," o caractere ",Object(m.b)("inlineCode",{parentName:"p"},";")," em seu final."))),Object(m.b)("h2",{id:"formul\xe1rio-de-edi\xe7\xe3o-de-dados-do-bloco-de-express\xe3o-matem\xe1tica"},"Formul\xe1rio de edi\xe7\xe3o de dados do bloco de Express\xe3o Matem\xe1tica"),Object(m.b)("p",null,"O formul\xe1rio de inser\xe7\xe3o e edi\xe7\xe3o das entradas do bloco e da express\xe3o matem\xe1tica gen\xe9rica, assim como ferramentas de aux\xedlio de sua constru\xe7\xe3o \xe9 apresentado na figura abaixo."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(p.a)("images/mathExpressionForm.png"),alt:"Formul\xe1rio de edi\xe7\xe3o de dados do bloco de express\xe3o matem\xe1tica no PSP-UFU",title:"Formul\xe1rio de edi\xe7\xe3o de dados do bloco de express\xe3o matem\xe1tica no PSP-UFU"}))),Object(m.b)("p",null,"No campo de \u201cVari\xe1veis de entrada\u201d \xe9 inserida uma lista com os nomes das entradas separados por espa\xe7os. Um n\xfamero qualquer de entrada pode ser definido nessa lista esses nomes s\xe3o apresentados no \xedcone gr\xe1fico presente no editor de controle, al\xe9m de serem destacados na express\xe3o inserida pelo usu\xe1rio. O n\xfamero de entradas e sa\xeddas se comporta de forma semelhante aos ",Object(m.b)("a",{parentName:"p",href:"sum"},"blocos somadores"),"."),Object(m.b)("p",null,"Abaixo do campo das vari\xe1veis de entrada est\xe1 presente o local para inser\xe7\xe3o da express\xe3o matem\xe1tica. A sintaxe da express\xe3o inserida pelo usu\xe1rio possui realce (por meio de diferentes formas e cores da fonte) para n\xfameros, operadores, vari\xe1veis de entrada, fun\xe7\xf5es e constantes, facilitando a cria\xe7\xe3o, manipula\xe7\xe3o e identifica\xe7\xe3o de erros de digita\xe7\xe3o e l\xf3gica."),Object(m.b)("div",{className:"admonition admonition-info alert alert--info"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"}))),"Informa\xe7\xe3o")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Como ferramenta de aux\xedlio ao usu\xe1rio foi desenvolvida uma verifica\xe7\xe3o da express\xe3o inserida."))),Object(m.b)("p",null,"Tal ferramenta ir\xe1 encontrar erros e indicar\xe1 ao usu\xe1rio qual o tipo do erro, al\xe9m da sua localiza\xe7\xe3o, destacando-o. A figura anterior exemplifica a identifica\xe7\xe3o autom\xe1tica do erro pelo PSP-UFU, assim como sua posi\xe7\xe3o na express\xe3o inserida."),Object(m.b)("h2",{id:"exemplo-de-express\xe3o-matem\xe1tica"},"Exemplo de Express\xe3o Matem\xe1tica"),Object(m.b)("p",null,"A corrente de campo em ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"p"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},"."),Object(m.b)("mi",{parentName:"mrow"},"u"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},".")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p.u.")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.19444em"}}),Object(m.b)("span",{parentName:"span",className:"mord 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Esses dados s\xe3o obtidos por meio do estudo de ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerFlow"}),"fluxo de carga"),"."),Object(m.b)("div",{className:"admonition admonition-info alert alert--info"},Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(m.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"}),Object(m.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"})))),"Informa\xe7\xe3o")),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(m.b)("p",{parentName:"div"},"Atualmente, o PSP-UFU fornece resultado para os seguintes tipos de falta:"),Object(m.b)("ul",{parentName:"div"},Object(m.b)("li",{parentName:"ul"},"Falta Trif\xe1sica (3F-T);"),Object(m.b)("li",{parentName:"ul"},"Falta Fase-Terra (F-T);"),Object(m.b)("li",{parentName:"ul"},"Falta Fase-Fase (F-F);"),Object(m.b)("li",{parentName:"ul"},"Falta Fase-Fase-Terra (F-F-T).")))),Object(m.b)("p",null,"Os modelos dos elementos el\xe9tricos que constituem um sistema de pot\xeancia para o estudo de curto-circuito s\xe3o semelhantes aos do fluxo de carga, apresentando algumas diverg\xeancias para as faltas desbalanceadas (F-T, F-F e F-F-T)."),Object(m.b)("p",null,"As faltas que ocorrem com maior frequ\xeancia em sistemas de pot\xeancia s\xe3o assim\xe9tricas. Como qualquer falta assim\xe9trica provoca fluxo de corrente desequilibrada \xe9 necess\xe1rio empregar o m\xe9todo das componentes sim\xe9tricas. Esse m\xe9todo permite o estudo de sistemas balanceados em conjunto cargas desbalanceadas."),Object(m.b)("h2",{id:"execu\xe7\xe3o-do-c\xe1lculo-de-curto-circuito-no-psp-ufu"},"Execu\xe7\xe3o do c\xe1lculo de curto-circuito no PSP-UFU"),Object(m.b)("p",null,"Existem duas formas de se calcular o curto-circuito no PSP-UFU:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Falta"),": Calcula a falta inserida nas ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"bus"}),"barras"),". Nesse tipo de c\xe1lculo \xe9 poss\xedvel calcular faltas ",Object(m.b)("em",{parentName:"li"},"shunt")," nos ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"bus"}),"barramentos")," balanceadas e desbalanceadas."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"N\xedvel de curto-circuito"),": Calcula o n\xedvel de curto-circuito (falta trif\xe1sica) em todos ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"bus"}),"barramentos")," do sistema.")),Object(m.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerEditor"}),"editor de pot\xeancia"),", a execu\xe7\xe3o do c\xe1lculo de curto-circuito \xe9 realizada no ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," clicando no bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"Falta"),". Para calcular o n\xedvel de curto-circuito (falta trif\xe1sica) em todos ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"bus"}),"barramentos")," do sistema, basta clicar no bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"N\xedvel de curto-circuito"),"."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(b.a)("images/menuSimulationFaulta.svg"),alt:"Execu\xe7\xe3o dos c\xe1lculos de curto-circuito",title:"Execu\xe7\xe3o dos c\xe1lculos de curto-circuito"}))),Object(m.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(m.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(m.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o")),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(m.b)("p",{parentName:"div"},"\xc9 poss\xedvel calcular as faltas sem a execu\xe7\xe3o do ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerFlow"}),"fluxo de carga"),", por\xe9m ",Object(m.b)("strong",{parentName:"p"},"n\xe3o \xe9 recomend\xe1vel"),", visto que os valores das correntes de falta s\xe3o significativamente alteradas."))),Object(m.b)("p",null,Object(m.b)("strong",{parentName:"p"},"Outra possibilidade")," \xe9 a execu\xe7\xe3o por meio do c\xe1lculo cont\xednuo, tamb\xe9m presente no ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," e seu acionamento \xe9 realizado co clicar no bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"Habilitar solu\xe7\xe3o"),". Com essa op\xe7\xe3o, os c\xe1lculos est\xe1ticos selecionados nas ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o")," s\xe3o automaticamente realizados ao modificar quaisquer par\xe2metros da rede, como dados el\xe9tricos e acionamento dos disjuntores dos elementos (remo\xe7\xe3o ou inser\xe7\xe3o)."),Object(m.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(m.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(m.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"})))),"Cuidado!")),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(m.b)("p",{parentName:"div"},"Os c\xe1lculos de curtos-circuitos n\xe3o s\xe3o habilitados por padr\xe3o no c\xe1lculo cont\xednuo e devem ser inseridos nas ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),"."))),Object(m.b)("p",null,"Os resultados do c\xe1lculo de curto-circuito s\xe3o exibidos nos ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"text"}),"elementos de texto vinculado"),", ao posicionar o mouse sobre os barramentos e em ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"tabularReport"}),"relat\xf3rios tabulares"),"."),Object(m.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-c\xe1lculo-de-curto-circuito"},"Erros comuns na execu\xe7\xe3o do c\xe1lculo de curto-circuito"),Object(m.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao calculo de curto-circuito."),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-falha-ao-inverter-a-matriz-admit\xe2ncia-de-sequ\xeancia-zero"},'A seguinte mensagem de erro \xe9 exibida: "Falha ao inverter a matriz admit\xe2ncia de sequ\xeancia zero"'),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Impossibilidade de circula\xe7\xe3o da corrente de sequ\xeancia zero"),". Caso o gerador n\xe3o seja aterrado, n\xe3o circular\xe1 corrente de sequ\xeancia zero por ele. Nesse caso, dependendo da conex\xe3o do transformador pr\xf3ximo ao gerador sem aterramento, a matriz admit\xe2ncia de sequ\xeancia zero \xe9 singular. Para contornar esse problema escolha uma das duas solu\xe7\xf5es abaixo:",Object(m.b)("ul",{parentName:"li"},Object(m.b)("li",{parentName:"ul"},'Marque a op\xe7\xe3o "Neutro aterrado" e insira um alto valor de reat\xe2ncia de aterramento (',Object(m.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(m.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"j"),Object(m.b)("mn",{parentName:"mrow"},"9999"),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"p"),Object(m.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"."),Object(m.b)("mi",{parentName:"mrow"},"u"),Object(m.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),".")),Object(m.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"j9999~p.u.")))),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.85396em",verticalAlign:"-0.19444em"}})),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05724em"}}),"j"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"9"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"9"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"9"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"9"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mspace nobreak"}),"\xa0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"p"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"."),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"u"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"."))))),", por exemplo);"),Object(m.b)("li",{parentName:"ul"},"Ou, na barra do gerador, insira um reator de baixo valor de pot\xeancia reativa (",Object(m.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(m.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"1"),Object(m.b)("mo",Object(s.a)({parentName:"mrow"},{separator:"true",lspace:"0em",rspace:"0em"}),","),Object(m.b)("mn",{parentName:"mrow"},"0"),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"v"),Object(m.b)("mi",{parentName:"mrow"},"a"),Object(m.b)("mi",{parentName:"mrow"},"r")),Object(m.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"1{,}0~var")))),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8388800000000001em",verticalAlign:"-0.19444em"}})),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mpunct"}),",")),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mspace nobreak"}),"\xa0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03588em"}}),"v"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"a"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.02778em"}}),"r"))))),", por exemplo).")))),Object(m.b)("h2",{id:"o-c\xe1lculo-de-curto-circuito"},"O c\xe1lculo de curto-circuito"),Object(m.b)("p",null,"Como j\xe1 foi apresentado anteriormente, as faltas que ocorrem com maior frequ\xeancia em sistemas de pot\xeancia s\xe3o assim\xe9tricas. Como qualquer falta assim\xe9trica provoca fluxo de corrente desequilibrada \xe9 necess\xe1rio empregar o m\xe9todo das componentes sim\xe9tricas. Esse m\xe9todo permite o estudo de sistemas balanceados com cargas desbalanceadas."),Object(m.b)("h3",{id:"m\xe9todo-das-componentes-sim\xe9tricas"},"M\xe9todo das componentes sim\xe9tricas"),Object(m.b)("p",null,"Esse m\xe9todo proposto por C. L. Fortescue, permite definir um sistema de n fasores desbalanceados em n \u2013 1 sistemas de n fases balanceados e um sistema de fase zero. O sistema de fase zero \xe9 definido por todas as fases de mesmo m\xf3dulo e \xe2ngulo.\nPara um sistema trif\xe1sico pode-se definir tr\xeas componentes de sequ\xeancia:"),Object(m.b)("ol",null,Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia positiva, constituindo em tr\xeas fasores iguais em m\xf3dulo, 120\xba defasados entre si, e tendo a mesma sequ\xeancia de fase que os fasores originais;"),Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia negativa, constituindo em tr\xeas fasores iguais em m\xf3dulo, 120\xba defasados entre si, e tendo a sequ\xeancia de fase oposta \xe0 dos fasores originais."),Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia zero, constituindo em tr\xeas fasores iguais em m\xf3dulo e com defasagem nula entre si.")),Object(m.b)("p",null,"Com isso pode-se decompor as tens\xf5es de fase em componentes sim\xe9tricas pelas seguintes equa\xe7\xf5es:"),Object(m.b)("div",{className:"math 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Nesse caso, ao criar um novo projeto a tela de trabalho n\xe3o \xe9 exibida ou \xe9 exibida uma tela preta. A solu\xe7\xe3o \xe9 alterar o renderizador para "Device Context". 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href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><article><header><h1 class="margin-bottom--sm blogPostTitle_GeHD">2019w42a-beta</h1><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_fNvV">October 12, 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Harmonic study implemented;</li><li>🔥 Autolayout from importation implemented;</li><li>🔥 Importation of Matpower files;</li><li>🐛 Several minor 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href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><article><header><h1 class="margin-bottom--sm blogPostTitle_GeHD">22019w44a-beta</h1><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_fNvV">October 31, 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Minor changes to 32bit MSW version for compatibility;</li><li>✔️ Travis CI (Continuous Integration) implemented (Linux).</li></ul></div><footer class="row 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col--offset-2"><article><header><h1 class="margin-bottom--sm blogPostTitle_1mse">2019w42a-beta</h1><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_3bQP">October 13, 2019  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>🔥 Harmonic study implemented;</li><li>🔥 Autolayout from importation implemented;</li><li>🔥 Importation of Matpower files;</li><li>🐛 Several minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a 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col--offset-2"><article><header><h1 class="margin-bottom--sm blogPostTitle_1mse">22019w44a-beta</h1><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_3bQP">November 1, 2019  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>✔️ Minor changes to 32bit MSW version for compatibility;</li><li>✔️ Travis CI (Continuous Integration) implemented (Linux).</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" 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href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><article><header><h1 class="margin-bottom--sm blogPostTitle_GeHD">2020w24a-beta</h1><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_fNvV">June 12, 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Device context render implemented (to avoid use OpenGL - some machines can&#x27;t use it properly);</li><li>✔️ Added Visual Studio solution to the GitHub project;</li><li>🐛 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href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><article><header><h1 class="margin-bottom--sm blogPostTitle_GeHD">2020w28a-beta</h1><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_fNvV">July 8, 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Voltage heatmap implemented (OpenGL enabled);</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div 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href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><article><header><h1 class="margin-bottom--sm blogPostTitle_GeHD">2020w31a-beta</h1><div class="margin-vert--md"><time datetime="2020-08-01T00:00:00.000Z" class="blogPostDate_fNvV">July 31, 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>);</li><li>✔️ New fancy <a href="https://thales1330.github.io/PSP/" target="_blank" rel="noopener noreferrer">PSP-UFU website</a> and <a href="https://thales1330.github.io/PSP/blog" target="_blank" rel="noopener noreferrer">changelog</a> using <a href="https://v2.docusaurus.io/" target="_blank" rel="noopener noreferrer">docusaurus</a>;</li><li>✔️ Text element performance improved for OpenGL render;</li><li>🐛 Several major and minor bug fixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2020-08-01-2020w31a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" 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diff --git a/docs/blog/2020w24a-beta/index.html b/docs/blog/2020w24a-beta/index.html
deleted file mode 100644
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some machines can&#x27;t use it properly);</li><li>✔️ Added Visual Studio solution to the GitHub project;</li><li>🐛 Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2020-06-13-2020w24a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="margin-vert--xl"><nav class="pagination-nav" aria-label="Blog post page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/blog/2020w28a-beta"><div class="pagination-nav__sublabel">Newer Post</div><div class="pagination-nav__label">« 2020w28a-beta</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/blog/2019w44a-beta"><div class="pagination-nav__sublabel">Older Post</div><div class="pagination-nav__label">22019w44a-beta »</div></a></div></nav></div></div></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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col--offset-2"><article><header><h1 class="margin-bottom--sm blogPostTitle_1mse">2020w28a-beta</h1><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_3bQP">July 9, 2020  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>🔥 Voltage heatmap implemented (OpenGL enabled);</li><li>🐛 Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2020-07-09-2020w28a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="margin-vert--xl"><nav class="pagination-nav" aria-label="Blog post page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/blog/2020w31a-beta"><div class="pagination-nav__sublabel">Newer Post</div><div class="pagination-nav__label">« 2020w31a-beta</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/blog/2020w24a-beta"><div 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class="DocSearch-Button-Key">⌘</span><span class="DocSearch-Button-Key">K</span></button></div></div><div role="presentation" class="navbar-sidebar__backdrop"></div><div class="navbar-sidebar"><div class="navbar-sidebar__brand"><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a></div><div class="navbar-sidebar__items"><div class="menu"><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" href="/PSP/docs/">User Guide</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link navbar__link--active" href="/PSP/blog">Changelog</a></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">PSP-UFU Source Code</a></li></ul></div></div></div></nav><div class="main-wrapper"><div class="container margin-vert--lg"><div class="row"><div class="col col--8 col--offset-2"><article><header><h1 class="margin-bottom--sm blogPostTitle_1mse">2020w31a-beta</h1><div class="margin-vert--md"><time datetime="2020-08-01T00:00:00.000Z" class="blogPostDate_3bQP">August 1, 2020  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>);</li><li>✔️ New fancy <a href="https://thales1330.github.io/PSP/" target="_blank" rel="noopener noreferrer">PSP-UFU website</a> and <a href="https://thales1330.github.io/PSP/blog" target="_blank" rel="noopener noreferrer">changelog</a> using <a href="https://v2.docusaurus.io/" target="_blank" rel="noopener noreferrer">docusaurus</a>;</li><li>✔️ Text element performance improved for OpenGL render;</li><li>🐛 Several major and minor bug fixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2020-08-01-2020w31a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="margin-vert--xl"><nav class="pagination-nav" aria-label="Blog post page navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/blog/2020w28a-beta"><div class="pagination-nav__sublabel">Older Post</div><div class="pagination-nav__label">2020w28a-beta »</div></a></div></nav></div></div></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. Built with Docusaurus.</div></div></div></footer></div>
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diff --git a/docs/blog/atom.xml b/docs/blog/atom.xml
new file mode 100644
index 0000000..9bf3ed0
--- /dev/null
+++ b/docs/blog/atom.xml
@@ -0,0 +1,45 @@
+<?xml version="1.0" encoding="utf-8"?>
+<feed xmlns="http://www.w3.org/2005/Atom">
+    <id>https://thales1330.github.io/blog</id>
+    <title>PSP-UFU Blog</title>
+    <updated>2020-08-01T00:00:00.000Z</updated>
+    <generator>https://github.com/jpmonette/feed</generator>
+    <link rel="alternate" href="https://thales1330.github.io/blog"/>
+    <subtitle>PSP-UFU Blog</subtitle>
+    <icon>https://thales1330.github.io/img/favicon.ico</icon>
+    <entry>
+        <title type="html"><![CDATA[2020w31a-beta]]></title>
+        <id>2020w31a-beta</id>
+        <link href="https://thales1330.github.io/PSP/blog/2020/08/01/2020w31a-beta"/>
+        <updated>2020-08-01T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- 🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on GitHub or Twitter);]]></summary>
+    </entry>
+    <entry>
+        <title type="html"><![CDATA[2020w28a-beta]]></title>
+        <id>2020w28a-beta</id>
+        <link href="https://thales1330.github.io/PSP/blog/2020/07/09/2020w28a-beta"/>
+        <updated>2020-07-09T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- 🔥 Voltage heatmap implemented (OpenGL enabled);]]></summary>
+    </entry>
+    <entry>
+        <title type="html"><![CDATA[2020w24a-beta]]></title>
+        <id>2020w24a-beta</id>
+        <link href="https://thales1330.github.io/PSP/blog/2020/06/13/2020w24a-beta"/>
+        <updated>2020-06-13T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- ✔️ Device context render implemented (to avoid use OpenGL - some machines can't use it properly);]]></summary>
+    </entry>
+    <entry>
+        <title type="html"><![CDATA[22019w44a-beta]]></title>
+        <id>22019w44a-beta</id>
+        <link href="https://thales1330.github.io/PSP/blog/2019/11/01/2019w44a-beta"/>
+        <updated>2019-11-01T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- ✔️ Minor changes to 32bit MSW version for compatibility;]]></summary>
+    </entry>
+    <entry>
+        <title type="html"><![CDATA[2019w42a-beta]]></title>
+        <id>2019w42a-beta</id>
+        <link href="https://thales1330.github.io/PSP/blog/2019/10/13/2019w42a-beta"/>
+        <updated>2019-10-13T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- 🔥 Harmonic study implemented;]]></summary>
+    </entry>
+</feed>
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+<link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">Blog | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="Blog | PSP-UFU"><meta data-react-helmet="true" name="description" content="Blog"><meta data-react-helmet="true" property="og:description" content="Blog"><meta data-react-helmet="true" property="og:url" content="https://thales1330.github.io/PSP/blog"><meta data-react-helmet="true" name="docsearch:language" content="en"><meta data-react-helmet="true" name="docsearch:docusaurus_tag" content="blog_posts_list"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/img/favicon.ico"><link data-react-helmet="true" rel="canonical" href="https://thales1330.github.io/PSP/blog"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/blog" hreflang="en"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/pt/blog" hreflang="pt"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/blog" hreflang="x-default"><link data-react-helmet="true" rel="preconnect" href="https://BH4D9OD16A-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="/PSP/assets/css/styles.73f64b8b.css">
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class="DocSearch-Button-Key">⌘</span><span class="DocSearch-Button-Key">K</span></button></div></div><div role="presentation" class="navbar-sidebar__backdrop"></div><div class="navbar-sidebar"><div class="navbar-sidebar__brand"><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a></div><div class="navbar-sidebar__items"><div class="menu"><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" href="/PSP/docs/">User Guide</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link navbar__link--active" href="/PSP/blog">Changelog</a></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">PSP-UFU Source Code</a></li></ul></div></div></div></nav><div class="main-wrapper"><div class="container margin-vert--lg"><div class="row"><main class="col col--8 col--offset-2"><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2020w31a-beta">2020w31a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-08-01T00:00:00.000Z" class="blogPostDate_3bQP">August 1, 2020  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>);</li><li>✔️ New fancy <a href="https://thales1330.github.io/PSP/" target="_blank" rel="noopener noreferrer">PSP-UFU website</a> and <a href="https://thales1330.github.io/PSP/blog" target="_blank" rel="noopener noreferrer">changelog</a> using <a href="https://v2.docusaurus.io/" target="_blank" rel="noopener noreferrer">docusaurus</a>;</li><li>✔️ Text element performance improved for OpenGL render;</li><li>🐛 Several major and minor bug fixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2020w28a-beta">2020w28a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_3bQP">July 9, 2020  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>🔥 Voltage heatmap implemented (OpenGL enabled);</li><li>🐛 Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2020w24a-beta">2020w24a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_3bQP">June 13, 2020  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>✔️ Device context render implemented (to avoid use OpenGL - some machines can&#x27;t use it properly);</li><li>✔️ Added Visual Studio solution to the GitHub project;</li><li>🐛 Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2019w44a-beta">22019w44a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_3bQP">November 1, 2019  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>✔️ Minor changes to 32bit MSW version for compatibility;</li><li>✔️ Travis CI (Continuous Integration) implemented (Linux).</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2019w42a-beta">2019w42a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_3bQP">October 13, 2019  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>🔥 Harmonic study implemented;</li><li>🔥 Autolayout from importation implemented;</li><li>🔥 Importation of Matpower files;</li><li>🐛 Several minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"></div></nav></main></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code 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class="main-wrapper blog-wrapper"><div class="container margin-vert--lg"><div class="row"><div class="col col--3"><div class="sidebar_2ahu thin-scrollbar"><h3 class="sidebarItemTitle_2hhb">Recent posts</h3><ul class="sidebarItemList_2xAf"><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2020/08/01/2020w31a-beta">2020w31a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2020/07/09/2020w28a-beta">2020w28a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2020/06/13/2020w24a-beta">2020w24a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2019/11/01/2019w44a-beta">22019w44a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2020/08/01/2020w31a-beta">2020w31a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-08-01T00:00:00.000Z" class="blogPostDate_fNvV">July 31, 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>);</li><li>✔️ New fancy <a href="https://thales1330.github.io/PSP/" target="_blank" rel="noopener noreferrer">PSP-UFU website</a> and <a href="https://thales1330.github.io/PSP/blog" target="_blank" rel="noopener noreferrer">changelog</a> using <a href="https://v2.docusaurus.io/" target="_blank" rel="noopener noreferrer">docusaurus</a>;</li><li>✔️ Text element performance improved for OpenGL render;</li><li>🐛 Several major and minor bug fixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2020/07/09/2020w28a-beta">2020w28a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_fNvV">July 8, 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Voltage heatmap implemented (OpenGL enabled);</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2020/06/13/2020w24a-beta">2020w24a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_fNvV">June 12, 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Device context render implemented (to avoid use OpenGL - some machines can&#x27;t use it properly);</li><li>✔️ Added Visual Studio solution to the GitHub project;</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2019/11/01/2019w44a-beta">22019w44a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_fNvV">October 31, 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Minor changes to 32bit MSW version for compatibility;</li><li>✔️ Travis CI (Continuous Integration) implemented (Linux).</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_fNvV">October 12, 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Harmonic study implemented;</li><li>🔥 Autolayout from importation implemented;</li><li>🔥 Importation of Matpower files;</li><li>🐛 Several minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"></div></nav></main></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 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+        <item>
+            <title><![CDATA[2020w31a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/blog/2020/08/01/2020w31a-beta</link>
+            <guid>2020w31a-beta</guid>
+            <pubDate>Sat, 01 Aug 2020 00:00:00 GMT</pubDate>
+            <description><![CDATA[- 🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on GitHub or Twitter);]]></description>
+        </item>
+        <item>
+            <title><![CDATA[2020w28a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/blog/2020/07/09/2020w28a-beta</link>
+            <guid>2020w28a-beta</guid>
+            <pubDate>Thu, 09 Jul 2020 00:00:00 GMT</pubDate>
+            <description><![CDATA[- 🔥 Voltage heatmap implemented (OpenGL enabled);]]></description>
+        </item>
+        <item>
+            <title><![CDATA[2020w24a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/blog/2020/06/13/2020w24a-beta</link>
+            <guid>2020w24a-beta</guid>
+            <pubDate>Sat, 13 Jun 2020 00:00:00 GMT</pubDate>
+            <description><![CDATA[- ✔️ Device context render implemented (to avoid use OpenGL - some machines can't use it properly);]]></description>
+        </item>
+        <item>
+            <title><![CDATA[22019w44a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/blog/2019/11/01/2019w44a-beta</link>
+            <guid>22019w44a-beta</guid>
+            <pubDate>Fri, 01 Nov 2019 00:00:00 GMT</pubDate>
+            <description><![CDATA[- ✔️ Minor changes to 32bit MSW version for compatibility;]]></description>
+        </item>
+        <item>
+            <title><![CDATA[2019w42a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/blog/2019/10/13/2019w42a-beta</link>
+            <guid>2019w42a-beta</guid>
+            <pubDate>Sun, 13 Oct 2019 00:00:00 GMT</pubDate>
+            <description><![CDATA[- 🔥 Harmonic study implemented;]]></description>
+        </item>
+    </channel>
+</rss>
\ No newline at end of file
diff --git a/docs/blog/tags/index.html b/docs/blog/tags/index.html
index 2f738ba..119e35f 100644
--- a/docs/blog/tags/index.html
+++ b/docs/blog/tags/index.html
@@ -1,27 +1,31 @@
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+<meta name="viewport" content="width=device-width,initial-scale=1">
+<meta name="generator" content="Docusaurus v2.0.0-alpha.71">
+<link rel="alternate" type="application/rss+xml" href="/PSP/blog/rss.xml" title="PSP-UFU Blog RSS Feed">
+<link rel="alternate" type="application/atom+xml" href="/PSP/blog/atom.xml" title="PSP-UFU Blog Atom Feed">
+<link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">Tags | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="Tags | PSP-UFU"><meta data-react-helmet="true" name="description" content="Blog Tags"><meta data-react-helmet="true" property="og:description" content="Blog Tags"><meta data-react-helmet="true" property="og:url" content="https://thales1330.github.io/PSP/blog/tags"><meta data-react-helmet="true" name="docsearch:language" content="en"><meta data-react-helmet="true" name="docsearch:docusaurus_tag" content="blog_tags_list"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/img/favicon.ico"><link data-react-helmet="true" rel="canonical" href="https://thales1330.github.io/PSP/blog/tags"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/blog/tags" hreflang="en"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/pt/blog/tags" hreflang="pt"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/blog/tags" hreflang="x-default"><link data-react-helmet="true" rel="preconnect" href="https://BH4D9OD16A-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="/PSP/assets/css/styles.73f64b8b.css">
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5.98-.043.029-.098.129-.033.207l.619.756c.08.096.206.059.256.023 2.51-1.73 3.661-4.515 2.869-6.683zm-7.386 3.188c-.966-.121-.944-.914-.944-1.453 0-.773.327-1.58.876-2.156a3.21 3.21 0 011.229-.799l.082 4.277a2.773 2.773 0 01-1.243.131zm2.427-.553l.046-4.109c.084-.004.166-.01.252-.01.773 0 1.494.145 1.885.361.391.217-1.023 2.713-2.183 3.758zm-8.95-7.668a.196.196 0 00-.196-.145h-1.95a.194.194 0 00-.194.144L.008 16.916c-.017.051-.011.076.062.076h1.733c.075 0 .099-.023.114-.072l1.008-3.318h3.496l1.008 3.318c.016.049.039.072.113.072h1.734c.072 0 .078-.025.062-.076-.014-.05-3.083-9.741-3.494-11.04zm-2.618 6.318l1.447-5.25 1.447 5.25H3.226z"></path></svg><span>English</span></span></a><ul class="dropdown__menu"><li><a href="/PSP/blog/tags" target="_self" rel="noopener noreferrer" class="dropdown__link dropdown__link--active">English</a></li><li><a href="/PSP/pt/blog/tags" target="_self" rel="noopener noreferrer" class="dropdown__link">Português</a></li></ul></div><a 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011.229-.799l.082 4.277a2.773 2.773 0 01-1.243.131zm2.427-.553l.046-4.109c.084-.004.166-.01.252-.01.773 0 1.494.145 1.885.361.391.217-1.023 2.713-2.183 3.758zm-8.95-7.668a.196.196 0 00-.196-.145h-1.95a.194.194 0 00-.194.144L.008 16.916c-.017.051-.011.076.062.076h1.733c.075 0 .099-.023.114-.072l1.008-3.318h3.496l1.008 3.318c.016.049.039.072.113.072h1.734c.072 0 .078-.025.062-.076-.014-.05-3.083-9.741-3.494-11.04zm-2.618 6.318l1.447-5.25 1.447 5.25H3.226z"></path></svg><span>Languages</span></span></a><ul class="menu__list"><li class="menu__list-item"><a href="/PSP/blog/tags" target="_self" rel="noopener noreferrer" class="menu__link dropdown__link--active">English</a></li><li class="menu__list-item"><a href="/PSP/pt/blog/tags" target="_self" rel="noopener noreferrer" class="menu__link">Português</a></li></ul></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">GitHub</a></li></ul></div></div></div></nav><div class="main-wrapper blog-wrapper"><div class="container margin-vert--lg"><div class="row"><div class="col col--3"><div class="sidebar_2ahu thin-scrollbar"><h3 class="sidebarItemTitle_2hhb">Recent posts</h3><ul class="sidebarItemList_2xAf"><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2020/08/01/2020w31a-beta">2020w31a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2020/07/09/2020w28a-beta">2020w28a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2020/06/13/2020w24a-beta">2020w24a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2019/11/01/2019w44a-beta">22019w44a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><h1>Tags</h1><div class="margin-vert--lg"><div><h3>P</h3><a class="padding-right--md" href="/PSP/blog/tags/psp-ufu">psp-ufu (5)</a><hr></div></div></main></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" 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Built with Docusaurus.</div></div></div></footer></div>
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diff --git a/docs/blog/tags/psp-ufu/index.html b/docs/blog/tags/psp-ufu/index.html
index 0b474ce..0268b83 100644
--- a/docs/blog/tags/psp-ufu/index.html
+++ b/docs/blog/tags/psp-ufu/index.html
@@ -1,39 +1,43 @@
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posts tagged with &quot;psp-ufu&quot;</h1><a href="/PSP/blog/tags">View All Tags</a><div class="margin-vert--xl"><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2020w31a-beta">2020w31a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-08-01T00:00:00.000Z" class="blogPostDate_3bQP">August 1, 2020  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>);</li><li>✔️ New fancy <a href="https://thales1330.github.io/PSP/" target="_blank" rel="noopener noreferrer">PSP-UFU website</a> and <a href="https://thales1330.github.io/PSP/blog" target="_blank" rel="noopener noreferrer">changelog</a> using <a href="https://v2.docusaurus.io/" target="_blank" rel="noopener noreferrer">docusaurus</a>;</li><li>✔️ Text element performance improved for OpenGL render;</li><li>🐛 Several major and minor bug fixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w31a-beta" href="/PSP/blog/2020w31a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2020w28a-beta">2020w28a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_3bQP">July 9, 2020  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>🔥 Voltage heatmap implemented (OpenGL enabled);</li><li>🐛 Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w28a-beta" href="/PSP/blog/2020w28a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2020w24a-beta">2020w24a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_3bQP">June 13, 2020  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>✔️ Device context render implemented (to avoid use OpenGL - some machines can&#x27;t use it properly);</li><li>✔️ Added Visual Studio solution to the GitHub project;</li><li>🐛 Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w24a-beta" href="/PSP/blog/2020w24a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2019w44a-beta">22019w44a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_3bQP">November 1, 2019  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>✔️ Minor changes to 32bit MSW version for compatibility;</li><li>✔️ Travis CI (Continuous Integration) implemented (Linux).</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 22019w44a-beta" href="/PSP/blog/2019w44a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_1mse"><a href="/PSP/blog/2019w42a-beta">2019w42a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_3bQP">October 13, 2019  · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>🔥 Harmonic study implemented;</li><li>🔥 Autolayout from importation implemented;</li><li>🔥 Importation of Matpower files;</li><li>🐛 Several minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2019w42a-beta" href="/PSP/blog/2019w42a-beta"><strong>Read More</strong></a></div></footer></article></div></main></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li 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011.229-.799l.082 4.277a2.773 2.773 0 01-1.243.131zm2.427-.553l.046-4.109c.084-.004.166-.01.252-.01.773 0 1.494.145 1.885.361.391.217-1.023 2.713-2.183 3.758zm-8.95-7.668a.196.196 0 00-.196-.145h-1.95a.194.194 0 00-.194.144L.008 16.916c-.017.051-.011.076.062.076h1.733c.075 0 .099-.023.114-.072l1.008-3.318h3.496l1.008 3.318c.016.049.039.072.113.072h1.734c.072 0 .078-.025.062-.076-.014-.05-3.083-9.741-3.494-11.04zm-2.618 6.318l1.447-5.25 1.447 5.25H3.226z"></path></svg><span>Languages</span></span></a><ul class="menu__list"><li class="menu__list-item"><a href="/PSP/blog/tags/psp-ufu" target="_self" rel="noopener noreferrer" class="menu__link dropdown__link--active">English</a></li><li class="menu__list-item"><a href="/PSP/pt/blog/tags/psp-ufu" target="_self" rel="noopener noreferrer" class="menu__link">Português</a></li></ul></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">GitHub</a></li></ul></div></div></div></nav><div class="main-wrapper blog-wrapper"><div class="container margin-vert--lg"><div class="row"><div class="col col--3"><div class="sidebar_2ahu thin-scrollbar"><h3 class="sidebarItemTitle_2hhb">Recent posts</h3><ul class="sidebarItemList_2xAf"><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2020/08/01/2020w31a-beta">2020w31a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2020/07/09/2020w28a-beta">2020w28a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2020/06/13/2020w24a-beta">2020w24a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2019/11/01/2019w44a-beta">22019w44a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><h1>5 posts tagged with &quot;psp-ufu&quot;</h1><a href="/PSP/blog/tags">View All Tags</a><div class="margin-vert--xl"><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2020/08/01/2020w31a-beta">2020w31a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-08-01T00:00:00.000Z" class="blogPostDate_fNvV">July 31, 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>);</li><li>✔️ New fancy <a href="https://thales1330.github.io/PSP/" target="_blank" rel="noopener noreferrer">PSP-UFU website</a> and <a href="https://thales1330.github.io/PSP/blog" target="_blank" rel="noopener noreferrer">changelog</a> using <a href="https://v2.docusaurus.io/" target="_blank" rel="noopener noreferrer">docusaurus</a>;</li><li>✔️ Text element performance improved for OpenGL render;</li><li>🐛 Several major and minor bug fixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w31a-beta" href="/PSP/blog/2020/08/01/2020w31a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2020/07/09/2020w28a-beta">2020w28a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_fNvV">July 8, 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Voltage heatmap implemented (OpenGL enabled);</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w28a-beta" href="/PSP/blog/2020/07/09/2020w28a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2020/06/13/2020w24a-beta">2020w24a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_fNvV">June 12, 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Device context render implemented (to avoid use OpenGL - some machines can&#x27;t use it properly);</li><li>✔️ Added Visual Studio solution to the GitHub project;</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w24a-beta" href="/PSP/blog/2020/06/13/2020w24a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2019/11/01/2019w44a-beta">22019w44a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_fNvV">October 31, 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Minor changes to 32bit MSW version for compatibility;</li><li>✔️ Travis CI (Continuous Integration) implemented (Linux).</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 22019w44a-beta" href="/PSP/blog/2019/11/01/2019w44a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_fNvV">October 12, 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Harmonic study implemented;</li><li>🔥 Autolayout from importation implemented;</li><li>🔥 Importation of Matpower files;</li><li>🐛 Several minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2019w42a-beta" href="/PSP/blog/2019/10/13/2019w42a-beta"><strong>Read More</strong></a></div></footer></article></div></main></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. Built with Docusaurus.</div></div></div></footer></div>
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Nele \xe9 poss\xedvel tamb\xe9m acessar aos controles das m\xe1quinas s\xedncronas manipulados pelo ',Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"controlEditor"}),"editor de controle"),"."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(b.a)("images/syncGeneratorStabForm.png"),alt:"Formul\xe1rio de estabilidade dos geradores s\xedncronos no PSP-UFU",title:"Formul\xe1rio de estabilidade dos geradores s\xedncronos no PSP-UFU"}))),Object(n.b)("p",null,'No formul\xe1rio de estabilidade pode ser observado o bot\xe3o "Chaveamento" na parte inferior esquerda do formul\xe1rio. 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Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"https://pt.wikipedia.org/wiki/Unicode"}),"Unicode"),"."),Object(n.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(n.b)("h4",{id:"pot\xeancia-nominal"},"Pot\xeancia nominal"),Object(n.b)("p",null,"Pot\xeancia nominal do gerador, inserida em MVA, kVA ou VA."),Object(n.b)("p",null,'Esse campo \xe9 especialmente importante caso a op\xe7\xe3o "Utilizar a pot\xeancia nominal como base" esteja marcada.'),Object(n.b)("h4",{id:"pot\xeancias-ativa-e-reativa"},"Pot\xeancias ativa e reativa"),Object(n.b)("p",null,"Pot\xeancias ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do gerador."),Object(n.b)("p",null,"Caso a barra conectada seja PV o valor de pot\xeancia reativa ser\xe1 ignorado e caso seja de refer\xeancia ambos os valores inseridos ser\xe3o desprezados."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"Caso mais de um gerador esteja conectado na mesma barra, os valores de pot\xeancia reativa (nas barras de refer\xeancia e PV) e ativa (nas barras de refer\xeancia) s\xe3o igualmente distribu\xeddas, respeitando os limites individuais de pot\xeancia reativa."))),Object(n.b)("h4",{id:"pot\xeancias-reativas-m\xe1xima-e-m\xednima"},"Pot\xeancias reativas m\xe1xima e m\xednima"),Object(n.b)("p",null,"Limites de pot\xeancia reativa m\xe1xima e m\xednima do gerador para controle de tens\xe3o em barras PV. 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New York: Wiley-IEEE Press, 1995.")))}l.isMDXComponent=!0},123:function(a,e,t){"use strict";var s=t(0),m=t(20);e.a=function(){const a=Object(s.useContext)(m.a);if(null===a)throw new Error("Docusaurus context not provided");return a}},124:function(a,e,t){"use strict";t.d(e,"a",(function(){return j})),t.d(e,"b",(function(){return i}));var s=t(0),m=t.n(s);function n(a,e,t){return e in a?Object.defineProperty(a,e,{value:t,enumerable:!0,configurable:!0,writable:!0}):a[e]=t,a}function b(a,e){var t=Object.keys(a);if(Object.getOwnPropertySymbols){var s=Object.getOwnPropertySymbols(a);e&&(s=s.filter((function(e){return Object.getOwnPropertyDescriptor(a,e).enumerable}))),t.push.apply(t,s)}return t}function p(a){for(var e=1;e<arguments.length;e++){var 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Barramento</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Condutor de baixa impedância ao qual vários circuitos elétricos podem ser conectados em pontos separados.
-Nota - Em muitos casos, o barramento consiste em uma barra. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=151-12-30" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="barramento-no-psp-ufu"></a>Barramento no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barramento-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento <strong>barramento</strong>, ou simplesmente <strong>barra</strong>, é um conector ou nó do diagrama unifilar do PSP-UFU. Essa barra pode representar um PAC (Ponto de Acoplamento Comum), um poste de distribuição, uma subestação, um barramento da subestação, entre inúmeros outros tipos pontos de análise e conexão entre elementos.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-dos-barramentos"></a>Formulário de edição dos barramentos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-barramentos" title="Direct link to heading">#</a></h2><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O barramento deve ser o <strong>primeiro elemento elétrico a ser inserido no diagrama de potência</strong>, uma vez que os demais componentes de potência são conectados nele.</p></div></div><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das barras:</p><div><center><img src="/PSP/images/busForm.png" alt="Formulário dos barramentos no PSP-UFU" title="Formulário dos barramentos no PSP-UFU"></center></div><p>Esse formulário é subdividido em quatro contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais da barra e informações do fluxo de carga;</li><li><strong>Falta</strong>: local onde o curto-circuito <em>shunt</em> deve ser inserido;</li><li><strong>Estabilidade</strong>: contendo opções de visualização de dados da barra em gráficos no tempo e inserção de faltas trifásicas no cálculo de estabilidade transitória;</li><li><strong>Qualidade de energia</strong>: contém a opção de de visualização da impedância harmônica vista pela barra.</li></ul><hr><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Falta</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Estabilidade</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Qualidade de energia</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="tensão-nominal"></a>Tensão nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Utilizado para cálculo da impedância base de alguns elementos conectados, além do cálculo da relação de transformação dos transformadores conectados entre barras. Sua unidade pode ser selecionada, sendo expressa em V ou em kV.</p><p>A modificação desse parâmetro irá alterar toda a tensão do trecho conectado por linhas elétricas, sendo emitido um alerta ao usuário.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="tensão-controlada"></a>Tensão controlada<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-controlada" title="Direct link to heading">#</a></h4><p>Caracteriza o barramento como <strong>barra de tensão controlada (<a href="/PSP/docs/powerFlow">Barra PV</a>)</strong>,</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Essa opção é somente válida caso alguma máquina síncrona esteja conectado, caso contrário esse valor será ignorado. Caso o limite de potência reativa da máquina síncrona conectada seja ultrapassado esse valor também é ignorado.</p></div></div><p>O valor poderá ser inserido em p.u. ou em volts (ou kV caso a tensão nominal esteja nesta unidade).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="barra-de-referência"></a>Barra de referência<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barra-de-referência" title="Direct link to heading">#</a></h4><p>Caracteriza o barramento como <strong>barra de referência (<a href="/PSP/docs/powerFlow">Barra de oscilação</a>)</strong>. Essa opção é somente válida caso esteja conectado um gerador síncrono, caso contrário uma mensagem de erro será exibida ao usuário ao realizar algum dos cálculos do programa.</p></div></div></div><hr><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Resumo de como definir o tipo de barra:</p><ul><li>Para definir o barramento como sendo uma <strong><a href="/PSP/docs/powerFlow">Barra PQ</a></strong>, as opções &quot;Barra de referência&quot; e &quot;Tensão controlada&quot; devem estar <strong>desmarcadas</strong>; </li><li>Para definir o barramento como sendo uma <strong><a href="/PSP/docs/powerFlow">Barra PV</a></strong>, deve-se <strong>marcar somente</strong> a opção &quot;Tensão controlada&quot;, mantendo a opção &quot;Barra de referência&quot; desmarcada;</li><li>Para definir o barramento como sendo uma <strong><a href="/PSP/docs/powerFlow">Barra de Referência</a></strong>, deve-se marcar a opção &quot;Barra de referência&quot;. Caso a opção &quot;Tensão controlada&quot; esteja desmarcada será admitido um valor de tensão controlada de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">1{,}0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></li></ul></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p><strong>O sistema deve possuir somente uma barra de referência</strong>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/bus.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/powerEditor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Editor de Potência</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/line"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Linha »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#barramento-no-psp-ufu" class="table-of-contents__link">Barramento no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-barramentos" class="table-of-contents__link">Formulário de edição dos barramentos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Barramento</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Condutor de baixa impedância ao qual vários circuitos elétricos podem ser conectados em pontos separados.
+Nota - Em muitos casos, o barramento consiste em uma barra. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=151-12-30" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barramento-no-psp-ufu"></a>Barramento no PSP-UFU<a class="hash-link" href="#barramento-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento <strong>barramento</strong>, ou simplesmente <strong>barra</strong>, é um conector ou nó do diagrama unifilar do PSP-UFU. Essa barra pode representar um PAC (Ponto de Acoplamento Comum), um poste de distribuição, uma subestação, um barramento da subestação, entre inúmeros outros tipos pontos de análise e conexão entre elementos.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-barramentos"></a>Formulário de edição dos barramentos<a class="hash-link" href="#formulário-de-edição-dos-barramentos" title="Direct link to heading">#</a></h2><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O barramento deve ser o <strong>primeiro elemento elétrico a ser inserido no diagrama de potência</strong>, uma vez que os demais componentes de potência são conectados nele.</p></div></div><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das barras:</p><div><center><img src="/PSP/images/busForm.png" alt="Formulário dos barramentos no PSP-UFU" title="Formulário dos barramentos no PSP-UFU"></center></div><p>Esse formulário é subdividido em quatro contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais da barra e informações do fluxo de carga;</li><li><strong>Falta</strong>: local onde o curto-circuito <em>shunt</em> deve ser inserido;</li><li><strong>Estabilidade</strong>: contendo opções de visualização de dados da barra em gráficos no tempo e inserção de faltas trifásicas no cálculo de estabilidade transitória;</li><li><strong>Qualidade de energia</strong>: contém a opção de de visualização da impedância harmônica vista pela barra.</li></ul><hr><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Qualidade de energia</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-nominal"></a>Tensão nominal<a class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Utilizado para cálculo da impedância base de alguns elementos conectados, além do cálculo da relação de transformação dos transformadores conectados entre barras. Sua unidade pode ser selecionada, sendo expressa em V ou em kV.</p><p>A modificação desse parâmetro irá alterar toda a tensão do trecho conectado por linhas elétricas, sendo emitido um alerta ao usuário.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-controlada"></a>Tensão controlada<a class="hash-link" href="#tensão-controlada" title="Direct link to heading">#</a></h4><p>Caracteriza o barramento como <strong>barra de tensão controlada (<a href="/PSP/docs/powerFlow">Barra PV</a>)</strong>,</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Essa opção é somente válida caso alguma máquina síncrona esteja conectado, caso contrário esse valor será ignorado. Caso o limite de potência reativa da máquina síncrona conectada seja ultrapassado esse valor também é ignorado.</p></div></div><p>O valor poderá ser inserido em p.u. ou em volts (ou kV caso a tensão nominal esteja nesta unidade).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barra-de-referência"></a>Barra de referência<a class="hash-link" href="#barra-de-referência" title="Direct link to heading">#</a></h4><p>Caracteriza o barramento como <strong>barra de referência (<a href="/PSP/docs/powerFlow">Barra de oscilação</a>)</strong>. Essa opção é somente válida caso esteja conectado um gerador síncrono, caso contrário uma mensagem de erro será exibida ao usuário ao realizar algum dos cálculos do programa.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserir-falta"></a>Inserir falta<a class="hash-link" href="#inserir-falta" title="Direct link to heading">#</a></h4><p>Indica se existe um curto-circuito na barra nos estudos de <a href="/PSP/docs/fault">curto-circuito</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Nota-se que essa opção irá inserir uma falta no cálculo de <a href="/PSP/docs/fault">curto-circuito</a> e <strong>não será considerado no estudo de estabilidade</strong>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tipo-de-falta"></a>Tipo de falta<a class="hash-link" href="#tipo-de-falta" title="Direct link to heading">#</a></h4><p>Seleciona o tipo de falha shunt da barra:</p><ul><li>trifásico para a terra</li><li>fase-fase</li><li>fase-fase-terra</li><li>fase-terra</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="local-da-falta"></a>Local da falta<a class="hash-link" href="#local-da-falta" title="Direct link to heading">#</a></h4><p>Seleciona a fase em que se situa a falta (ou combinação delas no caso de falha entre duas fases), sendo essa opção desabilitada para o tipo de falta trifásica.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-e-reatância-de-falta"></a>Resistência e reatância de falta<a class="hash-link" href="#resistência-e-reatância-de-falta" title="Direct link to heading">#</a></h4><p>Representam a impedância da falta. Curtos-circuitos francos (sem impedância de falta) são representados inserindo valor zero em ambos os campos.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-dados"></a>Imprimir dados<a class="hash-link" href="#imprimir-dados" title="Direct link to heading">#</a></h4><p>Mostra os dados da barra editada no gráfico gerado pelo estudo de estabilidade transitória. Os seguintes dados são exibidos:</p><ul><li>módulo da tensão do barramento</li><li>ângulo da tensão do barramento.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserir-falta-1"></a>Inserir falta<a class="hash-link" href="#inserir-falta-1" title="Direct link to heading">#</a></h4><p>Insere uma falta trifásica na barra no cálculo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Nota-se que essa opção irá inserir uma falta no cálculo de <a href="/PSP/docs/stability">estabilidade</a> e <strong>não será considerado no estudo de <a href="/PSP/docs/fault">curto-circuito</a></strong>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tempo"></a>Tempo<a class="hash-link" href="#tempo" title="Direct link to heading">#</a></h4><p>Instante no tempo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">t_{falta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) em que ocorre a falta durante os estudos de <a href="/PSP/docs/stability">estabilidade</a>. Esse campo e os próximos são somente acessados caso seja inserida uma falta no estudo de estabilidade.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="duração-da-falta"></a>Duração da falta<a class="hash-link" href="#duração-da-falta" title="Direct link to heading">#</a></h4><p>Duração da falta inserida (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta t_{falta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) no estudo de estabilidade. A falta irá iniciar no instante <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">t_{falta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e será interrompida em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">t_{falta}+\Delta t_{falta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-e-reatância-de-falta-1"></a>Resistência e reatância de falta<a class="hash-link" href="#resistência-e-reatância-de-falta-1" title="Direct link to heading">#</a></h4><p>Impedância de falta nos estudos de estabilidade. Uma falta franca é representada pela inserção de um valor de impedância shunt de valor muito baixo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j10^{-5} p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>).</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-impedância-harmônica-da-barra"></a>Imprimir impedância harmônica da barra<a class="hash-link" href="#imprimir-impedância-harmônica-da-barra" title="Direct link to heading">#</a></h4><p>Imprime a impedância harmônica vista por essa barra referente ao estudo de <a href="/PSP/docs/harmonics">Resposta na Frequência</a>.</p></div></div></div><hr><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Resumo de como definir o tipo de barra:</p><ul><li>Para definir o barramento como sendo uma <strong><a href="/PSP/docs/powerFlow">Barra PQ</a></strong>, as opções &quot;Barra de referência&quot; e &quot;Tensão controlada&quot; devem estar <strong>desmarcadas</strong>; </li><li>Para definir o barramento como sendo uma <strong><a href="/PSP/docs/powerFlow">Barra PV</a></strong>, deve-se <strong>marcar somente</strong> a opção &quot;Tensão controlada&quot;, mantendo a opção &quot;Barra de referência&quot; desmarcada;</li><li>Para definir o barramento como sendo uma <strong><a href="/PSP/docs/powerFlow">Barra de Referência</a></strong>, deve-se marcar a opção &quot;Barra de referência&quot;. Caso a opção &quot;Tensão controlada&quot; esteja desmarcada será admitido um valor de tensão controlada de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">1{,}0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></li></ul></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p><strong>O sistema deve possuir somente uma barra de referência</strong>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/bus.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/powerEditor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Editor de Potência</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/line"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Linha »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#barramento-no-psp-ufu" class="table-of-contents__link">Barramento no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-barramentos" class="table-of-contents__link">Formulário de edição dos barramentos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Ferramentas CAD</h1></header><div class="markdown"><blockquote><p>Desenho assistido por computador (DAC; em inglês: <em>computer aided design</em> - CAD) é o nome genérico de sistemas computacionais (<em>software</em>) utilizados pela engenharia, geologia, geografia, arquitetura e design para facilitar o projeto e desenho técnicos. No caso do <em>design</em>, este pode estar ligado especificamente a todas as suas vertentes (produtos como vestuário, eletroeletrônicos, automobilísticos, etc.), de modo que os jargões de cada especialidade são incorporados na interface de cada programa. <a href="https://pt.wikipedia.org/wiki/Desenho_assistido_por_computador" target="_blank" rel="noopener noreferrer">Wikipedia</a></p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="ferramentas-do-menu"></a>Ferramentas do menu<a aria-hidden="true" tabindex="-1" class="hash-link" href="#ferramentas-do-menu" title="Direct link to heading">#</a></h2><p>Todos os objetos criados contêm vários atributos gráficos, como: posição, orientação, cor, tamanho, etc. Para modificá-los foram inseridas algumas ferramentas intuitivas e de fácil utilização, as quais são: <a href="/PSP/docs/cadTools#selecionar"><strong>selecionar</strong></a>, <a href="/PSP/docs/cadTools#mover"><strong>mover</strong></a>, <a href="/PSP/docs/cadTools#girar"><strong>girar</strong></a>, <a href="/PSP/docs/cadTools#arrastar"><strong>arrastar</strong></a>, <a href="/PSP/docs/cadTools#zoom"><strong>zoom</strong></a>, <a href="/PSP/docs/cadTools#encaixar"><strong>encaixar</strong></a>, <a href="/PSP/docs/cadTools#copiar-e-colar"><strong>copiar, colar</strong></a> e <a href="/PSP/docs/cadTools#excluir"><strong>excluir</strong></a>.</p><p>Essas ferramentas podem ser acessadas por suas teclas de atalho assim como pelo <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ferramentas</a>.</p><p>Tais ferramentas permitem a <strong>personalização</strong>, <strong>navegação</strong> e auxiliam a <strong>criação</strong> de redes elétricas de potência, assim como diagramas de controle na plataforma.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="selecionar"></a>Selecionar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#selecionar" title="Direct link to heading">#</a></h3><p>A seleção de um único objeto é realizada clicando com o botão esquerdo do mouse sobre o elemento, que será identificado pelo contorno em <span style="background-color:#80bfff;border-radius:2px;color:#000;padding:0.2rem">azul</span> do objeto.</p><p>Para a seleção de vários objetos simultaneamente é necessário clicar com o botão esquerdo do mouse em um local vazio da tela e arrastá-lo, criando assim um <strong>retângulo de seleção</strong> que irá acompanhar o ponteiro do mouse enquanto o botão esquerdo estiver pressionado. Ao soltá-lo todos os objetos que intercederem o retângulo serão selecionados.</p><p>Mantendo pressionado a tecla Control (Ctrl) do teclado pode-se manter seleções anteriores enquanto novos elementos são selecionados.</p><p>Para desmarcar todos os objetos basta clicar em uma área sem objetos.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/select.mp4" type="video/mp4"><source src="/PSP/videos/select.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="mover"></a>Mover<a aria-hidden="true" tabindex="-1" class="hash-link" href="#mover" title="Direct link to heading">#</a></h3><p>Um elemento poderá ser movido para qualquer local por meio da ação <em>drag-and-drop</em> (clicando e arrastando) ou pela ferramenta mover.</p><p>Ao habilitar o “modo mover”, movendo todos os componentes selecionados à medida que a posição do mouse é alterada. Para desabilitar esse modo basta pressionar a tecla Escape do teclado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/move.mp4" type="video/mp4"><source src="/PSP/videos/move.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="girar"></a>Girar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#girar" title="Direct link to heading">#</a></h3><p>Alguns objetos poderão ser rotacionados através da ferramenta girar, acessado pela tecla de atalho “R”, girando o objeto no sentido horário, ou “Shift + R”, para o sentido anti-horário (essa opção irá afetar todos os objetos selecionados). O elemento também pode ser rotacionado pelo <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ferramentas</a> ou ao acessar o menu de contexto clicando com o botão direito sobre o elemento.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O menu de contexto somente será exibido se o elemento estiver selecionado.</p></div></div><p>Ao utilizar essa ferramenta os objetos irão girar 45º a partir do seu ponto de origem. Para atingir a angulação pretendida basta repetir o processo de rotação.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/rotate.mp4" type="video/mp4"><source src="/PSP/videos/rotate.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="arrastar"></a>Arrastar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#arrastar" title="Direct link to heading">#</a></h3><p>Três ferramentas muito úteis na navegação do circuito, principalmente em grandes redes, são: arrastar, zoom e encaixar.</p><p>A ferramenta arrastar possibilita mover todo o circuito <strong>preservando as posições relativas entre os elementos</strong>. Esta ferramenta pode ser acessada pelo submenu “Ferramentas” ou <strong>mantendo pressionada o <em>scroll</em> do mouse</strong>, ativando assim o “modo arrastar”. Ao ativá-lo todo o circuito será movido clicando e arrastando na direção desejada.</p><p>Para sair do “modo arrastar” basta pressionar a tecla Escape do teclado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/drag.mp4" type="video/mp4"><source src="/PSP/videos/drag.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="zoom"></a>Zoom<a aria-hidden="true" tabindex="-1" class="hash-link" href="#zoom" title="Direct link to heading">#</a></h3><p>O zoom será aplicado ao utilizar o <strong>scroll do mouse</strong> para mais ou menos zoom. A ferramenta zoom irá aproximar ou distanciar o circuito a partir da posição do ponteiro do mouse na tela para encontrar objetos ou destaca-los.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/zoom.mp4" type="video/mp4"><source src="/PSP/videos/zoom.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="encaixar"></a>Encaixar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#encaixar" title="Direct link to heading">#</a></h3><p>A ferramenta encaixar irá mover o circuito e aplicar o zoom necessário para que todos os componentes da rede sejam exibidos na tela. Esta ferramenta pode ser acessada através do submenu “Ferramentas” ou pela combinação de teclas “Shift + F”.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/fit.mp4" type="video/mp4"><source src="/PSP/videos/fit.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="copiar-e-colar"></a>Copiar e Colar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#copiar-e-colar" title="Direct link to heading">#</a></h3><p>Duas ferramentas importantes durante o processo de criação da rede elétrica de potência no <em>software</em> são copiar e colar. Qualquer circuito criado no PSP-UFU poderá ser duplicado completamente ou parcialmente no mesmo projeto ou em múltiplos através dessas ferramentas.</p><p>Para copiar basta selecionar os elementos que se deseja duplicar e acessar a ferramenta no submenu “Ferramentas” ou pressionando a combinação das teclas “Ctrl + C”, então os dados do circuito serão copiados para a área de transferência. Ao copiar um circuito ele poderá ser colado no mesmo projeto ou em um projeto distinto, utilizando o mesmo submenu ou pressionando a combinação das teclas “Ctrl + V”.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/copyPaste.mp4" type="video/mp4"><source src="/PSP/videos/copyPaste.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="excluir"></a>Excluir<a aria-hidden="true" tabindex="-1" class="hash-link" href="#excluir" title="Direct link to heading">#</a></h3><p>Qualquer objeto poderá ser excluído do projeto por meio da ferramenta deletar, presente no menu Ribbon, pela tecla de atalho Delete ou pelo acesso ao menu de contexto clicando com o botão direito sobre o elemento selecionado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/delete.mp4" type="video/mp4"><source src="/PSP/videos/delete.webm" type="video/webm"></video><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="personalização-gráfica-os-elementos"></a>Personalização gráfica os elementos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#personalização-gráfica-os-elementos" title="Direct link to heading">#</a></h2><p>Alguns elementos permitem sua personalização gráfica, como as barras e linhas.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="barra"></a>Barra<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barra" title="Direct link to heading">#</a></h3><p>No caso dos barramentos, pode-se alterar o <strong>comprimento da barra</strong> clicando e arrastando no <em>pickbox</em> exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="linha"></a>Linha<a aria-hidden="true" tabindex="-1" class="hash-link" href="#linha" title="Direct link to heading">#</a></h3><p>As linhas podem ser inseridas com <strong>&quot;nós&quot; (pontos de ancoragem)</strong>, personalizando a disposição das linhas no diagrama unifilar. Esses nós são inseridos durante o processo de criação do elemento, anteriormente à seleção da segunda barra.</p><p>Após a sua inserção, novos nós podem ser anexados pelo menu de contexto acessado ao clicar com o botão direito sobre a linha selecionada. Sua remoção também é alcançada por uma opção do menu de contexto.</p><p>Assim como nos barramentos, a alteração da posição dos nós é obtida clicando e arrastando o <em>pickbox</em> exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/graphicPropsChange.mp4" type="video/mp4"><source src="/PSP/videos/graphicPropsChange.webm" type="video/webm"></video><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="conexão-e-desconexão-de-elemetos-na-barra"></a>Conexão e desconexão de elemetos na barra<a aria-hidden="true" tabindex="-1" class="hash-link" href="#conexão-e-desconexão-de-elemetos-na-barra" title="Direct link to heading">#</a></h2><p>Os elementos podem ser desconectados e reconectados às barras após sua inserção somente pela ferramenta <em>drag-and-drop</em>. Para isso, o elemento deve ser selecionado e o seu nó de conexão deve ser arrastado para o local requerido.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O elemento a ser desconectado/reconectado <strong>deve ser previamente selecionado</strong> para efetuar a operação. Caso contrário a barra será movida ou nenhuma operação será realizada.</p></div></div><p>No caso de desconexão, o elemento será automaticamente removido da simulação, sendo indicado pela sua cor (o elemento passa a ter cor <span style="background-color:#7f7f7f;border-radius:2px;color:#000;padding:0.2rem">cinza</span>).</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Ao reconectar o elemento ao circuito, deve-se <strong>atentar para estado do seu &quot;disjuntor&quot;</strong>, indicado por um quadrado próximo ao seu nó de conexão. Um elemento reconectado volta ao circuito com seu disjuntor aberto (<span style="background-color:#ff1919;border-radius:2px;color:#000;padding:0.2rem">vermelho</span>), sendo necessário clicar sobre o mesmo a fim de inserir o elemento no circuito.</p></div></div><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/disconnect.mp4" type="video/mp4"><source src="/PSP/videos/disconnect.webm" type="video/webm"></video></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/cadTools.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/mainScreen"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Tela Principal</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/powerEditor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Editor de Potência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#ferramentas-do-menu" class="table-of-contents__link">Ferramentas do menu</a><ul><li><a href="#selecionar" class="table-of-contents__link">Selecionar</a></li><li><a href="#mover" class="table-of-contents__link">Mover</a></li><li><a href="#girar" class="table-of-contents__link">Girar</a></li><li><a href="#arrastar" class="table-of-contents__link">Arrastar</a></li><li><a href="#zoom" class="table-of-contents__link">Zoom</a></li><li><a href="#encaixar" class="table-of-contents__link">Encaixar</a></li><li><a href="#copiar-e-colar" class="table-of-contents__link">Copiar e Colar</a></li><li><a href="#excluir" class="table-of-contents__link">Excluir</a></li></ul></li><li><a href="#personalização-gráfica-os-elementos" class="table-of-contents__link">Personalização gráfica os elementos</a><ul><li><a href="#barra" class="table-of-contents__link">Barra</a></li><li><a href="#linha" class="table-of-contents__link">Linha</a></li></ul></li><li><a href="#conexão-e-desconexão-de-elemetos-na-barra" class="table-of-contents__link">Conexão e desconexão de elemetos na barra</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Ferramentas CAD</h1></header><div class="markdown"><blockquote><p>Desenho assistido por computador (DAC; em inglês: <em>computer aided design</em> - CAD) é o nome genérico de sistemas computacionais (<em>software</em>) utilizados pela engenharia, geologia, geografia, arquitetura e design para facilitar o projeto e desenho técnicos. No caso do <em>design</em>, este pode estar ligado especificamente a todas as suas vertentes (produtos como vestuário, eletroeletrônicos, automobilísticos, etc.), de modo que os jargões de cada especialidade são incorporados na interface de cada programa. <a href="https://pt.wikipedia.org/wiki/Desenho_assistido_por_computador" target="_blank" rel="noopener noreferrer">Wikipedia</a></p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="ferramentas-do-menu"></a>Ferramentas do menu<a class="hash-link" href="#ferramentas-do-menu" title="Direct link to heading">#</a></h2><p>Todos os objetos criados contêm vários atributos gráficos, como: posição, orientação, cor, tamanho, etc. Para modificá-los foram inseridas algumas ferramentas intuitivas e de fácil utilização, as quais são: <a href="/PSP/docs/cadTools#selecionar"><strong>selecionar</strong></a>, <a href="/PSP/docs/cadTools#mover"><strong>mover</strong></a>, <a href="/PSP/docs/cadTools#girar"><strong>girar</strong></a>, <a href="/PSP/docs/cadTools#arrastar"><strong>arrastar</strong></a>, <a href="/PSP/docs/cadTools#zoom"><strong>zoom</strong></a>, <a href="/PSP/docs/cadTools#encaixar"><strong>encaixar</strong></a>, <a href="/PSP/docs/cadTools#copiar-e-colar"><strong>copiar, colar</strong></a> e <a href="/PSP/docs/cadTools#excluir"><strong>excluir</strong></a>.</p><p>Essas ferramentas podem ser acessadas por suas teclas de atalho assim como pelo <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ferramentas</a>.</p><p>Tais ferramentas permitem a <strong>personalização</strong>, <strong>navegação</strong> e auxiliam a <strong>criação</strong> de redes elétricas de potência, assim como diagramas de controle na plataforma.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="selecionar"></a>Selecionar<a class="hash-link" href="#selecionar" title="Direct link to heading">#</a></h3><p>A seleção de um único objeto é realizada clicando com o botão esquerdo do mouse sobre o elemento, que será identificado pelo contorno em <span style="background-color:#80bfff;border-radius:2px;color:#000;padding:0.2rem">azul</span> do objeto.</p><p>Para a seleção de vários objetos simultaneamente é necessário clicar com o botão esquerdo do mouse em um local vazio da tela e arrastá-lo, criando assim um <strong>retângulo de seleção</strong> que irá acompanhar o ponteiro do mouse enquanto o botão esquerdo estiver pressionado. Ao soltá-lo todos os objetos que intercederem o retângulo serão selecionados.</p><p>Mantendo pressionado a tecla Control (Ctrl) do teclado pode-se manter seleções anteriores enquanto novos elementos são selecionados.</p><p>Para desmarcar todos os objetos basta clicar em uma área sem objetos.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/select.mp4" type="video/mp4"><source src="/PSP/videos/select.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="mover"></a>Mover<a class="hash-link" href="#mover" title="Direct link to heading">#</a></h3><p>Um elemento poderá ser movido para qualquer local por meio da ação <em>drag-and-drop</em> (clicando e arrastando) ou pela ferramenta mover.</p><p>Ao habilitar o “modo mover”, movendo todos os componentes selecionados à medida que a posição do mouse é alterada. Para desabilitar esse modo basta pressionar a tecla Escape do teclado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/move.mp4" type="video/mp4"><source src="/PSP/videos/move.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="girar"></a>Girar<a class="hash-link" href="#girar" title="Direct link to heading">#</a></h3><p>Alguns objetos poderão ser rotacionados através da ferramenta girar, acessado pela tecla de atalho “R”, girando o objeto no sentido horário, ou “Shift + R”, para o sentido anti-horário (essa opção irá afetar todos os objetos selecionados). O elemento também pode ser rotacionado pelo <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ferramentas</a> ou ao acessar o menu de contexto clicando com o botão direito sobre o elemento.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O menu de contexto somente será exibido se o elemento estiver selecionado.</p></div></div><p>Ao utilizar essa ferramenta os objetos irão girar 45º a partir do seu ponto de origem. Para atingir a angulação pretendida basta repetir o processo de rotação.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/rotate.mp4" type="video/mp4"><source src="/PSP/videos/rotate.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="arrastar"></a>Arrastar<a class="hash-link" href="#arrastar" title="Direct link to heading">#</a></h3><p>Três ferramentas muito úteis na navegação do circuito, principalmente em grandes redes, são: arrastar, zoom e encaixar.</p><p>A ferramenta arrastar possibilita mover todo o circuito <strong>preservando as posições relativas entre os elementos</strong>. Esta ferramenta pode ser acessada pelo submenu “Ferramentas” ou <strong>mantendo pressionada o <em>scroll</em> do mouse</strong>, ativando assim o “modo arrastar”. Ao ativá-lo todo o circuito será movido clicando e arrastando na direção desejada.</p><p>Para sair do “modo arrastar” basta pressionar a tecla Escape do teclado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/drag.mp4" type="video/mp4"><source src="/PSP/videos/drag.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="zoom"></a>Zoom<a class="hash-link" href="#zoom" title="Direct link to heading">#</a></h3><p>O zoom será aplicado ao utilizar o <strong>scroll do mouse</strong> para mais ou menos zoom. A ferramenta zoom irá aproximar ou distanciar o circuito a partir da posição do ponteiro do mouse na tela para encontrar objetos ou destaca-los.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/zoom.mp4" type="video/mp4"><source src="/PSP/videos/zoom.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="encaixar"></a>Encaixar<a class="hash-link" href="#encaixar" title="Direct link to heading">#</a></h3><p>A ferramenta encaixar irá mover o circuito e aplicar o zoom necessário para que todos os componentes da rede sejam exibidos na tela. Esta ferramenta pode ser acessada através do submenu “Ferramentas” ou pela combinação de teclas “Shift + F”.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/fit.mp4" type="video/mp4"><source src="/PSP/videos/fit.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="copiar-e-colar"></a>Copiar e Colar<a class="hash-link" href="#copiar-e-colar" title="Direct link to heading">#</a></h3><p>Duas ferramentas importantes durante o processo de criação da rede elétrica de potência no <em>software</em> são copiar e colar. Qualquer circuito criado no PSP-UFU poderá ser duplicado completamente ou parcialmente no mesmo projeto ou em múltiplos através dessas ferramentas.</p><p>Para copiar basta selecionar os elementos que se deseja duplicar e acessar a ferramenta no submenu “Ferramentas” ou pressionando a combinação das teclas “Ctrl + C”, então os dados do circuito serão copiados para a área de transferência. Ao copiar um circuito ele poderá ser colado no mesmo projeto ou em um projeto distinto, utilizando o mesmo submenu ou pressionando a combinação das teclas “Ctrl + V”.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/copyPaste.mp4" type="video/mp4"><source src="/PSP/videos/copyPaste.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="excluir"></a>Excluir<a class="hash-link" href="#excluir" title="Direct link to heading">#</a></h3><p>Qualquer objeto poderá ser excluído do projeto por meio da ferramenta deletar, presente no menu Ribbon, pela tecla de atalho Delete ou pelo acesso ao menu de contexto clicando com o botão direito sobre o elemento selecionado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/delete.mp4" type="video/mp4"><source src="/PSP/videos/delete.webm" type="video/webm"></video><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="personalização-gráfica-os-elementos"></a>Personalização gráfica os elementos<a class="hash-link" href="#personalização-gráfica-os-elementos" title="Direct link to heading">#</a></h2><p>Alguns elementos permitem sua personalização gráfica, como as barras e linhas.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barra"></a>Barra<a class="hash-link" href="#barra" title="Direct link to heading">#</a></h3><p>No caso dos barramentos, pode-se alterar o <strong>comprimento da barra</strong> clicando e arrastando no <em>pickbox</em> exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="linha"></a>Linha<a class="hash-link" href="#linha" title="Direct link to heading">#</a></h3><p>As linhas podem ser inseridas com <strong>&quot;nós&quot; (pontos de ancoragem)</strong>, personalizando a disposição das linhas no diagrama unifilar. Esses nós são inseridos durante o processo de criação do elemento, anteriormente à seleção da segunda barra.</p><p>Após a sua inserção, novos nós podem ser anexados pelo menu de contexto acessado ao clicar com o botão direito sobre a linha selecionada. Sua remoção também é alcançada por uma opção do menu de contexto.</p><p>Assim como nos barramentos, a alteração da posição dos nós é obtida clicando e arrastando o <em>pickbox</em> exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/graphicPropsChange.mp4" type="video/mp4"><source src="/PSP/videos/graphicPropsChange.webm" type="video/webm"></video><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="conexão-e-desconexão-de-elemetos-na-barra"></a>Conexão e desconexão de elemetos na barra<a class="hash-link" href="#conexão-e-desconexão-de-elemetos-na-barra" title="Direct link to heading">#</a></h2><p>Os elementos podem ser desconectados e reconectados às barras após sua inserção somente pela ferramenta <em>drag-and-drop</em>. Para isso, o elemento deve ser selecionado e o seu nó de conexão deve ser arrastado para o local requerido.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O elemento a ser desconectado/reconectado <strong>deve ser previamente selecionado</strong> para efetuar a operação. Caso contrário a barra será movida ou nenhuma operação será realizada.</p></div></div><p>No caso de desconexão, o elemento será automaticamente removido da simulação, sendo indicado pela sua cor (o elemento passa a ter cor <span style="background-color:#7f7f7f;border-radius:2px;color:#000;padding:0.2rem">cinza</span>).</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Ao reconectar o elemento ao circuito, deve-se <strong>atentar para estado do seu &quot;disjuntor&quot;</strong>, indicado por um quadrado próximo ao seu nó de conexão. Um elemento reconectado volta ao circuito com seu disjuntor aberto (<span style="background-color:#ff1919;border-radius:2px;color:#000;padding:0.2rem">vermelho</span>), sendo necessário clicar sobre o mesmo a fim de inserir o elemento no circuito.</p></div></div><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/disconnect.mp4" type="video/mp4"><source src="/PSP/videos/disconnect.webm" type="video/webm"></video></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/cadTools.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/mainScreen"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Tela Principal</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/powerEditor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Editor de Potência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#ferramentas-do-menu" class="table-of-contents__link">Ferramentas do menu</a><ul><li><a href="#selecionar" class="table-of-contents__link">Selecionar</a></li><li><a href="#mover" class="table-of-contents__link">Mover</a></li><li><a href="#girar" class="table-of-contents__link">Girar</a></li><li><a href="#arrastar" class="table-of-contents__link">Arrastar</a></li><li><a href="#zoom" class="table-of-contents__link">Zoom</a></li><li><a href="#encaixar" class="table-of-contents__link">Encaixar</a></li><li><a href="#copiar-e-colar" class="table-of-contents__link">Copiar e Colar</a></li><li><a href="#excluir" class="table-of-contents__link">Excluir</a></li></ul></li><li><a href="#personalização-gráfica-os-elementos" class="table-of-contents__link">Personalização gráfica os elementos</a><ul><li><a href="#barra" class="table-of-contents__link">Barra</a></li><li><a href="#linha" class="table-of-contents__link">Linha</a></li></ul></li><li><a href="#conexão-e-desconexão-de-elemetos-na-barra" class="table-of-contents__link">Conexão e desconexão de elemetos na barra</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Capacitor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Um dispositivo constituído essencialmente por dois eletrodos separados por um dielétrico. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=436-01-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="capacitor-no-psp-ufu"></a>Capacitor no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#capacitor-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento capacitor representa, geralmente, um banco de capacitores <em>shunt</em> no circuito do PSP-UFU.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/docs/bus">barramento</a> e um <a href="/PSP/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-dos-capacitores"></a>Formulário de edição dos capacitores<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-capacitores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos capacitores:</p><div><center><img src="/PSP/images/capacitorForm.png" alt="Formulário dos capacitores no PSP-UFU" title="Formulário dos capacitores no PSP-UFU"></center></div><p>Além do único contexto geral, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do capacitor durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/swCapacitor.png" alt="Formulário de chaveamento do capacitor" title="Formulário de chaveamento do capacitor"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Botão Estabilidade</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-reativa"></a>Potência reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base de potência do sistema). </p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/capacitor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/load"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Carga</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/inductor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Indutor »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#capacitor-no-psp-ufu" class="table-of-contents__link">Capacitor no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-capacitores" class="table-of-contents__link">Formulário de edição dos capacitores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Capacitor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Um dispositivo constituído essencialmente por dois eletrodos separados por um dielétrico. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=436-01-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="capacitor-no-psp-ufu"></a>Capacitor no PSP-UFU<a class="hash-link" href="#capacitor-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento capacitor representa, geralmente, um banco de capacitores <em>shunt</em> no circuito do PSP-UFU.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/docs/bus">barramento</a> e um <a href="/PSP/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-capacitores"></a>Formulário de edição dos capacitores<a class="hash-link" href="#formulário-de-edição-dos-capacitores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos capacitores:</p><div><center><img src="/PSP/images/capacitorForm.png" alt="Formulário dos capacitores no PSP-UFU" title="Formulário dos capacitores no PSP-UFU"></center></div><p>Além do único contexto geral, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do capacitor durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/swCapacitor.png" alt="Formulário de chaveamento do capacitor" title="Formulário de chaveamento do capacitor"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-reativa"></a>Potência reativa<a class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base de potência do sistema). </p></div><div role="tabpanel" hidden=""><p>O botão &quot;Estabilidade&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/capacitor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/load"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Carga</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/inductor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Indutor »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#capacitor-no-psp-ufu" class="table-of-contents__link">Capacitor no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-capacitores" class="table-of-contents__link">Formulário de edição dos capacitores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Constante</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>As constantes são valores numéricos reais que permanecem inalteradas em todo o período de simulação. Esses elementos possuem somente um nó de saída, o qual poderá ser conectado a qualquer outro nó de entrada dos elementos de controle.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Esse elemento pode, por exemplo, ser utilizado para definir valores alvos ou deslocamentos de sinal.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/const.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/rateLimiter"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Limitador de taxa</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/gain"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Ganho »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Constante</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>As constantes são valores numéricos reais que permanecem inalteradas em todo o período de simulação. Esses elementos possuem somente um nó de saída, o qual poderá ser conectado a qualquer outro nó de entrada dos elementos de controle.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Esse elemento pode, por exemplo, ser utilizado para definir valores alvos ou deslocamentos de sinal.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/const.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/rateLimiter"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Limitador de taxa</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/gain"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Ganho »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="docMainContainer_3EyW"><div class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Editor de Controle</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Os controles das <a href="/PSP/docs/syncGenerator">máquinas síncronas</a> são criados, modificados, importados e exportados em um editor distinto dos elementos de potência, apresentado na figura abaixo.</p><div><center><img src="/PSP/images/controlEditor.png" alt="Editor de elementos de controle do PSP-UFU" title="Editor de elementos de controle do PSP-UFU"></center></div><p>O acesso ao editor é realizado por meio de dois botões presentes no <a href="/PSP/docs/syncGenerator#formul%C3%A1rio-de-edi%C3%A7%C3%A3o-dos-geradores-s%C3%ADncronos">formulário de edição dos dados de estabilidade dos geradores síncronos</a>:</p><ul><li><a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Editar AVR</a></li><li><a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Editar regulador de velocidade</a></li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU a opção de editar o <strong>AVR</strong> engloba mais que somente o controle de tensão da máquina. Nele <strong>deve</strong> ser inserida a malha de controle da máquina assim como a <strong>excitatriz da máquina síncrona</strong>. Outras estratégias de controle (opcionais), como PSS (<em>Power System Stabilizer</em>) e/ou controles de sobre e sub excitação, são também implementadas em conjunto (como apresentado na figura anterior).</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Assim como no AVR, o <strong>Regulador de Velocidade</strong> engloba mais que a regulação primária da máquina. Nessa opção <strong>deve</strong> ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o <strong>modelo da turbina</strong>. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="elementos-de-controle"></a>Elementos de Controle<a aria-hidden="true" tabindex="-1" class="hash-link" href="#elementos-de-controle" title="Direct link to heading">#</a></h2><p>O acesso aos elementos de controle é realizado em uma janela (removível e encaixável) por meio de ícones relacionados.</p><p>Uma vez que o usuário clicar no ícone desejado, o elemento de controle acompanhará o ponteiro do mouse até ser efetivamente inserido na posição desejada ao clicar novamente na área de trabalho. Os elementos são então conectados por “linhas de conexão” inseridas ao clicar nos nós dos componentes previamente adicionados, permitindo a construção da rede de controle genérica.</p><p>As ferramentas de manipulação e navegação, como arrastar, mover e excluir são <a href="/PSP/docs/cadTools">herdadas</a> do <a href="/PSP/docs/powerEditor">editor de elementos de potência</a>, possuindo comportamento idêntico. A <strong>edição dos dados</strong> dos componentes inseridos também é realizada com <strong>duplo clique</strong> sobre o elemento inserido, exibindo um formulário de edição de dados.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/timelapseControlEditor.mp4" type="video/mp4"><source src="/PSP/videos/timelapseControlEditor.webm" type="video/webm"></video><p>Os seguintes blocos de controle estão presentes no PSP-UFU:</p><ul><li><a href="/PSP/docs/io">Entrada e Saída</a></li><li><a href="/PSP/docs/transferFunction">Função Transferência</a></li><li>Matemática<ul><li><a href="/PSP/docs/sum">Somador</a></li><li><a href="/PSP/docs/multiplier">Multiplicador</a></li><li><a href="/PSP/docs/divider">Divisor</a></li><li><a href="/PSP/docs/exponential">Exponencial</a></li></ul></li><li>Limitadores<ul><li><a href="/PSP/docs/limiter">Limitador absoluto</a></li><li><a href="/PSP/docs/rateLimiter">Limitador de taxa</a></li></ul></li><li>Constantes<ul><li><a href="/PSP/docs/const">Constante</a></li><li><a href="/PSP/docs/gain">Ganho</a></li></ul></li><li><a href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="inicialização-do-sistema-de-controle"></a>Inicialização do sistema de controle<a aria-hidden="true" tabindex="-1" class="hash-link" href="#inicialização-do-sistema-de-controle" title="Direct link to heading">#</a></h2><p>Caso as entradas do sistema de controle apresentem valores diferentes de zero ou a primeira operação entre elas não resulte em uma saída nula, o sistema necessita de inicialização, de forma a adequar os valores de entradas e saída dos blocos elementares e dos vetores de estado das <a href="/PSP/docs/transferFunction">funções transferências</a> presentes. Tal procedimento é realizado por meio da solução de toda rede de controle até que se obtenha sua convergência, ou seja, a diferença absoluta entre as mesmas saídas de uma solução anterior e uma atual deve ser nula ou muito próxima de zero.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>info</h5></div><div class="admonition-content"><p>O processo de <strong>inicialização é realizada automaticamente</strong> pelo PSP-UFU.</p><p>Uma vez que a inicialização é imposta pelas <a href="/PSP/docs/io">entradas</a> do controle, erros de convergência podem ocorrer devido à má parametrização dos elementos dos sistemas de <strong>potência e controle</strong>.</p></div></div><p>Para otimizar e melhorar a estabilidade do processo de inicialização utilizou-se um passo de integração variável dentro de limites, de forma que o passo aumenta em condições de diferenças menores entre as soluções do sistema de controle e diminui caso essa diferença se torne elevada. A implementação dessa abordagem reduziu significativamente o encerramento do processo com erro causado pela instabilidade numérica, além de acelerar a inicialização.</p><p>O fluxograma abaixo evidencia o processo de inicialização implementado no PSP-UFU:</p><div><center><img src="/PSP/images/controlInit.png" alt="Estrutura da inicialização do sistema de controle" title="Estrutura da inicialização do sistema de controle"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="teste-do-diagrama-de-controle"></a>Teste do diagrama de controle<a aria-hidden="true" tabindex="-1" class="hash-link" href="#teste-do-diagrama-de-controle" title="Direct link to heading">#</a></h2><p>Na parte inferior esquerda do Editor de Controle está presente o botão &quot;<strong>Testar Sistema...</strong>&quot;. O botão acessa o formulário de teste de controles, como é indicado na figura abaixo:</p><div><center><img src="/PSP/images/testControl.png" alt="Formulário de teste de controles" title="Formulário de teste de controles"></center></div><p>Nesse formulário é possível inserir o comportamento de <strong>todas as entradas do diagrama</strong>:</p><ul><li><strong>Tipo de entrada</strong>: Define um dos tipos de entradas possíveis no PSP-UFU: &quot;Passo (<em>Step</em>)&quot;, Rampa (<em>Ramp</em>) ou Quadrática (<em>Quadratic</em>);</li><li><strong>Tempo de início (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">t_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>)</strong>: Define o tempo de início na simulação da entrada definida. As entradas iniciais são sempre definidas como zero e assumem valores diferentes após o tempo de início;</li><li><strong>Inclinação (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span>)</strong>: Valor relacionado ao tipo de entrada:<ul><li>Para entrada do tipo &quot;Passo&quot; - O valor da inclinação define o valor final do passo:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>&lt;</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t &lt; t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li><li>Para entrada do tipo &quot;Rampa&quot; - define o coeficiente de inclinação da reta:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>&lt;</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>×</mo><mi>t</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t &lt; t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li><li>Para entrada do tipo &quot;Quadrática&quot; - define o coeficiente de crescimento da curva:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>&lt;</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>×</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t &lt; t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t^2 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li></ul></li><li><strong>Passo de integração (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>)</strong>: Passo do método de integração Trapezoidal Implícito;</li><li><strong>Tempo de simulação (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">t_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>)</strong>: Tempo total de simulação.</li></ul><p>Após a simulação os resultados são exibidos no <a href="/PSP/docs/graphViewer">visualizador de gráficos</a>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A mesma parametrização de entrada é aplicada em todos os blocos de <a href="/PSP/docs/io">entrada</a> do sistema.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>É possível aplicar <strong>várias entradas distintas</strong> (inclusive com condicionais!) utilizando o bloco de <a href="/PSP/docs/mathExpression">expressão matemática</a>.</p></div></div><p>Utilizando o bloco de <a href="/PSP/docs/mathExpression">expressão matemática</a> é possível inserir entradas complexas, como por exemplo testar o diagrama de blocos (AVR DC1C + PSS1A - <a href="https://doi.org/10.1109/IEEESTD.2016.7553421" target="_blank" rel="noopener noreferrer">IEEE Std. 421.5-2016</a>) da figura abaixo:</p><div><center><img src="/PSP/images/exControl.svg" alt="Sistema de controle sob teste" title="Sistema de controle sob teste"></center></div><div><center><img src="/PSP/images/compControl.svg" alt="Comparação do sistema de controle sob teste" title="Comparação do sistema de controle sob teste"></center></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/controlEditor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/indMotor"><div 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Editor de Controle</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Os controles das <a href="/PSP/docs/syncGenerator">máquinas síncronas</a> são criados, modificados, importados e exportados em um editor distinto dos elementos de potência, apresentado na figura abaixo.</p><div><center><img src="/PSP/images/controlEditor.png" alt="Editor de elementos de controle do PSP-UFU" title="Editor de elementos de controle do PSP-UFU"></center></div><p>O acesso ao editor é realizado por meio de dois botões presentes no <a href="/PSP/docs/syncGenerator#formul%C3%A1rio-de-edi%C3%A7%C3%A3o-dos-geradores-s%C3%ADncronos">formulário de edição dos dados de estabilidade dos geradores síncronos</a>:</p><ul><li><a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Editar AVR</a></li><li><a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Editar regulador de velocidade</a></li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU a opção de editar o <strong>AVR</strong> engloba mais que somente o controle de tensão da máquina. Nele <strong>deve</strong> ser inserida a malha de controle da máquina assim como a <strong>excitatriz da máquina síncrona</strong>. Outras estratégias de controle (opcionais), como PSS (<em>Power System Stabilizer</em>) e/ou controles de sobre e sub excitação, são também implementadas em conjunto (como apresentado na figura anterior).</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Assim como no AVR, o <strong>Regulador de Velocidade</strong> engloba mais que a regulação primária da máquina. Nessa opção <strong>deve</strong> ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o <strong>modelo da turbina</strong>. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="elementos-de-controle"></a>Elementos de Controle<a class="hash-link" href="#elementos-de-controle" title="Direct link to heading">#</a></h2><p>O acesso aos elementos de controle é realizado em uma janela (removível e encaixável) por meio de ícones relacionados.</p><p>Uma vez que o usuário clicar no ícone desejado, o elemento de controle acompanhará o ponteiro do mouse até ser efetivamente inserido na posição desejada ao clicar novamente na área de trabalho. Os elementos são então conectados por “linhas de conexão” inseridas ao clicar nos nós dos componentes previamente adicionados, permitindo a construção da rede de controle genérica.</p><p>As ferramentas de manipulação e navegação, como arrastar, mover e excluir são <a href="/PSP/docs/cadTools">herdadas</a> do <a href="/PSP/docs/powerEditor">editor de elementos de potência</a>, possuindo comportamento idêntico. A <strong>edição dos dados</strong> dos componentes inseridos também é realizada com <strong>duplo clique</strong> sobre o elemento inserido, exibindo um formulário de edição de dados.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/timelapseControlEditor.mp4" type="video/mp4"><source src="/PSP/videos/timelapseControlEditor.webm" type="video/webm"></video><p>Os seguintes blocos de controle estão presentes no PSP-UFU:</p><ul><li><a href="/PSP/docs/io">Entrada e Saída</a></li><li><a href="/PSP/docs/transferFunction">Função Transferência</a></li><li>Matemática<ul><li><a href="/PSP/docs/sum">Somador</a></li><li><a href="/PSP/docs/multiplier">Multiplicador</a></li><li><a href="/PSP/docs/divider">Divisor</a></li><li><a href="/PSP/docs/exponential">Exponencial</a></li></ul></li><li>Limitadores<ul><li><a href="/PSP/docs/limiter">Limitador absoluto</a></li><li><a href="/PSP/docs/rateLimiter">Limitador de taxa</a></li></ul></li><li>Constantes<ul><li><a href="/PSP/docs/const">Constante</a></li><li><a href="/PSP/docs/gain">Ganho</a></li></ul></li><li><a href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inicialização-do-sistema-de-controle"></a>Inicialização do sistema de controle<a class="hash-link" href="#inicialização-do-sistema-de-controle" title="Direct link to heading">#</a></h2><p>Caso as entradas do sistema de controle apresentem valores diferentes de zero ou a primeira operação entre elas não resulte em uma saída nula, o sistema necessita de inicialização, de forma a adequar os valores de entradas e saída dos blocos elementares e dos vetores de estado das <a href="/PSP/docs/transferFunction">funções transferências</a> presentes. Tal procedimento é realizado por meio da solução de toda rede de controle até que se obtenha sua convergência, ou seja, a diferença absoluta entre as mesmas saídas de uma solução anterior e uma atual deve ser nula ou muito próxima de zero.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>info</h5></div><div class="admonition-content"><p>O processo de <strong>inicialização é realizada automaticamente</strong> pelo PSP-UFU.</p><p>Uma vez que a inicialização é imposta pelas <a href="/PSP/docs/io">entradas</a> do controle, erros de convergência podem ocorrer devido à má parametrização dos elementos dos sistemas de <strong>potência e controle</strong>.</p></div></div><p>Para otimizar e melhorar a estabilidade do processo de inicialização utilizou-se um passo de integração variável dentro de limites, de forma que o passo aumenta em condições de diferenças menores entre as soluções do sistema de controle e diminui caso essa diferença se torne elevada. A implementação dessa abordagem reduziu significativamente o encerramento do processo com erro causado pela instabilidade numérica, além de acelerar a inicialização.</p><p>O fluxograma abaixo evidencia o processo de inicialização implementado no PSP-UFU:</p><div><center><img src="/PSP/images/controlInit.png" alt="Estrutura da inicialização do sistema de controle" title="Estrutura da inicialização do sistema de controle"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="teste-do-diagrama-de-controle"></a>Teste do diagrama de controle<a class="hash-link" href="#teste-do-diagrama-de-controle" title="Direct link to heading">#</a></h2><p>Na parte inferior esquerda do Editor de Controle está presente o botão &quot;<strong>Testar Sistema...</strong>&quot;. O botão acessa o formulário de teste de controles, como é indicado na figura abaixo:</p><div><center><img src="/PSP/images/testControl.png" alt="Formulário de teste de controles" title="Formulário de teste de controles"></center></div><p>Nesse formulário é possível inserir o comportamento de <strong>todas as entradas do diagrama</strong>:</p><ul><li><strong>Tipo de entrada</strong>: Define um dos tipos de entradas possíveis no PSP-UFU: &quot;Passo (<em>Step</em>)&quot;, Rampa (<em>Ramp</em>) ou Quadrática (<em>Quadratic</em>);</li><li><strong>Tempo de início (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">t_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>)</strong>: Define o tempo de início na simulação da entrada definida. As entradas iniciais são sempre definidas como zero e assumem valores diferentes após o tempo de início;</li><li><strong>Inclinação (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span>)</strong>: Valor relacionado ao tipo de entrada:<ul><li>Para entrada do tipo &quot;Passo&quot; - O valor da inclinação define o valor final do passo:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>&lt;</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t &lt; t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li><li>Para entrada do tipo &quot;Rampa&quot; - define o coeficiente de inclinação da reta:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>&lt;</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>×</mo><mi>t</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t &lt; t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li><li>Para entrada do tipo &quot;Quadrática&quot; - define o coeficiente de crescimento da curva:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>&lt;</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>×</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t &lt; t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t^2 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li></ul></li><li><strong>Passo de integração (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>)</strong>: Passo do método de integração Trapezoidal Implícito;</li><li><strong>Tempo de simulação (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">t_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>)</strong>: Tempo total de simulação.</li></ul><p>Após a simulação os resultados são exibidos no <a href="/PSP/docs/graphViewer">visualizador de gráficos</a>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A mesma parametrização de entrada é aplicada em todos os blocos de <a href="/PSP/docs/io">entrada</a> do sistema.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>É possível aplicar <strong>várias entradas distintas</strong> (inclusive com condicionais!) utilizando o bloco de <a href="/PSP/docs/mathExpression">expressão matemática</a>.</p></div></div><p>Utilizando o bloco de <a href="/PSP/docs/mathExpression">expressão matemática</a> é possível inserir entradas complexas, como por exemplo testar o diagrama de blocos (AVR DC1C + PSS1A - <a href="https://doi.org/10.1109/IEEESTD.2016.7553421" target="_blank" rel="noopener noreferrer">IEEE Std. 421.5-2016</a>) da figura abaixo:</p><div><center><img src="/PSP/images/exControl.svg" alt="Sistema de controle sob teste" title="Sistema de controle sob teste"></center></div><div><center><img src="/PSP/images/compControl.svg" alt="Comparação do sistema de controle sob teste" title="Comparação do sistema de controle sob teste"></center></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/controlEditor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/indMotor"><div 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class="docMainContainer_3EyW"><div class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Divisor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Esse elemento é bastante simples e cumpre com a função matemáticas de seu nome.</p><p>Possui duas entradas de valores reais e sua saída é computada na sequência do nó superior pelo inferior, ou seja, a saída será o valor presente no nó 1 dividido pelo nó 2.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Nós não conectados no divisor são considerados como valor nulo. Caso a divisão possua numerador zero, o valor da saída é definida como sendo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">10^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span></span>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/divider.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/multiplier"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Multiplicador</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/exponential"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Exponencial »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Divisor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Esse elemento é bastante simples e cumpre com a função matemáticas de seu nome.</p><p>Possui duas entradas de valores reais e sua saída é computada na sequência do nó superior pelo inferior, ou seja, a saída será o valor presente no nó 1 dividido pelo nó 2.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Nós não conectados no divisor são considerados como valor nulo. Caso a divisão possua numerador zero, o valor da saída é definida como sendo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">10^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span></span>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/divider.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/multiplier"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Multiplicador</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/exponential"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Exponencial »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="docMainContainer_3EyW"><div class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Exponencial</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>A função exponencial dos elementos de controle realiza essa operação com uma entrada real, obedecendo a expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mi mathvariant="bold">A</mi><msup><mi>e</mi><mrow><mi mathvariant="bold">B</mi><msub><mi>u</mi><mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">y_n = \bold{A} e^{\bold{B} u_n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8932769999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">A</span></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8932769999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">B</span></span><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a entrada e a saída do bloco exponencial, respectivamente;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\bold{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">A</span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">B</mi></mrow><annotation encoding="application/x-tex">\bold{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">B</span></span></span></span></span></span>	são constantes definidas pelo usuário.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-de-dados-do-bloco-exponencial"></a>Formulário de edição de dados do bloco Exponencial<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-do-bloco-exponencial" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco exponencial.</p><div><center><img src="/PSP/images/expForm.png" alt="Formulário de edição de dados do bloco exponencial no PSP-UFU" title="Formulário de edição de dados do bloco exponencial no PSP-UFU"></center></div><p>Esse bloco não linear é definido pelas constantes A e B, inseridas pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tais blocos são úteis na representação de não linearidades, como, por exemplo, modelagem da saturação de máquinas de corrente contínuas presentes em alguns reguladores automáticos de tensão.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/exponential.md" target="_blank" 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Exponencial</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>A função exponencial dos elementos de controle realiza essa operação com uma entrada real, obedecendo a expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mi mathvariant="bold">A</mi><msup><mi>e</mi><mrow><mi mathvariant="bold">B</mi><msub><mi>u</mi><mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">y_n = \bold{A} e^{\bold{B} u_n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8932769999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">A</span></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8932769999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">B</span></span><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a entrada e a saída do bloco exponencial, respectivamente;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\bold{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">A</span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">B</mi></mrow><annotation encoding="application/x-tex">\bold{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">B</span></span></span></span></span></span>	são constantes definidas pelo usuário.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-bloco-exponencial"></a>Formulário de edição de dados do bloco Exponencial<a class="hash-link" href="#formulário-de-edição-de-dados-do-bloco-exponencial" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco exponencial.</p><div><center><img src="/PSP/images/expForm.png" alt="Formulário de edição de dados do bloco exponencial no PSP-UFU" title="Formulário de edição de dados do bloco exponencial no PSP-UFU"></center></div><p>Esse bloco não linear é definido pelas constantes A e B, inseridas pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tais blocos são úteis na representação de não linearidades, como, por exemplo, modelagem da saturação de máquinas de corrente contínuas presentes em alguns reguladores automáticos de tensão.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/exponential.md" 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class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Curto-Circuito</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O principal objetivo da análise de curto-circuito é o cálculo das correntes e tensões de falta para especificação de disjuntores, transformadores de corrente e a parametrização de relés de proteção. De 70 a 80% das faltas em linhas de transmissão são entre uma fase e terra, as quais ocorrem devido ao centelhamento de apenas uma fase da linha para a torre e então para a terra.O menor número de faltas, cerca de 5%, envolve todas as três fases, chamadas de faltas trifásicas. Os outros tipos de faltas envolvem duas fases e duas fases e a terra.</p><p>Todas essas falhas, exceto a trifásica, são assimétricas e causam desequilíbrio entre as fases.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="cálculo-de-curto-circuito-no-psp-ufu"></a>Cálculo de Curto-Circuito no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#cálculo-de-curto-circuito-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O primeiro estágio do cálculo de curto-circuito é a determinação das tensões pré-falta, das potências de geração e cargas do sistema. Esses dados são obtidos por meio do estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Atualmente, o PSP-UFU fornece resultado para os seguintes tipos de falta:</p><ul><li>Falta Trifásica (3F-T);</li><li>Falta Fase-Terra (F-T);</li><li>Falta Fase-Fase (F-F);</li><li>Falta Fase-Fase-Terra (F-F-T).</li></ul></div></div><p>Os modelos dos elementos elétricos que constituem um sistema de potência para o estudo de curto-circuito são semelhantes aos do fluxo de carga, apresentando algumas divergências para as faltas desbalanceadas (F-T, F-F e F-F-T).</p><p>As faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados em conjunto cargas desbalanceadas.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="execução-do-cálculo-de-curto-circuito-no-psp-ufu"></a>Execução do cálculo de curto-circuito no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#execução-do-cálculo-de-curto-circuito-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Existem duas formas de se calcular o curto-circuito no PSP-UFU:</p><ul><li><strong>Falta</strong>: Calcula a falta inserida nas <a href="/PSP/docs/bus">barras</a>. Nesse tipo de cálculo é possível calcular faltas <em>shunt</em> nos <a href="/PSP/docs/bus">barramentos</a> balanceadas e desbalanceadas.</li><li><strong>Nível de curto-circuito</strong>: Calcula o nível de curto-circuito (falta trifásica) em todos <a href="/PSP/docs/bus">barramentos</a> do sistema.</li></ul><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do cálculo de curto-circuito é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Falta</strong>. Para calcular o nível de curto-circuito (falta trifásica) em todos <a href="/PSP/docs/bus">barramentos</a> do sistema, basta clicar no botão <strong>Nível de curto-circuito</strong>.</p><div><center><img src="/PSP/images/menuSimulationFaulta.svg" alt="Execução dos cálculos de curto-circuito" title="Execução dos cálculos de curto-circuito"></center></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>É possível calcular as faltas sem a execução do <a href="/PSP/docs/powerFlow">fluxo de carga</a>, porém <strong>não é recomendável</strong>, visto que os valores das correntes de falta são significativamente alteradas.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Os cálculos de curtos-circuitos não são habilitados por padrão no cálculo contínuo e devem ser inseridos nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p></div></div><p>Os resultados do cálculo de curto-circuito são exibidos nos <a href="/PSP/docs/text">elementos de texto vinculado</a>, ao posicionar o mouse sobre os barramentos e em <a href="/PSP/docs/tabularReport">relatórios tabulares</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="erros-comuns-na-execução-do-cálculo-de-curto-circuito"></a>Erros comuns na execução do cálculo de curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#erros-comuns-na-execução-do-cálculo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao calculo de curto-circuito.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="a-seguinte-mensagem-de-erro-é-exibida-falha-ao-inverter-a-matriz-admitância-de-sequência-zero"></a>A seguinte mensagem de erro é exibida: &quot;Falha ao inverter a matriz admitância de sequência zero&quot;<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-falha-ao-inverter-a-matriz-admitância-de-sequência-zero" title="Direct link to heading">#</a></h4><ul><li><strong>Impossibilidade de circulação da corrente de sequência zero</strong>. Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a matriz admitância de sequência zero é singular. Para contornar esse problema escolha uma das duas soluções abaixo:<ul><li>Marque a opção &quot;Neutro aterrado&quot; e insira um alto valor de reatância de aterramento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>9999</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j9999~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo);</li><li>Ou, na barra do gerador, insira um reator de baixo valor de potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">1{,}0~var</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault" style="margin-right:0.03588em">v</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>, por exemplo).</li></ul></li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="o-cálculo-de-curto-circuito"></a>O cálculo de curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#o-cálculo-de-curto-circuito" title="Direct link to heading">#</a></h2><p>Como já foi apresentado anteriormente, as faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados com cargas desbalanceadas.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="método-das-componentes-simétricas"></a>Método das componentes simétricas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#método-das-componentes-simétricas" title="Direct link to heading">#</a></h3><p>Esse método proposto por C. L. Fortescue, permite definir um sistema de n fasores desbalanceados em n – 1 sistemas de n fases balanceados e um sistema de fase zero. O sistema de fase zero é definido por todas as fases de mesmo módulo e ângulo.
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href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Curto-Circuito</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O principal objetivo da análise de curto-circuito é o cálculo das correntes e tensões de falta para especificação de disjuntores, transformadores de corrente e a parametrização de relés de proteção. De 70 a 80% das faltas em linhas de transmissão são entre uma fase e terra, as quais ocorrem devido ao centelhamento de apenas uma fase da linha para a torre e então para a terra.O menor número de faltas, cerca de 5%, envolve todas as três fases, chamadas de faltas trifásicas. Os outros tipos de faltas envolvem duas fases e duas fases e a terra.</p><p>Todas essas falhas, exceto a trifásica, são assimétricas e causam desequilíbrio entre as fases.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="cálculo-de-curto-circuito-no-psp-ufu"></a>Cálculo de Curto-Circuito no PSP-UFU<a class="hash-link" href="#cálculo-de-curto-circuito-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O primeiro estágio do cálculo de curto-circuito é a determinação das tensões pré-falta, das potências de geração e cargas do sistema. Esses dados são obtidos por meio do estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Atualmente, o PSP-UFU fornece resultado para os seguintes tipos de falta:</p><ul><li>Falta Trifásica (3F-T);</li><li>Falta Fase-Terra (F-T);</li><li>Falta Fase-Fase (F-F);</li><li>Falta Fase-Fase-Terra (F-F-T).</li></ul></div></div><p>Os modelos dos elementos elétricos que constituem um sistema de potência para o estudo de curto-circuito são semelhantes aos do fluxo de carga, apresentando algumas divergências para as faltas desbalanceadas (F-T, F-F e F-F-T).</p><p>As faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados em conjunto cargas desbalanceadas.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="execução-do-cálculo-de-curto-circuito-no-psp-ufu"></a>Execução do cálculo de curto-circuito no PSP-UFU<a class="hash-link" href="#execução-do-cálculo-de-curto-circuito-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Existem duas formas de se calcular o curto-circuito no PSP-UFU:</p><ul><li><strong>Falta</strong>: Calcula a falta inserida nas <a href="/PSP/docs/bus">barras</a>. Nesse tipo de cálculo é possível calcular faltas <em>shunt</em> nos <a href="/PSP/docs/bus">barramentos</a> balanceadas e desbalanceadas.</li><li><strong>Nível de curto-circuito</strong>: Calcula o nível de curto-circuito (falta trifásica) em todos <a href="/PSP/docs/bus">barramentos</a> do sistema.</li></ul><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do cálculo de curto-circuito é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Falta</strong>. Para calcular o nível de curto-circuito (falta trifásica) em todos <a href="/PSP/docs/bus">barramentos</a> do sistema, basta clicar no botão <strong>Nível de curto-circuito</strong>.</p><div><center><img src="/PSP/images/menuSimulationFaulta.svg" alt="Execução dos cálculos de curto-circuito" title="Execução dos cálculos de curto-circuito"></center></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>É possível calcular as faltas sem a execução do <a href="/PSP/docs/powerFlow">fluxo de carga</a>, porém <strong>não é recomendável</strong>, visto que os valores das correntes de falta são significativamente alteradas.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Os cálculos de curtos-circuitos não são habilitados por padrão no cálculo contínuo e devem ser inseridos nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p></div></div><p>Os resultados do cálculo de curto-circuito são exibidos nos <a href="/PSP/docs/text">elementos de texto vinculado</a>, ao posicionar o mouse sobre os barramentos e em <a href="/PSP/docs/tabularReport">relatórios tabulares</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="erros-comuns-na-execução-do-cálculo-de-curto-circuito"></a>Erros comuns na execução do cálculo de curto-circuito<a class="hash-link" href="#erros-comuns-na-execução-do-cálculo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao calculo de curto-circuito.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-falha-ao-inverter-a-matriz-admitância-de-sequência-zero"></a>A seguinte mensagem de erro é exibida: &quot;Falha ao inverter a matriz admitância de sequência zero&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-falha-ao-inverter-a-matriz-admitância-de-sequência-zero" title="Direct link to heading">#</a></h4><ul><li><strong>Impossibilidade de circulação da corrente de sequência zero</strong>. Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a matriz admitância de sequência zero é singular. Para contornar esse problema escolha uma das duas soluções abaixo:<ul><li>Marque a opção &quot;Neutro aterrado&quot; e insira um alto valor de reatância de aterramento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>9999</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j9999~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo);</li><li>Ou, na barra do gerador, insira um reator de baixo valor de potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">1{,}0~var</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault" style="margin-right:0.03588em">v</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>, por exemplo).</li></ul></li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="o-cálculo-de-curto-circuito"></a>O cálculo de curto-circuito<a class="hash-link" href="#o-cálculo-de-curto-circuito" title="Direct link to heading">#</a></h2><p>Como já foi apresentado anteriormente, as faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados com cargas desbalanceadas.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="método-das-componentes-simétricas"></a>Método das componentes simétricas<a class="hash-link" href="#método-das-componentes-simétricas" title="Direct link to heading">#</a></h3><p>Esse método proposto por C. L. Fortescue, permite definir um sistema de n fasores desbalanceados em n – 1 sistemas de n fases balanceados e um sistema de fase zero. O sistema de fase zero é definido por todas as fases de mesmo módulo e ângulo.
 Para um sistema trifásico pode-se definir três componentes de sequência:</p><ol><li>Componentes de sequência positiva, constituindo em três fasores iguais em módulo, 120º defasados entre si, e tendo a mesma sequência de fase que os fasores originais;</li><li>Componentes de sequência negativa, constituindo em três fasores iguais em módulo, 120º defasados entre si, e tendo a sequência de fase oposta à dos fasores originais.</li><li>Componentes de sequência zero, constituindo em três fasores iguais em módulo e com defasagem nula entre si.</li></ol><p>Com isso pode-se decompor as tensões de fase em componentes simétricas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>a</mi></msub><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>b</mi></msub><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>b</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>b</mi><mn>2</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>b</mi><mn>0</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>c</mi></msub><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>c</mi><mn>2</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>c</mi><mn>0</mn></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \dot{V}_a = \dot{V}_{a1} + \dot{V}_{a2} + \dot{V}_{a0}\\ \dot{V}_b = \dot{V}_{b1} + \dot{V}_{b2} + \dot{V}_{b0}\\ \dot{V}_c = \dot{V}_{c1} + \dot{V}_{c2} + \dot{V}_{c0} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" 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class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A figura abaixo apresenta um exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados.</p><div><center><img src="/PSP/images/fortescue.svg" alt="Exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados" title="Exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados"></center></div><p>Para simplificar os cálculos adota-se um operador “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63056em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span>”, com o intuito de indicar a rotação de um fasor. Tal operador é um número complexo de módulo unitário e ângulo de 120º:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mn>1</mn><mi mathvariant="normal">∠</mi><mn>12</mn><msup><mn>0</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>=</mo><mn>1</mn><msup><mi>e</mi><mrow><mi>j</mi><mn>2</mn><mi>π</mi><mi mathvariant="normal">/</mi><mn>3</mn></mrow></msup><mo>=</mo><mo>−</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>866</mn></mrow><annotation encoding="application/x-tex">\overline{a} = 1 \angle 120^{\circ} = 1 e^{j2\pi/3} = -0{,}5 + j0{,}866</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63056em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.724115em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">∠</span><span class="mord">1</span><span class="mord">2</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.724115em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mtight">2</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">π</span><span class="mord mtight">/</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">−</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">8</span><span class="mord">6</span><span class="mord">6</span></span></span></span></span></div><p>Com isso pode-se utilizar as equações (de tensão apresentadas  em conjunto com o operador “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63056em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span>” para construir a seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>a</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>b</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>c</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mover><mover><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msup></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo stretchy="true">⏞</mo></mover><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow></mover><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \dot{V}_a\\ \dot{V}_b\\ \dot{V}_c \end{bmatrix} = \overbrace{ \begin{bmatrix} 1 &amp; 1 &amp; 1\\ 1 &amp; \overline{a}^2 &amp; \overline{a}\\ 1 &amp; \overline{a} &amp; \overline{a}^2 \end{bmatrix} }^{\left[ \bold{A} \right]} \begin{bmatrix} \dot{V}_{a0}\\ \dot{V}_{a1}\\ \dot{V}_{a2} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.8405700000000005em;vertical-align:-1.6702850000000007em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.170285em"><span style="top:-4.250095em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.9699049999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.6897149999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6702850000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:5.2692950000000005em;vertical-align:-1.6702850000000007em"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.59901em"><span style="top:-4.6990099999999995em"><span class="pstrut" style="height:4.6990099999999995em"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.69901em"><span style="top:-4.05101em"><span class="pstrut" style="height:4.05101em"></span><span class="mord"><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span></span></span><span class="svg-align" style="top:-6.202019999999999em"><span class="pstrut" style="height:4.05101em"></span><span class="stretchy" style="height:0.548em;min-width:1.6em"><span class="brace-left" style="height:0.548em"><svg width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
 -45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7
  5-6 9-10 13-.7 1-7.3 1-20 1H6z"></path></svg></span><span class="brace-center" style="height:0.548em"><svg width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M200428 334
@@ -28,16 +30,16 @@ c-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14
  12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z"></path></svg></span><span class="brace-right" style="height:0.548em"><svg width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M400000 542l
 -6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
 s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
-c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span style="top:-7.773019999999999em"><span class="pstrut" style="height:4.6990099999999995em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="minner mtight"><span class="mopen mtight delimcenter" style="top:0em"><span class="mtight">[</span></span><span class="mord mtight"><span class="mord mathbf mtight">A</span></span><span class="mclose mtight delimcenter" style="top:0em"><span class="mtight">]</span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.170285em"><span style="top:-4.250095em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.9699049999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.6897149999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6702850000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Considerando a matriz quadrada da equação anterior sendo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left[ \bold{A} \right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathbf">A</span></span><span class="mclose delimcenter" style="top:0em">]</span></span></span></span></span></span>, pode-se encontrar as componentes simétricas pré-multiplicando ambos os lados dessa mesma equação por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\left[ \bold{A} \right]^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.204008em;vertical-align:-0.25em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathbf">A</span></span><span class="mclose delimcenter" style="top:0em">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954008em"><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span>.</p><p>Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas faltas assimétricas (F-T, F-F e F-F-T) é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As informações a respeito das particularidades dos modelos para o estudo de curto-circuito são apresentados individualmente nos <a href="/PSP/docs/powerEditor#editando-dados-el%C3%A9tricos">elementos de potência</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="equações-do-curto-circuito"></a>Equações do curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#equações-do-curto-circuito" title="Direct link to heading">#</a></h3><p>Primeiramente será tratado o equacionamento para faltas balanceadas e então os estudos serão estendidos para as faltas desbalanceadas por meio da utilização do método das componentes simétricas.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="faltas-balanceadas"></a>Faltas balanceadas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#faltas-balanceadas" title="Direct link to heading">#</a></h4><p>Utiliza-se da matriz impedância de barras para o cálculo de curto-circuito, definida pela seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Z</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\dot{V}] = [Z_{bus}][\dot{I}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Z</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[Z_{bus}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span> é a inversa da matriz admitância de barras, chamada de matriz impedância de barras.</li></ul><p>Por meio da expansão da equação anterior é possível calcular a corrente de falta trifásica na barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi></msub><mo>=</mo><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\dot{I}_f = \frac{\dot{E}_i}{\overline{z}_{ii}+\overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.569298em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\dot{I}_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> é a corrente de falta trifásica na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\dot{E}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão pré-falta na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{ii}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a impedância equivalente de Thevenin vista pela barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>, retirada da matriz impedância</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{f}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.916668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> é a impedância de falta</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="faltas-desbalanceadas"></a>Faltas desbalanceadas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#faltas-desbalanceadas" title="Direct link to heading">#</a></h4><p>O desenvolvimento das equações do cálculo de curto-circuito para faltas desbalanceadas é realizado seguindo o seguinte procedimento:</p><ol><li>Definir os diagramas no ponto da falta, mostrando as conexões de todas fases para a falta. Assume-se que apenas impedâncias balanceadas estão presentes em ambos os lados do ponto da falta e o equivalente Thevenin até esse ponto é conhecido;</li><li>Escrever as condições de contorno relacionando as tensões e corrente conhecidas para o tipo de falta estudada;</li><li>Transformar as correntes e tensões do item 2 de a-b-c para o sistema de coordenadas 0-1-2;</li><li>Encontrar a corrente do curto-circuito em estudo baseado no seguinte sistema de equações (para a fase A):<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>a</mi></msub><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>0</mn></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \dot{V}_{a1} = \dot{E}_a - \dot{I}_{a1} \overline{z}_1\\ \dot{V}_{a2} = - \dot{I}_{a2} \overline{z}_2\\ \dot{V}_{a0} = - \dot{I}_{a0} \overline{z}_0 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li></ol><p>A tabela abaixo apresenta as equações para as faltas desbalanceadas após a execução do procedimento apresenteado:</p><table><thead><tr><th><strong>Falta</strong></th><th align="center"><strong>Seq. Positiva (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th><th align="center"><strong>Seq. Negativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th><th align="center"><strong>Seq. Zero (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th></tr></thead><tbody><tr><td><strong>F-T</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6078659999999996em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td></tr><tr><td><strong>F-F</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6078659999999996em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">- \dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span></td></tr><tr><td><strong>F-F-T</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo fence="true">)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i \left( \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f} \right)}{\overline{z}_{ii}^{1} \overline{z}_{ii}^{2} + 3 \overline{z}_{ii}^{2} \overline{z}_{f} + \overline{z}_{ii}^{2} \overline{z}_{ii}^{0} + 3 \overline{z}_{ii}^{1} \overline{z}_{f} + \overline{z}_{ii}^{1} \overline{z}_{ii}^{0} }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.670876em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6602000000000001em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7400100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mo>−</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">- \dfrac{\dot{E}_i - \overline{z}_{ii}^{1} \dot{I}_{f}^{1}}{\overline{z}_{ii}^{2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.700974em;vertical-align:-0.971568em"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.729406em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" 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style="border-bottom-width:0.04em"></span></span><span style="top:-3.809216em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.971568em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mo>−</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">- \dfrac{\dot{E}_i - \overline{z}_{ii}^{1} \dot{I}_{f}^{1}}{\overline{z}_{ii}^{0} + 3 \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.740082em;vertical-align:-1.010676em"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.729406em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.809216em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td></tr></tbody></table><p>Para obter os valores em a-b-c é usada a equação matricial apresentada anteriormente, encerrando o cálculo de curto-circuito.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.</li><li>ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: <a href="https://doi.org/10.1109/9780470545577" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/9780470545577</a></li><li>FORTESCUE, C. L. Method of Symmetrical Coordinates Applied to the Solution of Polyphase Networks. Trans. AIEE, v. 37, p.1027-1140, 1918. doi: <a href="https://doi.org/10.1109/T-AIEE.1918.4765570" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/T-AIEE.1918.4765570</a></li><li>ANDERSON, P. M. Analysis of faulted power systems. New York: IEEE Press, 1995.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/fault.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/powerFlow"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Fluxo de Potência</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/harmonics"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Harmônicos »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#cálculo-de-curto-circuito-no-psp-ufu" class="table-of-contents__link">Cálculo de Curto-Circuito no PSP-UFU</a></li><li><a href="#execução-do-cálculo-de-curto-circuito-no-psp-ufu" class="table-of-contents__link">Execução do cálculo de curto-circuito no PSP-UFU</a><ul><li><a href="#erros-comuns-na-execução-do-cálculo-de-curto-circuito" class="table-of-contents__link">Erros comuns na execução do cálculo de curto-circuito</a></li></ul></li><li><a href="#o-cálculo-de-curto-circuito" class="table-of-contents__link">O cálculo de curto-circuito</a><ul><li><a href="#método-das-componentes-simétricas" class="table-of-contents__link">Método das componentes simétricas</a></li><li><a href="#equações-do-curto-circuito" class="table-of-contents__link">Equações do curto-circuito</a></li></ul></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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+c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span style="top:-7.773019999999999em"><span class="pstrut" style="height:4.6990099999999995em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="minner mtight"><span class="mopen mtight delimcenter" style="top:0em"><span class="mtight">[</span></span><span class="mord mtight"><span class="mord mathbf mtight">A</span></span><span class="mclose mtight delimcenter" style="top:0em"><span class="mtight">]</span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.170285em"><span style="top:-4.250095em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.9699049999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.6897149999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6702850000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Considerando a matriz quadrada da equação anterior sendo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left[ \bold{A} \right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathbf">A</span></span><span class="mclose delimcenter" style="top:0em">]</span></span></span></span></span></span>, pode-se encontrar as componentes simétricas pré-multiplicando ambos os lados dessa mesma equação por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\left[ \bold{A} \right]^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.204008em;vertical-align:-0.25em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathbf">A</span></span><span class="mclose delimcenter" style="top:0em">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954008em"><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span>.</p><p>Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas faltas assimétricas (F-T, F-F e F-F-T) é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As informações a respeito das particularidades dos modelos para o estudo de curto-circuito são apresentados individualmente nos <a href="/PSP/docs/powerEditor#editando-dados-el%C3%A9tricos">elementos de potência</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="equações-do-curto-circuito"></a>Equações do curto-circuito<a class="hash-link" href="#equações-do-curto-circuito" title="Direct link to heading">#</a></h3><p>Primeiramente será tratado o equacionamento para faltas balanceadas e então os estudos serão estendidos para as faltas desbalanceadas por meio da utilização do método das componentes simétricas.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="faltas-balanceadas"></a>Faltas balanceadas<a class="hash-link" href="#faltas-balanceadas" title="Direct link to heading">#</a></h4><p>Utiliza-se da matriz impedância de barras para o cálculo de curto-circuito, definida pela seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Z</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\dot{V}] = [Z_{bus}][\dot{I}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Z</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[Z_{bus}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span> é a inversa da matriz admitância de barras, chamada de matriz impedância de barras.</li></ul><p>Por meio da expansão da equação anterior é possível calcular a corrente de falta trifásica na barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi></msub><mo>=</mo><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\dot{I}_f = \frac{\dot{E}_i}{\overline{z}_{ii}+\overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.569298em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\dot{I}_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> é a corrente de falta trifásica na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\dot{E}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão pré-falta na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{ii}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a impedância equivalente de Thevenin vista pela barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>, retirada da matriz impedância</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{f}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.916668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> é a impedância de falta</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="faltas-desbalanceadas"></a>Faltas desbalanceadas<a class="hash-link" href="#faltas-desbalanceadas" title="Direct link to heading">#</a></h4><p>O desenvolvimento das equações do cálculo de curto-circuito para faltas desbalanceadas é realizado seguindo o seguinte procedimento:</p><ol><li>Definir os diagramas no ponto da falta, mostrando as conexões de todas fases para a falta. Assume-se que apenas impedâncias balanceadas estão presentes em ambos os lados do ponto da falta e o equivalente Thevenin até esse ponto é conhecido;</li><li>Escrever as condições de contorno relacionando as tensões e corrente conhecidas para o tipo de falta estudada;</li><li>Transformar as correntes e tensões do item 2 de a-b-c para o sistema de coordenadas 0-1-2;</li><li>Encontrar a corrente do curto-circuito em estudo baseado no seguinte sistema de equações (para a fase A):<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>a</mi></msub><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>0</mn></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \dot{V}_{a1} = \dot{E}_a - \dot{I}_{a1} \overline{z}_1\\ \dot{V}_{a2} = - \dot{I}_{a2} \overline{z}_2\\ \dot{V}_{a0} = - \dot{I}_{a0} \overline{z}_0 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li></ol><p>A tabela abaixo apresenta as equações para as faltas desbalanceadas após a execução do procedimento apresenteado:</p><table><thead><tr><th><strong>Falta</strong></th><th align="center"><strong>Seq. Positiva (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th><th align="center"><strong>Seq. Negativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th><th align="center"><strong>Seq. Zero (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th></tr></thead><tbody><tr><td><strong>F-T</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6078659999999996em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td></tr><tr><td><strong>F-F</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6078659999999996em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">- \dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span></td></tr><tr><td><strong>F-F-T</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo fence="true">)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i \left( \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f} \right)}{\overline{z}_{ii}^{1} \overline{z}_{ii}^{2} + 3 \overline{z}_{ii}^{2} \overline{z}_{f} + \overline{z}_{ii}^{2} \overline{z}_{ii}^{0} + 3 \overline{z}_{ii}^{1} \overline{z}_{f} + \overline{z}_{ii}^{1} \overline{z}_{ii}^{0} }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.670876em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6602000000000001em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7400100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mo>−</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">- \dfrac{\dot{E}_i - \overline{z}_{ii}^{1} \dot{I}_{f}^{1}}{\overline{z}_{ii}^{2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.700974em;vertical-align:-0.971568em"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.729406em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.809216em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.971568em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mo>−</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">- \dfrac{\dot{E}_i - \overline{z}_{ii}^{1} \dot{I}_{f}^{1}}{\overline{z}_{ii}^{0} + 3 \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.740082em;vertical-align:-1.010676em"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.729406em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.809216em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td></tr></tbody></table><p>Para obter os valores em a-b-c é usada a equação matricial apresentada anteriormente, encerrando o cálculo de curto-circuito.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.</li><li>ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: <a href="https://doi.org/10.1109/9780470545577" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/9780470545577</a></li><li>FORTESCUE, C. L. Method of Symmetrical Coordinates Applied to the Solution of Polyphase Networks. Trans. AIEE, v. 37, p.1027-1140, 1918. doi: <a href="https://doi.org/10.1109/T-AIEE.1918.4765570" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/T-AIEE.1918.4765570</a></li><li>ANDERSON, P. M. Analysis of faulted power systems. New York: IEEE Press, 1995.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/fault.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/powerFlow"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Fluxo de Potência</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/harmonics"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Harmônicos »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#cálculo-de-curto-circuito-no-psp-ufu" class="table-of-contents__link">Cálculo de Curto-Circuito no PSP-UFU</a></li><li><a href="#execução-do-cálculo-de-curto-circuito-no-psp-ufu" class="table-of-contents__link">Execução do cálculo de curto-circuito no PSP-UFU</a><ul><li><a href="#erros-comuns-na-execução-do-cálculo-de-curto-circuito" class="table-of-contents__link">Erros comuns na execução do cálculo de curto-circuito</a></li></ul></li><li><a href="#o-cálculo-de-curto-circuito" class="table-of-contents__link">O cálculo de curto-circuito</a><ul><li><a href="#método-das-componentes-simétricas" class="table-of-contents__link">Método das componentes simétricas</a></li><li><a href="#equações-do-curto-circuito" class="table-of-contents__link">Equações do curto-circuito</a></li></ul></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Ganho</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O bloco de ganho multiplica o valor de entrada por uma constante definida pelo usuário, possuindo comportamento idêntico à combinação de uma <a href="/PSP/docs/const">constante</a> e um bloco <a href="/PSP/docs/multiplier">multiplicador</a>.</p><p>Esse valor de ganho permanece fixo durante todo o tempo de simulação e é identificado no elemento gráfico exibido no editor de elementos de controle.</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/gain.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" 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docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Visualizador de Gráficos</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Para exibição dos resultados originados dos cálculos de <a href="/PSP/docs/stability">estabilidade</a> e <a href="/PSP/docs/harmonics">varredura de frequências</a>, desenvolveu-se um visualizador de gráficos, como mostra a figura abaixo:</p><div><center><img src="/PSP/images/graphViewer.png" alt="Visualizador de gráficos do PSP-UFU" title="Visualizador de gráficos do PSP-UFU"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="criação-manipulação-e-navegação-de-curvas"></a>Criação, manipulação e navegação de curvas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#criação-manipulação-e-navegação-de-curvas" title="Direct link to heading">#</a></h2><p>Tal visualizador utiliza a biblioteca <a href="https://wxmathplot.sourceforge.io/" target="_blank" rel="noopener noreferrer">wxMathPlot</a>, e permite ao usuário imprimir todas as curvas no tempo originadas dos resultados de <a href="/PSP/docs/stability">estabilidade</a> e <a href="/PSP/docs/harmonics">varredura de frequências</a>, acessados em dados hierárquicos em forma de árvore, identificados pelo tipo e nome do elemento de potência, além do tipo de curva.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As curvas podem ser plotadas com <strong>clique duplo</strong> sobre ela ou ao selecionar a caixa &quot;Plotar&quot; nas propriedades do traço.</p></div></div><p>Os gráficos impressos podem ser caracterizados pela edição de suas propriedades do traço, ou seja, pode-se alterar a cor, espessura e tipo de traço (seis tipos disponíveis). Por padrão o eixo vertical é o tempo (em estabilidade) ou frequência (varredura de frequências), porém o usuário pode alterar para qualquer curva disponível.</p><p>Várias propriedades gerais do gráfico podem ser editadas: título do gráfico, rótulos dos eixos x e y, margens e limites inferiores e superiores de ambos os eixos.</p><p>O menu <strong>Visualização</strong> permite a inserção de exibição de legenda, linhas de grade e coordenadas do ponteiro do mouse (utilizada para destacar pontos na curva), além da possibilidade de alternar o tema de impressão entre escuro e claro, facilitando a visualização ou exportação do gráfico.</p><p>No menu <strong>Arquivo</strong>, pode-se salvar o gráfico na forma de imagem (<em>.png</em>, <em>.jpg</em> ou <em>.bmp</em>) ou enviá-lo diretamente para a área de transferência. No mesmo menu também é permitido a exportação no formato <em><a href="https://en.wikipedia.org/wiki/Comma-separated_values" target="_blank" rel="noopener noreferrer">CSV (Comma-separated Values)</a></em>, o qual poderá ser utilizado em outros visualizadores de gráficos, como o Excel ou o <a href="http://www.gnuplot.info/" target="_blank" rel="noopener noreferrer">gnuplot</a>.</p><p>A área de plotagem permite a navegação utilizando ferramentas de zoom, por área de seleção e comandos de mouse, e arrasto das curvas impressas.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Ao clicar com o botão direito sobre a área de plotagem as opções de manipulação são exibidas por meio de um menu de contexto.</p></div></div><p>Os seguintes <strong>comandos de mouse</strong> são utilizados para navegação das curvas:</p><table><thead><tr><th>Comando</th><th>Ação</th></tr></thead><tbody><tr><td>Botão esquerdo pressionado + Selecionar área</td><td>Seleciona a área de zoom</td></tr><tr><td>Botão direito pressionado + Mover mouse</td><td>Arrasta a curva</td></tr><tr><td>Roda do mouse</td><td>Arrasto vertical</td></tr><tr><td>Roda do mouse + Shift</td><td>Arrasto horizontal</td></tr><tr><td>Roda do mouse + Ctrl</td><td>Mais/menos zoom</td></tr></tbody></table></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/graphViewer.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/tabularReport"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Relatórios Tabulares</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/heatmap"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Mapa de Tensão »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#criação-manipulação-e-navegação-de-curvas" class="table-of-contents__link">Criação, manipulação e navegação de curvas</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="menu__link" tabindex="0" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Visualizador de Gráficos</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Para exibição dos resultados originados dos cálculos de <a href="/PSP/docs/stability">estabilidade</a> e <a href="/PSP/docs/harmonics">varredura de frequências</a>, desenvolveu-se um visualizador de gráficos, como mostra a figura abaixo:</p><div><center><img src="/PSP/images/graphViewer.png" alt="Visualizador de gráficos do PSP-UFU" title="Visualizador de gráficos do PSP-UFU"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="criação-manipulação-e-navegação-de-curvas"></a>Criação, manipulação e navegação de curvas<a class="hash-link" href="#criação-manipulação-e-navegação-de-curvas" title="Direct link to heading">#</a></h2><p>Tal visualizador utiliza a biblioteca <a href="https://wxmathplot.sourceforge.io/" target="_blank" rel="noopener noreferrer">wxMathPlot</a>, e permite ao usuário imprimir todas as curvas no tempo originadas dos resultados de <a href="/PSP/docs/stability">estabilidade</a> e <a href="/PSP/docs/harmonics">varredura de frequências</a>, acessados em dados hierárquicos em forma de árvore, identificados pelo tipo e nome do elemento de potência, além do tipo de curva.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As curvas podem ser plotadas com <strong>clique duplo</strong> sobre ela ou ao selecionar a caixa &quot;Plotar&quot; nas propriedades do traço.</p></div></div><p>Os gráficos impressos podem ser caracterizados pela edição de suas propriedades do traço, ou seja, pode-se alterar a cor, espessura e tipo de traço (seis tipos disponíveis). Por padrão o eixo vertical é o tempo (em estabilidade) ou frequência (varredura de frequências), porém o usuário pode alterar para qualquer curva disponível.</p><p>Várias propriedades gerais do gráfico podem ser editadas: título do gráfico, rótulos dos eixos x e y, margens e limites inferiores e superiores de ambos os eixos.</p><p>O menu <strong>Visualização</strong> permite a inserção de exibição de legenda, linhas de grade e coordenadas do ponteiro do mouse (utilizada para destacar pontos na curva), além da possibilidade de alternar o tema de impressão entre escuro e claro, facilitando a visualização ou exportação do gráfico.</p><p>No menu <strong>Arquivo</strong>, pode-se salvar o gráfico na forma de imagem (<em>.png</em>, <em>.jpg</em> ou <em>.bmp</em>) ou enviá-lo diretamente para a área de transferência. No mesmo menu também é permitido a exportação no formato <em><a href="https://en.wikipedia.org/wiki/Comma-separated_values" target="_blank" rel="noopener noreferrer">CSV (Comma-separated Values)</a></em>, o qual poderá ser utilizado em outros visualizadores de gráficos, como o Excel ou o <a href="http://www.gnuplot.info/" target="_blank" rel="noopener noreferrer">gnuplot</a>.</p><p>A área de plotagem permite a navegação utilizando ferramentas de zoom, por área de seleção e comandos de mouse, e arrasto das curvas impressas.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Ao clicar com o botão direito sobre a área de plotagem as opções de manipulação são exibidas por meio de um menu de contexto.</p></div></div><p>Os seguintes <strong>comandos de mouse</strong> são utilizados para navegação das curvas:</p><table><thead><tr><th>Comando</th><th>Ação</th></tr></thead><tbody><tr><td>Botão esquerdo pressionado + Selecionar área</td><td>Seleciona a área de zoom</td></tr><tr><td>Botão direito pressionado + Mover mouse</td><td>Arrasta a curva</td></tr><tr><td>Roda do mouse</td><td>Arrasto vertical</td></tr><tr><td>Roda do mouse + Shift</td><td>Arrasto horizontal</td></tr><tr><td>Roda do mouse + Ctrl</td><td>Mais/menos zoom</td></tr></tbody></table></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/graphViewer.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/tabularReport"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Relatórios Tabulares</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/heatmap"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Mapa de Tensão »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#criação-manipulação-e-navegação-de-curvas" class="table-of-contents__link">Criação, manipulação e navegação de curvas</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Fonte de Corrente Harmônica</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>(<strong>Fonte de perturbação harmônica</strong>) Parte do sistema elétrico de potência ou das instalações a ele conectadas, que causa distorção harmônica da forma de onda da corrente e / ou tensão. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=614-01-16" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="fonte-de-corrente-harmônica-no-psp-ufu"></a>Fonte de corrente harmônica no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#fonte-de-corrente-harmônica-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As fontes de corrente harmônica são responsáveis pela injeção de correntes harmônicas e são utilizadas pela ferramenta de cálculo de <a href="/PSP/docs/harmonics">distorções harmônicas</a>. Uma lista de correntes harmônicas pode ser inserida em um mesmo elemento, conforme é exibido na o formulário de edição de dados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A presença da fonte de corrente harmônica <strong>não é considerada</strong> nos estudos de <a href="/PSP/docs/powerFlow">fluxo de carga</a>, <a href="/PSP/docs/fault">curto-circuito</a> e <a href="/PSP/docs/stability">estabilidade</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-das-fontes-de-corrente-harmônica"></a>Formulário de edição das fontes de corrente harmônica<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-das-fontes-de-corrente-harmônica" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados da fonte de corrente harmônica:</p><div><center><img src="/PSP/images/harmSourceForm.png" alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU"></center></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="ordem"></a>Ordem<a aria-hidden="true" tabindex="-1" class="hash-link" href="#ordem" title="Direct link to heading">#</a></h4><p>Especifica a ordem harmônica a ser inserida na lista de correntes harmônicas do elemento. Caso a ordem harmônica já exista na lista, ela será substituída pelos novos parâmetros.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="corrente"></a>Corrente<a aria-hidden="true" tabindex="-1" class="hash-link" href="#corrente" title="Direct link to heading">#</a></h4><p>Módulo da corrente harmônica.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="unidade"></a>Unidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#unidade" title="Direct link to heading">#</a></h4><p>Unidade do módulo de corrente harmônica, em A ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="ângulo"></a>Ângulo<a aria-hidden="true" tabindex="-1" class="hash-link" href="#ângulo" title="Direct link to heading">#</a></h4><p>Ângulo da corrente harmônica, em graus.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="adicionar--remover"></a>Adicionar / Remover<a aria-hidden="true" tabindex="-1" class="hash-link" href="#adicionar--remover" title="Direct link to heading">#</a></h4><p>Insere ou modifica a nova ordem harmônica ao clicar em “Adicionar”. Retira a ordem harmônica selecionada da lista ao clicar em “Remover”.</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/harmSource.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/inductor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Indutor</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/syncGenerator"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Gerador Síncrono »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#fonte-de-corrente-harmônica-no-psp-ufu" class="table-of-contents__link">Fonte de corrente harmônica no PSP-UFU</a></li><li><a href="#formulário-de-edição-das-fontes-de-corrente-harmônica" class="table-of-contents__link">Formulário de edição das fontes de corrente harmônica</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Fonte de Corrente Harmônica</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>(<strong>Fonte de perturbação harmônica</strong>) Parte do sistema elétrico de potência ou das instalações a ele conectadas, que causa distorção harmônica da forma de onda da corrente e / ou tensão. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=614-01-16" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="fonte-de-corrente-harmônica-no-psp-ufu"></a>Fonte de corrente harmônica no PSP-UFU<a class="hash-link" href="#fonte-de-corrente-harmônica-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As fontes de corrente harmônica são responsáveis pela injeção de correntes harmônicas e são utilizadas pela ferramenta de cálculo de <a href="/PSP/docs/harmonics">distorções harmônicas</a>. Uma lista de correntes harmônicas pode ser inserida em um mesmo elemento, conforme é exibido na o formulário de edição de dados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A presença da fonte de corrente harmônica <strong>não é considerada</strong> nos estudos de <a href="/PSP/docs/powerFlow">fluxo de carga</a>, <a href="/PSP/docs/fault">curto-circuito</a> e <a href="/PSP/docs/stability">estabilidade</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-das-fontes-de-corrente-harmônica"></a>Formulário de edição das fontes de corrente harmônica<a class="hash-link" href="#formulário-de-edição-das-fontes-de-corrente-harmônica" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados da fonte de corrente harmônica:</p><div><center><img src="/PSP/images/harmSourceForm.png" alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU"></center></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="ordem"></a>Ordem<a class="hash-link" href="#ordem" title="Direct link to heading">#</a></h4><p>Especifica a ordem harmônica a ser inserida na lista de correntes harmônicas do elemento. Caso a ordem harmônica já exista na lista, ela será substituída pelos novos parâmetros.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="corrente"></a>Corrente<a class="hash-link" href="#corrente" title="Direct link to heading">#</a></h4><p>Módulo da corrente harmônica.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="unidade"></a>Unidade<a class="hash-link" href="#unidade" title="Direct link to heading">#</a></h4><p>Unidade do módulo de corrente harmônica, em A ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="ângulo"></a>Ângulo<a class="hash-link" href="#ângulo" title="Direct link to heading">#</a></h4><p>Ângulo da corrente harmônica, em graus.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="adicionar--remover"></a>Adicionar / Remover<a class="hash-link" href="#adicionar--remover" title="Direct link to heading">#</a></h4><p>Insere ou modifica a nova ordem harmônica ao clicar em “Adicionar”. Retira a ordem harmônica selecionada da lista ao clicar em “Remover”.</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/harmSource.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/inductor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Indutor</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/syncGenerator"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Gerador Síncrono »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#fonte-de-corrente-harmônica-no-psp-ufu" class="table-of-contents__link">Fonte de corrente harmônica no PSP-UFU</a></li><li><a href="#formulário-de-edição-das-fontes-de-corrente-harmônica" class="table-of-contents__link">Formulário de edição das fontes de corrente harmônica</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Harmônicos</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Duas ferramentas relacionadas a harmônicos foram desenvolvidas no PSP-UFU:</p><ul><li>Distorção Harmônica Total de Tensão (THD, do inglês <em>Total Harmonic Distortion</em>);</li><li>Resposta na Frequência.</li></ul><p>A ferramenta de <strong><a href="https://en.wikipedia.org/wiki/Total_harmonic_distortion" target="_blank" rel="noopener noreferrer">Distorção Harmônica Total</a></strong> calcula as tensões harmônicas causadas por <a href="/PSP/docs/harmSource">fontes de corrente harmônicas</a>, assim como o THD de todos os barramentos do sistema.</p><p>A ferramenta de <strong>Resposta na Frequência</strong> (ou análise de varredura de frequência) envolve a variação da impedância da rede em um espectro de frequências observado a partir de um certo barramento.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A análise de varredura de frequência é amplamente usada no projeto de filtros harmônicos.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/docs/bus">barramento</a> e um <a href="/PSP/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Na versão atual do programa não são consideradas as alterações das resistências do sistema causado pelo efeito pelicular. Versões futuras irão contemplar tal característica.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="execução-do-cálculo-de-harmônicos-no-psp-ufu"></a>Execução do cálculo de harmônicos no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#execução-do-cálculo-de-harmônicos-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do cálculo das distorções harmônicas é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Distorções Harmônicas</strong>. Para acessar a ferramenta de de varredura de frequência, basta clicar no botão <strong>Resposta na Frequência</strong>.</p><div><center><img src="/PSP/images/menuSimulationHamonics.svg" alt="Acesso às ferramentas de estudo harmônico" title="Acesso às ferramentas de estudo harmônico"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="distorções-harmônicas"></a>Distorções Harmônicas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#distorções-harmônicas" title="Direct link to heading">#</a></h3><p>Ao clicar sobre o botão &quot;Distorções Harmônicas&quot; as distorções causadas pelas <a href="/PSP/docs/harmSource">fontes de corrente harmônica</a> são calculadas em todos os barramentos do sistema.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso não forem inseridas <a href="/PSP/docs/harmSource">fontes de corrente harmônica</a> no sistema de potência, a distorção de tensão de todas as barras será <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">0{,}0~\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.94444em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord">%</span></span></span></span></span>.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Os cálculos dos níveis de THD <strong>não são habilitados por padrão</strong> no cálculo contínuo e devem ser inseridos nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p></div></div><p>Os resultados das distorções harmônicas são exibidos nos <a href="/PSP/docs/text">elementos de texto vinculados</a> e ao posicionar o mouse sobre um barramento.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="resposta-na-frequência"></a>Resposta na Frequência<a aria-hidden="true" tabindex="-1" class="hash-link" href="#resposta-na-frequência" title="Direct link to heading">#</a></h3><p>Ao clicar sobre o botão &quot;Resposta na Frequência&quot; será exibido um formulário para inserção dos parâmetros da ferramenta:</p><div><center><img src="/PSP/images/injHarmCurrent.png" alt="Acesso à ferramentas resposta na frequência" title="Acesso à ferramentas resposta na frequência"></center></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="frequência-inicial"></a>Frequência inicial<a aria-hidden="true" tabindex="-1" class="hash-link" href="#frequência-inicial" title="Direct link to heading">#</a></h4><p>Define a frequência inicial da varredura.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="frequência-final"></a>Frequência final<a aria-hidden="true" tabindex="-1" class="hash-link" href="#frequência-final" title="Direct link to heading">#</a></h4><p>Define a frequência final da varredura.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="passo-de-frequência"></a>Passo de frequência<a aria-hidden="true" tabindex="-1" class="hash-link" href="#passo-de-frequência" title="Direct link to heading">#</a></h4><p>Define o passo de incremento da frequência. Passos menores irão gerar gráficos mais precisos, porém podem aumentar muito o tempo de execução.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="corrente-injetada"></a>Corrente injetada<a aria-hidden="true" tabindex="-1" class="hash-link" href="#corrente-injetada" title="Direct link to heading">#</a></h4><p>Define o barramento o qual será injetada a corrente harmônica para análise.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Esse campo pode ser entendido como: &quot;<em>A resposta das impedâncias harmônicas vistas pelos barramentos do sistema caso tenha uma fonte de corrente harmônica na barra selecionada</em>&quot;.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="botão-executar"></a>Botão Executar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#botão-executar" title="Direct link to heading">#</a></h4><p>Ao clicar no botão executar a varredura será executada e os resultados exibidos no <a href="/PSP/docs/graphViewer">vizualizador de gráficos</a>, como mostra a imagem abaixo.</p><div><center><img src="/PSP/images/freqScanPlot.png" alt="Exemplo de resposta na frequência" title="Exemplo de resposta na frequência"></center></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Para visualizar a varredura da impedância harmônica de um barramento, essa opção deve ser habilitada em seu <a href="/PSP/docs/bus#imprimir-imped%C3%A2ncia-harm%C3%B4nica-da-barra">formulário de edição de dados</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="cálculos-das-ferramentas-harmônicas"></a>Cálculos das ferramentas harmônicas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#cálculos-das-ferramentas-harmônicas" title="Direct link to heading">#</a></h2><p>Ambos cálculos são realizados a partir da seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><msup><mo stretchy="false">]</mo><mi>h</mi></msup><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{I}]^h= [Y_{bus}]^h [\dot{V}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{I}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span>	Vetor das correntes harmônicas injetadas nas barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[Y_{bus}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span>	Matriz admitância harmônica de barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{V}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span>	Vetor das tensões harmônicas nas barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	Ordem harmônica</li></ul><p>O vetor das correntes harmônicas injetadas é definido no programa utilizando um elemento de potência, chamado “fonte de corrente harmônica”, em que o usuário pode criar uma lista de correntes injetadas (em A e/ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) em um barramento.</p><p>O programa define automaticamente as ordens harmônicas na simulação analisando todas as listas de fontes de corrente harmônicas previamente aos cálculos. Com isso são calculadas as admitâncias harmônicas necessárias de cada elemento pela multiplicação das reatâncias indutivas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>l</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">x_{l}^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.097216em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>) e susceptâncias capacitivas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>b</mi><mi>c</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">b_{c}^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.061108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span>) fundamentais de cada elemento pelas ordens harmônicas:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>l</mi><mi>h</mi></msubsup><mo>=</mo><mi>h</mi><mo>×</mo><msubsup><mi>x</mi><mi>l</mi><mn>1</mn></msubsup><mspace linebreak="newline"></mspace><msubsup><mi>b</mi><mi>c</mi><mi>h</mi></msubsup><mo>=</mo><mi>h</mi><mo>×</mo><msubsup><mi>b</mi><mi>c</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">x_{l}^h=h×x_l^1\\ b_{c}^h=h×b_c^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1111079999999998em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1111079999999998em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Uma vez calculadas as admitâncias harmônicas, são utilizados os mesmos modelos e algoritmos convencionais para construção da <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow><mi>h</mi></msubsup></mrow><annotation encoding="application/x-tex">Y_{bus}^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.132216em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.4168920000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>A equação matricial é resolvida usando o método de eliminação gaussiana para evitar a inversão de cada matriz de admitância harmônica. Este procedimento torna o cálculo das tensões harmônicas computacionalmente eficiente.</p><p>Uma vez calculadas as tensões harmônicas, a THD em uma barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span> pode ser definida por:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>THD</mtext><mi>i</mi></msub><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>h</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></munderover><msubsup><mi>V</mi><mi>i</mi><mi>h</mi></msubsup></mrow><msubsup><mi>V</mi><mi>i</mi><mn>1</mn></msubsup></mfrac></mrow><annotation encoding="application/x-tex">\text{THD}_i=\dfrac{\sum_{h=2}^{n} V_{i}^h}{V_i^1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord text"><span class="mord">THD</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.5016819999999997em;vertical-align:-0.9628639999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.538818em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="mrel mtight">=</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9628639999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Na ferramenta de Resposta na Frequência uma corrente senoidal é injetada na barra em uma faixa de frequências e o conjunto de equações matriciais é usado para calcular a resposta da tensão. Este cálculo é repetido em etapas discretas cobrindo o espectro de frequência especificado.</p><p>O programa usa a matriz admitância de sequência positiva e uma corrente injetada de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">1{,}0 p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> também de sequência positiva na equação matricial para calcular diretamente as impedâncias em p.u.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>ARRILLAGA, J.; WATSON, N. R. Power System Harmonics. John Wiley &amp; Sons; Chichester, 2003. doi: <a href="https://doi.org/10.1002/0470871229" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/0470871229</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/harmonics.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/fault"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Curto-Circuito</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/stability"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Estabilidade »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#execução-do-cálculo-de-harmônicos-no-psp-ufu" class="table-of-contents__link">Execução do cálculo de harmônicos no PSP-UFU</a><ul><li><a href="#distorções-harmônicas" class="table-of-contents__link">Distorções Harmônicas</a></li><li><a href="#resposta-na-frequência" class="table-of-contents__link">Resposta na Frequência</a></li></ul></li><li><a href="#cálculos-das-ferramentas-harmônicas" class="table-of-contents__link">Cálculos das ferramentas harmônicas</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/docs/transformer">Transformador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/load">Carga</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/capacitor">Capacitor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/inductor">Indutor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmSource">Fonte de Corrente Harmônica</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/syncGenerator">Gerador Síncrono</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/syncMotor">Compensador Síncrono</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/indMotor">Motor de Indução</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" 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href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Simulations</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Harmônicos</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Duas ferramentas relacionadas a harmônicos foram desenvolvidas no PSP-UFU:</p><ul><li>Distorção Harmônica Total de Tensão (THD, do inglês <em>Total Harmonic Distortion</em>);</li><li>Resposta na Frequência.</li></ul><p>A ferramenta de <strong><a href="https://en.wikipedia.org/wiki/Total_harmonic_distortion" target="_blank" rel="noopener noreferrer">Distorção Harmônica Total</a></strong> calcula as tensões harmônicas causadas por <a href="/PSP/docs/harmSource">fontes de corrente harmônicas</a>, assim como o THD de todos os barramentos do sistema.</p><p>A ferramenta de <strong>Resposta na Frequência</strong> (ou análise de varredura de frequência) envolve a variação da impedância da rede em um espectro de frequências observado a partir de um certo barramento.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A análise de varredura de frequência é amplamente usada no projeto de filtros harmônicos.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/docs/bus">barramento</a> e um <a href="/PSP/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Na versão atual do programa não são consideradas as alterações das resistências do sistema causado pelo efeito pelicular. Versões futuras irão contemplar tal característica.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="execução-do-cálculo-de-harmônicos-no-psp-ufu"></a>Execução do cálculo de harmônicos no PSP-UFU<a class="hash-link" href="#execução-do-cálculo-de-harmônicos-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do cálculo das distorções harmônicas é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Distorções Harmônicas</strong>. Para acessar a ferramenta de de varredura de frequência, basta clicar no botão <strong>Resposta na Frequência</strong>.</p><div><center><img src="/PSP/images/menuSimulationHamonics.svg" alt="Acesso às ferramentas de estudo harmônico" title="Acesso às ferramentas de estudo harmônico"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="distorções-harmônicas"></a>Distorções Harmônicas<a class="hash-link" href="#distorções-harmônicas" title="Direct link to heading">#</a></h3><p>Ao clicar sobre o botão &quot;Distorções Harmônicas&quot; as distorções causadas pelas <a href="/PSP/docs/harmSource">fontes de corrente harmônica</a> são calculadas em todos os barramentos do sistema.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso não forem inseridas <a href="/PSP/docs/harmSource">fontes de corrente harmônica</a> no sistema de potência, a distorção de tensão de todas as barras será <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">0{,}0~\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.94444em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord">%</span></span></span></span></span>.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Os cálculos dos níveis de THD <strong>não são habilitados por padrão</strong> no cálculo contínuo e devem ser inseridos nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p></div></div><p>Os resultados das distorções harmônicas são exibidos nos <a href="/PSP/docs/text">elementos de texto vinculados</a> e ao posicionar o mouse sobre um barramento.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resposta-na-frequência"></a>Resposta na Frequência<a class="hash-link" href="#resposta-na-frequência" title="Direct link to heading">#</a></h3><p>Ao clicar sobre o botão &quot;Resposta na Frequência&quot; será exibido um formulário para inserção dos parâmetros da ferramenta:</p><div><center><img src="/PSP/images/injHarmCurrent.png" alt="Acesso à ferramentas resposta na frequência" title="Acesso à ferramentas resposta na frequência"></center></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="frequência-inicial"></a>Frequência inicial<a class="hash-link" href="#frequência-inicial" title="Direct link to heading">#</a></h4><p>Define a frequência inicial da varredura.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="frequência-final"></a>Frequência final<a class="hash-link" href="#frequência-final" title="Direct link to heading">#</a></h4><p>Define a frequência final da varredura.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="passo-de-frequência"></a>Passo de frequência<a class="hash-link" href="#passo-de-frequência" title="Direct link to heading">#</a></h4><p>Define o passo de incremento da frequência. Passos menores irão gerar gráficos mais precisos, porém podem aumentar muito o tempo de execução.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="corrente-injetada"></a>Corrente injetada<a class="hash-link" href="#corrente-injetada" title="Direct link to heading">#</a></h4><p>Define o barramento o qual será injetada a corrente harmônica para análise.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Esse campo pode ser entendido como: &quot;<em>A resposta das impedâncias harmônicas vistas pelos barramentos do sistema caso tenha uma fonte de corrente harmônica na barra selecionada</em>&quot;.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="botão-executar"></a>Botão Executar<a class="hash-link" href="#botão-executar" title="Direct link to heading">#</a></h4><p>Ao clicar no botão executar a varredura será executada e os resultados exibidos no <a href="/PSP/docs/graphViewer">vizualizador de gráficos</a>, como mostra a imagem abaixo.</p><div><center><img src="/PSP/images/freqScanPlot.png" alt="Exemplo de resposta na frequência" title="Exemplo de resposta na frequência"></center></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Para visualizar a varredura da impedância harmônica de um barramento, essa opção deve ser habilitada em seu <a href="/PSP/docs/bus#imprimir-imped%C3%A2ncia-harm%C3%B4nica-da-barra">formulário de edição de dados</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="cálculos-das-ferramentas-harmônicas"></a>Cálculos das ferramentas harmônicas<a class="hash-link" href="#cálculos-das-ferramentas-harmônicas" title="Direct link to heading">#</a></h2><p>Ambos cálculos são realizados a partir da seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><msup><mo stretchy="false">]</mo><mi>h</mi></msup><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{I}]^h= [Y_{bus}]^h [\dot{V}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{I}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span>	Vetor das correntes harmônicas injetadas nas barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[Y_{bus}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span>	Matriz admitância harmônica de barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{V}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span>	Vetor das tensões harmônicas nas barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	Ordem harmônica</li></ul><p>O vetor das correntes harmônicas injetadas é definido no programa utilizando um elemento de potência, chamado “fonte de corrente harmônica”, em que o usuário pode criar uma lista de correntes injetadas (em A e/ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) em um barramento.</p><p>O programa define automaticamente as ordens harmônicas na simulação analisando todas as listas de fontes de corrente harmônicas previamente aos cálculos. Com isso são calculadas as admitâncias harmônicas necessárias de cada elemento pela multiplicação das reatâncias indutivas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>l</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">x_{l}^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.097216em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>) e susceptâncias capacitivas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>b</mi><mi>c</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">b_{c}^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.061108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span>) fundamentais de cada elemento pelas ordens harmônicas:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>l</mi><mi>h</mi></msubsup><mo>=</mo><mi>h</mi><mo>×</mo><msubsup><mi>x</mi><mi>l</mi><mn>1</mn></msubsup><mspace linebreak="newline"></mspace><msubsup><mi>b</mi><mi>c</mi><mi>h</mi></msubsup><mo>=</mo><mi>h</mi><mo>×</mo><msubsup><mi>b</mi><mi>c</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">x_{l}^h=h×x_l^1\\ b_{c}^h=h×b_c^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1111079999999998em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1111079999999998em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Uma vez calculadas as admitâncias harmônicas, são utilizados os mesmos modelos e algoritmos convencionais para construção da <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow><mi>h</mi></msubsup></mrow><annotation encoding="application/x-tex">Y_{bus}^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.132216em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.4168920000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>A equação matricial é resolvida usando o método de eliminação gaussiana para evitar a inversão de cada matriz de admitância harmônica. Este procedimento torna o cálculo das tensões harmônicas computacionalmente eficiente.</p><p>Uma vez calculadas as tensões harmônicas, a THD em uma barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span> pode ser definida por:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>THD</mtext><mi>i</mi></msub><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>h</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></munderover><msubsup><mi>V</mi><mi>i</mi><mi>h</mi></msubsup></mrow><msubsup><mi>V</mi><mi>i</mi><mn>1</mn></msubsup></mfrac></mrow><annotation encoding="application/x-tex">\text{THD}_i=\dfrac{\sum_{h=2}^{n} V_{i}^h}{V_i^1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord text"><span class="mord">THD</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.5016819999999997em;vertical-align:-0.9628639999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.538818em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="mrel mtight">=</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9628639999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Na ferramenta de Resposta na Frequência uma corrente senoidal é injetada na barra em uma faixa de frequências e o conjunto de equações matriciais é usado para calcular a resposta da tensão. Este cálculo é repetido em etapas discretas cobrindo o espectro de frequência especificado.</p><p>O programa usa a matriz admitância de sequência positiva e uma corrente injetada de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">1{,}0 p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> também de sequência positiva na equação matricial para calcular diretamente as impedâncias em p.u.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>ARRILLAGA, J.; WATSON, N. R. Power System Harmonics. John Wiley &amp; Sons; Chichester, 2003. doi: <a href="https://doi.org/10.1002/0470871229" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/0470871229</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/harmonics.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/fault"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Curto-Circuito</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/stability"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Estabilidade »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#execução-do-cálculo-de-harmônicos-no-psp-ufu" class="table-of-contents__link">Execução do cálculo de harmônicos no PSP-UFU</a><ul><li><a href="#distorções-harmônicas" class="table-of-contents__link">Distorções Harmônicas</a></li><li><a href="#resposta-na-frequência" class="table-of-contents__link">Resposta na Frequência</a></li></ul></li><li><a href="#cálculos-das-ferramentas-harmônicas" class="table-of-contents__link">Cálculos das ferramentas harmônicas</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Mapa de Tensão</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>A ferramenta Mapa de Tensão utiliza o concento de <a href="https://en.wikipedia.org/wiki/Heat_map" target="_blank" rel="noopener noreferrer">mapa de calor</a> para exibir os valores de tensão de todas as barras do sistema por meio de cores.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/heatmap.mp4" type="video/mp4"><source src="/PSP/videos/heatmap.webm" type="video/webm"></video><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Essa ferramenta é útil para identificação de barras com sub e sobretensões no sistema de potência, assim como o efeito da inserção / remoção de elementos do sistema.</p></div></div><p>A legenda que correlaciona os valores do módulo da tensão com as cores são exibidas à esquerda da área de trabalho.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="habilitando-a-ferramenta-mapa-de-tensão"></a>Habilitando a ferramenta Mapa de Tensão<a aria-hidden="true" tabindex="-1" class="hash-link" href="#habilitando-a-ferramenta-mapa-de-tensão" title="Direct link to heading">#</a></h2><p>A ferramenta pode ser habilitada ou desabilitada no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ribbon <strong>Ferramentas</strong></a> clicando no botão <strong>Mapa de tensão</strong> ou pela tecla de atalho &quot;Ctrl + Shift + H&quot;.</p><p>As configurações da ferramenta podem ser também acessadas no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ribbon <strong>Ferramentas</strong></a> clicando no botão <strong>Configurações do projeto</strong></p><div><center><img src="/PSP/images/heatmapForm.png" alt="Configurações da ferramenta Mapa de Tensão" title="Configurações da ferramenta Mapa de Tensão"></center></div><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Na versão atual do programa, somente as opções do mapa de tensão são exibidas nas configurações de projeto.</p></div></div><p>Os limites superiores e inferiores, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> são editados nesse formulário.</p><p>Alternativamente, o programa pode calcular automaticamente os limites. Nesse caso, o limite superior será a maior tensão do circuito e o limite inferior será a menor tensão do circuito.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O cálculo automático dos limites pode prejudicar a identificação de barras com sub e sobretensões no sistema.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/heatmap.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/graphViewer"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Visualizador de Gráficos</div></a></div><div class="pagination-nav__item pagination-nav__item--next"></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#habilitando-a-ferramenta-mapa-de-tensão" class="table-of-contents__link">Habilitando a ferramenta Mapa de Tensão</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Mapa de Tensão</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>A ferramenta Mapa de Tensão utiliza o concento de <a href="https://en.wikipedia.org/wiki/Heat_map" target="_blank" rel="noopener noreferrer">mapa de calor</a> para exibir os valores de tensão de todas as barras do sistema por meio de cores.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/heatmap.mp4" type="video/mp4"><source src="/PSP/videos/heatmap.webm" type="video/webm"></video><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Essa ferramenta é útil para identificação de barras com sub e sobretensões no sistema de potência, assim como o efeito da inserção / remoção de elementos do sistema.</p></div></div><p>A legenda que correlaciona os valores do módulo da tensão com as cores são exibidas à esquerda da área de trabalho.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="habilitando-a-ferramenta-mapa-de-tensão"></a>Habilitando a ferramenta Mapa de Tensão<a class="hash-link" href="#habilitando-a-ferramenta-mapa-de-tensão" title="Direct link to heading">#</a></h2><p>A ferramenta pode ser habilitada ou desabilitada no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ribbon <strong>Ferramentas</strong></a> clicando no botão <strong>Mapa de tensão</strong> ou pela tecla de atalho &quot;Ctrl + Shift + H&quot;.</p><p>As configurações da ferramenta podem ser também acessadas no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ribbon <strong>Ferramentas</strong></a> clicando no botão <strong>Configurações do projeto</strong></p><div><center><img src="/PSP/images/heatmapForm.png" alt="Configurações da ferramenta Mapa de Tensão" title="Configurações da ferramenta Mapa de Tensão"></center></div><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Na versão atual do programa, somente as opções do mapa de tensão são exibidas nas configurações de projeto.</p></div></div><p>Os limites superiores e inferiores, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> são editados nesse formulário.</p><p>Alternativamente, o programa pode calcular automaticamente os limites. Nesse caso, o limite superior será a maior tensão do circuito e o limite inferior será a menor tensão do circuito.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O cálculo automático dos limites pode prejudicar a identificação de barras com sub e sobretensões no sistema.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/heatmap.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/graphViewer"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Visualizador de Gráficos</div></a></div><div class="pagination-nav__item pagination-nav__item--next"></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#habilitando-a-ferramenta-mapa-de-tensão" class="table-of-contents__link">Habilitando a ferramenta Mapa de Tensão</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Motor de Indução</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Uma máquina assíncrona da qual apenas um enrolamento é energizado. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=411-31-10" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="motor-de-indução-trifásico-no-psp-ufu"></a>Motor de indução trifásico no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#motor-de-indução-trifásico-no-psp-ufu" title="Direct link to heading">#</a></h2><p>No PSP-UFU, os motores de indução são contemplados nos estudos de <a href="/PSP/docs/powerFLow">fluxo de carga</a> e no estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os dados de estabilidade da máquina de indução são utilizados em conjunto com o fluxo de carga, <strong>calculando de forma correta a potência reativa das máquinas</strong> e consequentemente a tensão no barramento conectado.</p></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao marcar a opção &quot;Calcular a potência reativa no fluxo de carga&quot;, devem ser inseridos os dados corretos na aba &quot;Estabilidade&quot;, caso contrário, será atribuída uma potência reativa incorreta no motor.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga"></a>Motor de indução trifásico no cálculo do fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>A figura abaixo apresenta o modelo do motor de indução trifásico (MIT) de gaiola simples.</p><div><center><img src="/PSP/images/indMotorModel.svg" alt="Circuito equivalente do motor de indução" title="Circuito equivalente do motor de indução"></center></div><p>As potências ativa e reativa podem ser calculadas em relação às variáveis e parâmetros do motor em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> como:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mfrac><mrow><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">{</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mo fence="true">}</mo></mrow></mrow><mrow><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"></mspace><mi>Q</mi><mo>=</mo><mfrac><mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">{</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mo>−</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mo fence="true">}</mo></mrow></mrow><mrow><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">P = \frac{V^2 \left\{ \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]+K_1 \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\ Q = \frac{-V^2 \left\{ K_1 \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]- \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\ ~\\ K_1=x_2+x_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.884028em;vertical-align:-1.2940180000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em"><span style="top:-2.11em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.284008em"><span class="pstrut" style="height:3.054008em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7940180000000003em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">{</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">}</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2940180000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.884028em;vertical-align:-1.2940180000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em"><span style="top:-2.11em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.284008em"><span class="pstrut" style="height:3.054008em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7940180000000003em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">{</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">}</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2940180000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Como pode ser observado nas equações acima, existem quatro variáveis e somente duas equações. Na prática, as variáveis podem ser reduzidas a três, uma vez que o módulo da tensão (V) é obtido nos resultados do <a href="/PSP/docs/powerFlow">fluxo de potência</a>. Para resolver as equações é necessário definir uma variável adicional. A variável escolhida como fixa no PSP-UFU é a potência ativa (P), por fornecer resultados numericamente corretos e adequados para motores em situações de estabilidade.</p><p>Portanto, nesse modelo estático, a potência ativa é mantida constante durante o cálculo do fluxo de carga e o escorregamento (s) é atualizado em cada iteração. A equação da potência ativa pode ser reescrita em relação ao escorregamento:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mi>A</mi><mo>+</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mi>B</mi><mo>+</mo><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)^2 A + \left(\frac{r_2}{s} \right) B + C = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.040008em;vertical-align:-0.686em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">s</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3540079999999999em"><span style="top:-3.6029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.8359999999999999em;vertical-align:-0.686em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">s</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></div><p>Em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>P</mi><mrow><mo fence="true">(</mo><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>K</mi><mn>3</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><msub><mi>r</mi><mn>1</mn></msub><mspace linebreak="newline"></mspace><mi>B</mi><mo>=</mo><mn>2</mn><mi>P</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>2</mn></msub><mo>+</mo><msub><mi>K</mi><mn>3</mn></msub><msub><mi>K</mi><mn>4</mn></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">(</mo><msub><mi>K</mi><mn>2</mn></msub><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><msub><mi>K</mi><mn>3</mn></msub><mo fence="true">)</mo></mrow><mspace linebreak="newline"></mspace><mi>C</mi><mo>=</mo><mi>P</mi><mrow><mo fence="true">(</mo><msubsup><mi>K</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>K</mi><mn>4</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><msub><mi>K</mi><mn>1</mn></msub><msub><mi>K</mi><mn>4</mn></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>3</mn></msub><mo>=</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>4</mn></msub><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A = P \left( r_1^2 + K_3^2 \right) - V^2 r_1\\ B = 2P(r_1 K_2 + K_3 K_4) - V^2 \left(K_2 + K_1 K_3 \right)\\ C = P \left( K_2^2 + K_4^2 \right) - V^2 K_1 K_4\\ K_2 = -x_1 K_1 -x_2 x_m\\ K_3 = x_m + x_1\\ K_4 = r_1 K_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.2141179999999998em;vertical-align:-0.35001em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.0141079999999998em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.2141179999999998em;vertical-align:-0.35001em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.0141079999999998em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Esse modelo pode ser inserido na solução do <a href="/PSP/docs/powerFlow">fluxo de carga</a> seguindo os seguintes passos:</p><ol><li>As constantes <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">K_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">K_4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são inicialmente calculadas . Esses valores são mantidos constantes durante toda a solução;</li><li>Em cada iteração são calculados os coeficientes <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span></span></span></span></span> utilizando o valor atualizado de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>;</li><li>A equação quadrática é resolvida e dois valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span> são obtidos, em que o maior deles é escolhido por estar na região estável da característica toque-escorregamento do motor;</li><li>Utilizando o novo valor de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span>, a potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span>) é obtida. O vetor de potências é então atualizado e os procedimentos convencionais de solução do fluxo de potência são realizados.</li></ol><p>Os passos de 2 a 4 são repetidos até que se obtenha a convergência.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU, os motores de indução não são considerados no cálculo de curto-circuito.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="motor-de-indução-trifásico-no-estudo-de-estabilidade"></a>Motor de indução trifásico no estudo de estabilidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#motor-de-indução-trifásico-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>Uma importante carga dinâmica são os motores de indução, uma vez que correspondem a uma parcela significativa das cargas presentes no sistema elétrico. O <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">modelo da máquina de indução apresentado anteriormente</a>, a qual pode ser utilizada tanto como motor quanto como gerador, é bem estabelecida na literatura.</p><p>Como descrito na <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">seção anterior</a>, a inicialização dessa máquina é realizada em conjunto com o fluxo de potência, visto que a potência reativa exigida pela máquina de indução é dependente dos parâmetros do motor, assim como a tensão do seu barramento. Essa abordagem é necessária, pois métodos convencionais conduzem a resultados errôneos em sistemas altamente carregados.</p><p>É necessário expressar a equação de movimento da máquina de indução em termos de torque e não potência, como é realizado com as máquinas síncronas. A simetria do rotor também faz com que a posição angular não seja importante e o escorregamento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">s</span></span></span></span></span>) é utilizado no lugar da velocidade (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span></span></span></span>), em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>=</mo><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><msub><mi mathvariant="normal">Ω</mi><mn>0</mn></msub><mo>−</mo><mi>ω</mi></mrow><msub><mi mathvariant="normal">Ω</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">s = frac{\Omega_0 - \omega}{\Omega_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mord mathdefault">a</span><span class="mord mathdefault">c</span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Desprezando as perdas por atrito e ventilação e a potência no eixo suave, as equações mecânicas do motor são expressas da seguinte forma:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>=</mo><mi>A</mi><mo>−</mo><mi>B</mi><mi>s</mi><mo>+</mo><mi>C</mi><msup><mi>s</mi><mn>2</mn></msup><mspace linebreak="newline"></mspace><msub><mi>T</mi><mi>e</mi></msub><mo>=</mo><mfrac><mrow><mi>R</mi><mi>e</mi><mrow><mo fence="true">{</mo><mover accent="true"><mi>E</mi><mo>˙</mo></mover><msup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mo>∗</mo></msup><mo fence="true">}</mo></mrow></mrow><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>s</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo fence="true">(</mo><msub><mi>T</mi><mi>m</mi></msub><mo>−</mo><msub><mi>T</mi><mi>e</mi></msub><mo fence="true">)</mo></mrow><mrow><mn>2</mn><mi>H</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">T_m = A -Bs + Cs^2\\ T_e = \frac{Re\left\{ \dot{E}\dot{I}^* \right\}}{\Omega_b}\\ \frac{ds}{dt} = \frac{\left( T_m - T_e \right)}{2H}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.02602em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.19002em"><span style="top:-2.464em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.38em"><span class="pstrut" style="height:3.15em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-4.1900200000000005em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">{</span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">}</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">T_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o torque mecânico;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">T_e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o torque elétrico;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span></span></span> é a inércia do conjunto motor - carga mecânica</li></ul><p>Os termos <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span></span></span></span></span> são termos que definem o comportamento do torque mecânico da carga de acordo com o escorregamento. O torque mecânico normalmente varia com a <strong>velocidade</strong>, podendo ser expressa proporcionalmente com a seguinte equação quadrática:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>∝</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>ω</mi><mo>+</mo><mi>c</mi><msup><mi>ω</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">T_m \propto a + b\omega + c\omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">b</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord mathdefault">c</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mo>∝</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>B</mi><mo>∝</mo><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>C</mi><mo>∝</mo><mi>c</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} A \propto a + b + c \\ B \propto b + 2c \\ C \propto c \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">c</span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">2</span><span class="mord mathdefault">c</span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>As equações elétricas do motor de indução de gaiola simples são baseadas no circuito equivalente da <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">figura anterior</a>. De forma semelhante ao modelo transitório da máquina síncrona, o motor de indução pode ser modelado pelo circuito equivalente de Thevenin de tensão transitória <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>E</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">E&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> atrás de uma resistência do estator <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">r_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e uma reatância transitória <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">x&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>. A reatância transitória aparente de rotor bloqueado é dada por:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">x&#x27; = x_1 + \frac{x_2 x_m}{x_2 + x_m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.801892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.94356em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A constante de tempo de circuito aberto do rotor (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">T_0&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.24810799999999997em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4518920000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em"><span></span></span></span></span></span></span></span></span></span></span>) é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><mrow><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><msub><mi>r</mi><mn>2</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">T_0&#x27; = \frac{x_2 + x_m}{\Omega_b r_2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.09633em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603300000000002em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>E a reatância de circuito aberto é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">x_0 = x_1 + x_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Uma vez que as reatâncias não são afetadas pela posição do rotor, as EADs do motor de indução podem ser expressar diretamente por componentes reais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>) e imaginárias (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">m</span></span></span></span></span>). Portanto, a descrição completa desse modelo é representada pelo seguinte sistema de equações algébrico-diferenciais:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>r</mi></msub><mo>−</mo><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>I</mi><mi>r</mi></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>I</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><msub><mi>V</mi><mi>m</mi></msub><mo>−</mo><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>I</mi><mi>m</mi></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>I</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mi>s</mi><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><mfrac><mrow><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>m</mi></msub></mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mi>s</mi><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><mfrac><mrow><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>i</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>r</mi></msub></mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac></mrow><annotation encoding="application/x-tex">V_r - E_r&#x27; = r_1 I_r - x&#x27; I_m\\ V_m - E_m&#x27; = r_1 I_m - x&#x27; I_m\\ \frac{dE_r&#x27;}{dt} = \Omega_b s E_m&#x27; - \frac{E_r&#x27; + \left( x_0 - x&#x27; \right) I_m}{T_0&#x27;}\\ \frac{dE_m&#x27;}{dt} = \Omega_b s E_r&#x27; - \frac{E_m&#x27; i \left( x_0 - x&#x27; \right) I_r}{T_0&#x27;}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.9518920000000001em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.9518920000000001em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.3811999999999998em;vertical-align:-0.9523079999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.433692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9523079999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.3811999999999998em;vertical-align:-0.9523079999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.433692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9523079999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-dos-geradores-síncronos"></a>Formulário de edição dos geradores síncronos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-geradores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos motores de indução:</p><div><center><img src="/PSP/images/indMotorForm.png" alt="Formulário dos motores de indução no PSP-UFU" title="Formulário dos motores de indução no PSP-UFU"></center></div><p>No formulário pode ser observado o botão &quot;Chaveamento&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do motor durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/indMotorSw.png" alt="Formulário de chaveamento do motores de indução" title="Formulário de chaveamento do motores de indução"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Estabilidade</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Botão Chaveamento</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do gerador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) e reativa (inserida em var, kvar, Mvar ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) do motor.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso a opção “Calcular a potência reativa no fluxo de carga” esteja ativada, o campo de potência reativa é desativado para edição.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="calcular-a-potência-reativa-no-fluxo-de-carga"></a>Calcular a potência reativa no fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#calcular-a-potência-reativa-no-fluxo-de-carga" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar os dados fornecidos no formulário de estabilidade para calcular a potência reativa do motor durante o processo iterativo do fluxo de carga.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Caso essa opção não seja utilizada o motor será considerado uma <a href="/PSP/docs/load">carga de potência constante</a> no estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>.</p><p>A não utilização dessa opção poderá gerar erros de regime permamente no estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a potência base do sistema.</p></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>SÁNCHEZ, J. C.; OLIVARES, T. I. A.; ORTIZ, G. R.; VEGA, D. R. Induction Motor Static Models for Power Flow and Voltage Stability Studies. In: IEEE Power and Energy Society General Meeting, 2012, San Diego. doi: <a href="https://doi.org/10.1109/PESGM.2012.6345618" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/PESGM.2012.6345618</a></li><li>IEEE Std 399-1997. IEEE Recommended Practice for Industrial and Commercial Power Systems Analysis (Brown Book). IEEE, New York, ago. 1998. doi: <a href="https://doi.org/10.1109/IEEESTD.1998.88568" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/IEEESTD.1998.88568</a></li><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/indMotor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/syncMotor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Compensador Síncrono</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/controlEditor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Editor de Controle »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#motor-de-indução-trifásico-no-psp-ufu" class="table-of-contents__link">Motor de indução trifásico no PSP-UFU</a><ul><li><a href="#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga" class="table-of-contents__link">Motor de indução trifásico no cálculo do fluxo de carga</a></li><li><a href="#motor-de-indução-trifásico-no-estudo-de-estabilidade" class="table-of-contents__link">Motor de indução trifásico no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-dos-geradores-síncronos" class="table-of-contents__link">Formulário de edição dos geradores síncronos</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Motor de Indução</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Uma máquina assíncrona da qual apenas um enrolamento é energizado. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=411-31-10" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="motor-de-indução-trifásico-no-psp-ufu"></a>Motor de indução trifásico no PSP-UFU<a class="hash-link" href="#motor-de-indução-trifásico-no-psp-ufu" title="Direct link to heading">#</a></h2><p>No PSP-UFU, os motores de indução são contemplados nos estudos de <a href="/PSP/docs/powerFLow">fluxo de carga</a> e no estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os dados de estabilidade da máquina de indução são utilizados em conjunto com o fluxo de carga, <strong>calculando de forma correta a potência reativa das máquinas</strong> e consequentemente a tensão no barramento conectado.</p></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao marcar a opção &quot;Calcular a potência reativa no fluxo de carga&quot;, devem ser inseridos os dados corretos na aba &quot;Estabilidade&quot;, caso contrário, será atribuída uma potência reativa incorreta no motor.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga"></a>Motor de indução trifásico no cálculo do fluxo de carga<a class="hash-link" href="#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>A figura abaixo apresenta o modelo do motor de indução trifásico (MIT) de gaiola simples.</p><div><center><img src="/PSP/images/indMotorModel.svg" alt="Circuito equivalente do motor de indução" title="Circuito equivalente do motor de indução"></center></div><p>As potências ativa e reativa podem ser calculadas em relação às variáveis e parâmetros do motor em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> como:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mfrac><mrow><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">{</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mo fence="true">}</mo></mrow></mrow><mrow><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"></mspace><mi>Q</mi><mo>=</mo><mfrac><mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">{</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mo>−</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mo fence="true">}</mo></mrow></mrow><mrow><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">P = \frac{V^2 \left\{ \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]+K_1 \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\ Q = \frac{-V^2 \left\{ K_1 \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]- \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\ ~\\ K_1=x_2+x_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.884028em;vertical-align:-1.2940180000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em"><span style="top:-2.11em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.284008em"><span class="pstrut" style="height:3.054008em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7940180000000003em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">{</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">}</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2940180000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.884028em;vertical-align:-1.2940180000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em"><span style="top:-2.11em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.284008em"><span class="pstrut" style="height:3.054008em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7940180000000003em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">{</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">}</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2940180000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Como pode ser observado nas equações acima, existem quatro variáveis e somente duas equações. Na prática, as variáveis podem ser reduzidas a três, uma vez que o módulo da tensão (V) é obtido nos resultados do <a href="/PSP/docs/powerFlow">fluxo de potência</a>. Para resolver as equações é necessário definir uma variável adicional. A variável escolhida como fixa no PSP-UFU é a potência ativa (P), por fornecer resultados numericamente corretos e adequados para motores em situações de estabilidade.</p><p>Portanto, nesse modelo estático, a potência ativa é mantida constante durante o cálculo do fluxo de carga e o escorregamento (s) é atualizado em cada iteração. A equação da potência ativa pode ser reescrita em relação ao escorregamento:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mi>A</mi><mo>+</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mi>B</mi><mo>+</mo><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)^2 A + \left(\frac{r_2}{s} \right) B + C = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.040008em;vertical-align:-0.686em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">s</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3540079999999999em"><span style="top:-3.6029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.8359999999999999em;vertical-align:-0.686em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">s</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></div><p>Em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>P</mi><mrow><mo fence="true">(</mo><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>K</mi><mn>3</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><msub><mi>r</mi><mn>1</mn></msub><mspace linebreak="newline"></mspace><mi>B</mi><mo>=</mo><mn>2</mn><mi>P</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>2</mn></msub><mo>+</mo><msub><mi>K</mi><mn>3</mn></msub><msub><mi>K</mi><mn>4</mn></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">(</mo><msub><mi>K</mi><mn>2</mn></msub><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><msub><mi>K</mi><mn>3</mn></msub><mo fence="true">)</mo></mrow><mspace linebreak="newline"></mspace><mi>C</mi><mo>=</mo><mi>P</mi><mrow><mo fence="true">(</mo><msubsup><mi>K</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>K</mi><mn>4</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><msub><mi>K</mi><mn>1</mn></msub><msub><mi>K</mi><mn>4</mn></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>3</mn></msub><mo>=</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>4</mn></msub><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A = P \left( r_1^2 + K_3^2 \right) - V^2 r_1\\ B = 2P(r_1 K_2 + K_3 K_4) - V^2 \left(K_2 + K_1 K_3 \right)\\ C = P \left( K_2^2 + K_4^2 \right) - V^2 K_1 K_4\\ K_2 = -x_1 K_1 -x_2 x_m\\ K_3 = x_m + x_1\\ K_4 = r_1 K_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.2141179999999998em;vertical-align:-0.35001em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.0141079999999998em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.2141179999999998em;vertical-align:-0.35001em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.0141079999999998em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Esse modelo pode ser inserido na solução do <a href="/PSP/docs/powerFlow">fluxo de carga</a> seguindo os seguintes passos:</p><ol><li>As constantes <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">K_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">K_4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são inicialmente calculadas . Esses valores são mantidos constantes durante toda a solução;</li><li>Em cada iteração são calculados os coeficientes <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span></span></span></span></span> utilizando o valor atualizado de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>;</li><li>A equação quadrática é resolvida e dois valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span> são obtidos, em que o maior deles é escolhido por estar na região estável da característica toque-escorregamento do motor;</li><li>Utilizando o novo valor de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span>, a potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span>) é obtida. O vetor de potências é então atualizado e os procedimentos convencionais de solução do fluxo de potência são realizados.</li></ol><p>Os passos de 2 a 4 são repetidos até que se obtenha a convergência.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU, os motores de indução não são considerados no cálculo de curto-circuito.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="motor-de-indução-trifásico-no-estudo-de-estabilidade"></a>Motor de indução trifásico no estudo de estabilidade<a class="hash-link" href="#motor-de-indução-trifásico-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>Uma importante carga dinâmica são os motores de indução, uma vez que correspondem a uma parcela significativa das cargas presentes no sistema elétrico. O <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">modelo da máquina de indução apresentado anteriormente</a>, a qual pode ser utilizada tanto como motor quanto como gerador, é bem estabelecida na literatura.</p><p>Como descrito na <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">seção anterior</a>, a inicialização dessa máquina é realizada em conjunto com o fluxo de potência, visto que a potência reativa exigida pela máquina de indução é dependente dos parâmetros do motor, assim como a tensão do seu barramento. Essa abordagem é necessária, pois métodos convencionais conduzem a resultados errôneos em sistemas altamente carregados.</p><p>É necessário expressar a equação de movimento da máquina de indução em termos de torque e não potência, como é realizado com as máquinas síncronas. A simetria do rotor também faz com que a posição angular não seja importante e o escorregamento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">s</span></span></span></span></span>) é utilizado no lugar da velocidade (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span></span></span></span>), em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>=</mo><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><msub><mi mathvariant="normal">Ω</mi><mn>0</mn></msub><mo>−</mo><mi>ω</mi></mrow><msub><mi mathvariant="normal">Ω</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">s = frac{\Omega_0 - \omega}{\Omega_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mord mathdefault">a</span><span class="mord mathdefault">c</span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Desprezando as perdas por atrito e ventilação e a potência no eixo suave, as equações mecânicas do motor são expressas da seguinte forma:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>=</mo><mi>A</mi><mo>−</mo><mi>B</mi><mi>s</mi><mo>+</mo><mi>C</mi><msup><mi>s</mi><mn>2</mn></msup><mspace linebreak="newline"></mspace><msub><mi>T</mi><mi>e</mi></msub><mo>=</mo><mfrac><mrow><mi>R</mi><mi>e</mi><mrow><mo fence="true">{</mo><mover accent="true"><mi>E</mi><mo>˙</mo></mover><msup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mo>∗</mo></msup><mo fence="true">}</mo></mrow></mrow><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>s</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo fence="true">(</mo><msub><mi>T</mi><mi>m</mi></msub><mo>−</mo><msub><mi>T</mi><mi>e</mi></msub><mo fence="true">)</mo></mrow><mrow><mn>2</mn><mi>H</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">T_m = A -Bs + Cs^2\\ T_e = \frac{Re\left\{ \dot{E}\dot{I}^* \right\}}{\Omega_b}\\ \frac{ds}{dt} = \frac{\left( T_m - T_e \right)}{2H}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.02602em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.19002em"><span style="top:-2.464em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.38em"><span class="pstrut" style="height:3.15em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-4.1900200000000005em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">{</span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">}</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">T_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o torque mecânico;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">T_e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o torque elétrico;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span></span></span> é a inércia do conjunto motor - carga mecânica</li></ul><p>Os termos <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span></span></span></span></span> são termos que definem o comportamento do torque mecânico da carga de acordo com o escorregamento. O torque mecânico normalmente varia com a <strong>velocidade</strong>, podendo ser expressa proporcionalmente com a seguinte equação quadrática:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>∝</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>ω</mi><mo>+</mo><mi>c</mi><msup><mi>ω</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">T_m \propto a + b\omega + c\omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">b</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord mathdefault">c</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mo>∝</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>B</mi><mo>∝</mo><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>C</mi><mo>∝</mo><mi>c</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} A \propto a + b + c \\ B \propto b + 2c \\ C \propto c \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">c</span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">2</span><span class="mord mathdefault">c</span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>As equações elétricas do motor de indução de gaiola simples são baseadas no circuito equivalente da <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">figura anterior</a>. De forma semelhante ao modelo transitório da máquina síncrona, o motor de indução pode ser modelado pelo circuito equivalente de Thevenin de tensão transitória <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>E</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">E&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> atrás de uma resistência do estator <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">r_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e uma reatância transitória <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">x&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>. A reatância transitória aparente de rotor bloqueado é dada por:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">x&#x27; = x_1 + \frac{x_2 x_m}{x_2 + x_m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.801892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.94356em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A constante de tempo de circuito aberto do rotor (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">T_0&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.24810799999999997em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4518920000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em"><span></span></span></span></span></span></span></span></span></span></span>) é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><mrow><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><msub><mi>r</mi><mn>2</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">T_0&#x27; = \frac{x_2 + x_m}{\Omega_b r_2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.09633em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603300000000002em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>E a reatância de circuito aberto é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">x_0 = x_1 + x_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Uma vez que as reatâncias não são afetadas pela posição do rotor, as EADs do motor de indução podem ser expressar diretamente por componentes reais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>) e imaginárias (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">m</span></span></span></span></span>). Portanto, a descrição completa desse modelo é representada pelo seguinte sistema de equações algébrico-diferenciais:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>r</mi></msub><mo>−</mo><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>I</mi><mi>r</mi></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>I</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><msub><mi>V</mi><mi>m</mi></msub><mo>−</mo><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>I</mi><mi>m</mi></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>I</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mi>s</mi><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><mfrac><mrow><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>m</mi></msub></mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mi>s</mi><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><mfrac><mrow><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>i</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>r</mi></msub></mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac></mrow><annotation encoding="application/x-tex">V_r - E_r&#x27; = r_1 I_r - x&#x27; I_m\\ V_m - E_m&#x27; = r_1 I_m - x&#x27; I_m\\ \frac{dE_r&#x27;}{dt} = \Omega_b s E_m&#x27; - \frac{E_r&#x27; + \left( x_0 - x&#x27; \right) I_m}{T_0&#x27;}\\ \frac{dE_m&#x27;}{dt} = \Omega_b s E_r&#x27; - \frac{E_m&#x27; i \left( x_0 - x&#x27; \right) I_r}{T_0&#x27;}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.9518920000000001em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.9518920000000001em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.3811999999999998em;vertical-align:-0.9523079999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.433692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9523079999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.3811999999999998em;vertical-align:-0.9523079999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.433692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9523079999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-geradores-síncronos"></a>Formulário de edição dos geradores síncronos<a class="hash-link" href="#formulário-de-edição-dos-geradores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos motores de indução:</p><div><center><img src="/PSP/images/indMotorForm.png" alt="Formulário dos motores de indução no PSP-UFU" title="Formulário dos motores de indução no PSP-UFU"></center></div><p>No formulário pode ser observado o botão &quot;Chaveamento&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do motor durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/indMotorSw.png" alt="Formulário de chaveamento do motores de indução" title="Formulário de chaveamento do motores de indução"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Chaveamento</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do gerador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) e reativa (inserida em var, kvar, Mvar ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) do motor.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso a opção “Calcular a potência reativa no fluxo de carga” esteja ativada, o campo de potência reativa é desativado para edição.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="calcular-a-potência-reativa-no-fluxo-de-carga"></a>Calcular a potência reativa no fluxo de carga<a class="hash-link" href="#calcular-a-potência-reativa-no-fluxo-de-carga" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar os dados fornecidos no formulário de estabilidade para calcular a potência reativa do motor durante o processo iterativo do fluxo de carga.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Caso essa opção não seja utilizada o motor será considerado uma <a href="/PSP/docs/load">carga de potência constante</a> no estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>.</p><p>A não utilização dessa opção poderá gerar erros de regime permamente no estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a potência base do sistema.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-dados-da-máquina-de-indução"></a>Imprimir dados da máquina de indução<a class="hash-link" href="#imprimir-dados-da-máquina-de-indução" title="Direct link to heading">#</a></h4><p>Exibe os dados do gerador síncrono nos gráficos no tempo. Os seguintes dados são exibidos:</p><ul><li>tensão terminal</li><li>potências ativa e reativas</li><li>corrente, torque elétrico</li><li>torque mecânico</li><li>velocidade</li><li>escorregamento</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inércia"></a>Inércia<a class="hash-link" href="#inércia" title="Direct link to heading">#</a></h4><p>Inércia do conjunto motor-carga, em s.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-e-reatância-do-estator"></a>Resistência e reatância do estator<a class="hash-link" href="#resistência-e-reatância-do-estator" title="Direct link to heading">#</a></h4><p>Resistência e reatância do estator do motor de indução, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-e-reatância-do-rotor"></a>Resistência e reatância do rotor<a class="hash-link" href="#resistência-e-reatância-do-rotor" title="Direct link to heading">#</a></h4><p>Resistência e reatância do rotor do motor de indução, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatância-de-magnetização"></a>Reatância de magnetização<a class="hash-link" href="#reatância-de-magnetização" title="Direct link to heading">#</a></h4><p>Reatância do ramo de magnetização do motor de indução, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-fator-de-gaiola"></a>Utilizar fator de gaiola<a class="hash-link" href="#utilizar-fator-de-gaiola" title="Direct link to heading">#</a></h4><p>Permite a inserção de um fator de gaiola (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">K_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) no motor de indução. Tal fator eleva o valor da resistência do rotor em condições de grande escorregamento, como ocorre em motores de gaiola dupla ou gaiola profunda. A resistência do rotor segue a seguinte equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>2</mn></msub><mo>=</mo><msub><mi>r</mi><msub><mn>2</mn><mn>0</mn></msub></msub><mrow><mo fence="true">(</mo><mn>1</mn><mo>+</mo><msub><mi>K</mi><mi>g</mi></msub><mi>s</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">r_2 = r_{2_0} \left( 1+K_g s \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><msub><mn>2</mn><mn>0</mn></msub></msub></mrow><annotation encoding="application/x-tex">r_{2_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68066em;vertical-align:-0.2501em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em"><span></span></span></span></span></span></span></span></span></span></span> é a resistência do rotor inserida no formulário.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="característica-da-carga"></a>Característica da carga<a class="hash-link" href="#característica-da-carga" title="Direct link to heading">#</a></h4><p>Descreve a característica do torque da carga mecânica no eixo do motor seguindo a equação do torque mecânico.</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>∝</mo><mi mathvariant="bold">a</mi><mo>+</mo><mi mathvariant="bold">b</mi><mi>ω</mi><mo>+</mo><mi mathvariant="bold">c</mi><msup><mi>ω</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">T_m \propto \bold{a} + \bold{b}\omega + \bold{c}\omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord"><span class="mord mathbf">a</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord"><span class="mord mathbf">b</span></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">c</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A soma dos coeficientes (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a + b + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">c</span></span></span></span></span>) deve ser unitária.</p></div></div></div><div role="tabpanel" hidden=""><p>O botão &quot;Chaveamento&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do gerador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>SÁNCHEZ, J. C.; OLIVARES, T. I. A.; ORTIZ, G. R.; VEGA, D. R. Induction Motor Static Models for Power Flow and Voltage Stability Studies. In: IEEE Power and Energy Society General Meeting, 2012, San Diego. doi: <a href="https://doi.org/10.1109/PESGM.2012.6345618" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/PESGM.2012.6345618</a></li><li>IEEE Std 399-1997. IEEE Recommended Practice for Industrial and Commercial Power Systems Analysis (Brown Book). IEEE, New York, ago. 1998. doi: <a href="https://doi.org/10.1109/IEEESTD.1998.88568" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/IEEESTD.1998.88568</a></li><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/indMotor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/syncMotor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Compensador Síncrono</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/controlEditor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Editor de Controle »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#motor-de-indução-trifásico-no-psp-ufu" class="table-of-contents__link">Motor de indução trifásico no PSP-UFU</a><ul><li><a href="#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga" class="table-of-contents__link">Motor de indução trifásico no cálculo do fluxo de carga</a></li><li><a href="#motor-de-indução-trifásico-no-estudo-de-estabilidade" class="table-of-contents__link">Motor de indução trifásico no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-dos-geradores-síncronos" class="table-of-contents__link">Formulário de edição dos geradores síncronos</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">PSP-UFU</h1></header><div class="markdown"><div class="admonition admonition-important alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>important</h5></div><div class="admonition-content"><p>This user guide was written in Brazilian Portuguese. If you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="sobre-o-psp-ufu"></a>Sobre o PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#sobre-o-psp-ufu" title="Direct link to heading">#</a></h2><p>O PSP-UFU (Plataforma de Sistemas de Potência da Universidade Federal de Uberlândia) é um software <strong>multi-plataforma</strong>, <strong>multilíngue</strong>, <strong>livre e de código aberto (FOSS)</strong> com <strong>recursos avançados de GUI (<em>Graphical User Interface</em>)</strong> e <strong>ferramentas CAD (<em>Computer Aided Design</em>)</strong> para estudos de sistemas elétricos de potência.</p><p>O software permite a construção de qualquer rede de transmissão elétrica e sistemas de controle através da inserção de elementos visuais. Para a visualização dos resultados, o programa oferece <strong>elementos de texto vinculados</strong> na tela principal e também <strong>editores de tabelas e gráficos</strong>.</p><p>O PSP-UFU tem como objetivo fornecer ferramentas eficientes de simulação para fins de <strong>pesquisa</strong> e <strong>educação</strong>, além de <strong>aplicações industriais</strong> em sistemas elétricos de potência.</p><p>Resumidamente, o <em>software</em> pode executar os seguintes estudos:</p><ul><li>Fluxo de potência</li><li>Cálculo de curto-circuito</li><li>Harmônicos</li><li>Estabilidade transitória e dinâmica</li></ul><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="publicações"></a>Publicações<a aria-hidden="true" tabindex="-1" class="hash-link" href="#publicações" title="Direct link to heading">#</a></h3><p>Os artigos abaixo possuem maiores detalhes a respeito do PSP-UFU:</p><blockquote><p>Oliveira, T. L., Guimarães, G. C., &amp; Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. <em>International Transactions on Electrical Energy Systems</em>, e12185. doi: <a href="https://doi.org/10.1002/2050-7038.12185" target="_blank" rel="noopener noreferrer">10.1002/2050-7038.12185</a></p></blockquote><blockquote><p>Oliveira, T. L., Guimarães, G. C., Silva, L. R., &amp; Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. <em>The International Journal of Electrical Engineering &amp; Education</em>, 0020720919879058. doi: <a href="https://doi.org/10.1177/0020720919879058" target="_blank" rel="noopener noreferrer">10.1177/0020720919879058</a></p></blockquote><p>A Tese de Doutorado abaixo detalha em profundidade os cálculos envolvidos, arquitetura de software e ferramentas implementadas no PSP-UFU:</p><blockquote><p>Oliveira, T. L. (2019). Desenvolvimento de um programa computacional livre, gráfico, e multiplataforma para analisar sistemas elétricos de potência em regime permanente e dinâmico. doi: <a href="https://dx.doi.org/10.14393/ufu.te.2019.2444" target="_blank" rel="noopener noreferrer">10.14393/ufu.te.2019.2444</a></p></blockquote><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="contribuindo-com-o-projeto"></a>Contribuindo com o projeto<a aria-hidden="true" tabindex="-1" class="hash-link" href="#contribuindo-com-o-projeto" title="Direct link to heading">#</a></h3><p>Se você está interessado em contribuir com o PSP-UFU de alguma forma (desenvolvimento, críticas, sugestões, etc.), entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="desenvolvedores"></a>Desenvolvedor<del>es</del><a aria-hidden="true" tabindex="-1" class="hash-link" href="#desenvolvedores" title="Direct link to heading">#</a></h3><p><a href="https://github.com/Thales1330" target="_blank" rel="noopener noreferrer">Thales Lima Oliveira</a></p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/aboutPSP.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/installation"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Instalação »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#sobre-o-psp-ufu" class="table-of-contents__link">Sobre o PSP-UFU</a><ul><li><a href="#publicações" class="table-of-contents__link">Publicações</a></li><li><a href="#contribuindo-com-o-projeto" class="table-of-contents__link">Contribuindo com o projeto</a></li><li><a href="#desenvolvedores" class="table-of-contents__link">Desenvolvedor<del>es</del></a></li></ul></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">PSP-UFU</h1></header><div class="markdown"><div class="admonition admonition-important alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>important</h5></div><div class="admonition-content"><p>Most of this user guide was written in Brazilian Portuguese. If you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="about-psp-ufu"></a>About PSP-UFU<a class="hash-link" href="#about-psp-ufu" title="Direct link to heading">#</a></h2><p>PSP-UFU (Power Systems Platform of Federal University of Uberlândia) is a <strong>multi-platform</strong>, <strong>multilingual</strong>, <strong>free and open source (FOSS)</strong> software with <strong>advanced GUI (Graphical User Interface) features</strong> and <strong>CAD (Computer Aided Design) tools</strong> for studies of electrical power systems.</p><p>The software allows the construction of any electrical transmission network and control systems through the deployment of visual elements. For viewing the results, the program offers <strong>linked text elements</strong> on the main screen and also <strong>table and graph editors</strong>.</p><p>The PSP-UFU aims to provide efficient simulation tools for <strong>research</strong> and <strong>education</strong> purposes, in addition to <strong>industrial applications</strong> in electrical power systems.</p><p>Briefly, software can perform the following studies:</p><ul><li>Power flow</li><li>Short circuit calculation</li><li>Harmonics</li><li>Transient and dynamic stability</li></ul><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="papers"></a>Papers<a class="hash-link" href="#papers" title="Direct link to heading">#</a></h3><p>The articles below have more details about the PSP-UFU:</p><blockquote><p>Oliveira, T. L., Guimarães, G. C., &amp; Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. <em>International Transactions on Electrical Energy Systems</em>, e12185. doi: <a href="https://doi.org/10.1002/2050-7038.12185" target="_blank" rel="noopener noreferrer">10.1002/2050-7038.12185</a></p></blockquote><blockquote><p>Oliveira, T. L., Guimarães, G. C., Silva, L. R., &amp; Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. <em>The International Journal of Electrical Engineering &amp; Education</em>, 0020720919879058. doi: <a href="https://doi.org/10.1177/0020720919879058" target="_blank" rel="noopener noreferrer">10.1177/0020720919879058</a></p></blockquote><p>The PhD. Thesis below details in depth the calculations involved, software architecture and tools implemented in the PSP-UFU:</p><blockquote><p>Oliveira, T. L. (2019). Desenvolvimento de um programa computacional livre, gráfico, e multiplataforma para analisar sistemas elétricos de potência em regime permanente e dinâmico. doi: <a href="https://dx.doi.org/10.14393/ufu.te.2019.2444" target="_blank" rel="noopener noreferrer">10.14393/ufu.te.2019.2444</a></p></blockquote><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="contributing-to-the-project"></a>Contributing to the project<a class="hash-link" href="#contributing-to-the-project" title="Direct link to heading">#</a></h3><p>If you are interested in contributing to PSP-UFU in any way (development, criticism, suggestions, etc.), contact us at <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="developers"></a>Developer<del>s</del><a class="hash-link" href="#developers" title="Direct link to heading">#</a></h3><p><a href="https://github.com/Thales1330" target="_blank" rel="noopener noreferrer">Thales Lima Oliveira</a></p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/aboutPSP.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/installation"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Installation »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#about-psp-ufu" class="table-of-contents__link">About PSP-UFU</a><ul><li><a href="#papers" class="table-of-contents__link">Papers</a></li><li><a href="#contributing-to-the-project" class="table-of-contents__link">Contributing to the project</a></li><li><a href="#developers" class="table-of-contents__link">Developer<del>s</del></a></li></ul></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Indutor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Um reator destinado à conexão de derivação em uma rede para compensar a corrente capacitiva. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=421-01-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="indutor-no-psp-ufu"></a>Indutor no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#indutor-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento indutor (ou reator) representa, geralmente, reator <em>shunt</em> no circuito do PSP-UFU.</p><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-dos-indutores"></a>Formulário de edição dos indutores<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-indutores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos indutores:</p><div><center><img src="/PSP/images/indForm.png" alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU"></center></div><p>Além do único contexto geral, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do indutor durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/indSw.png" alt="Formulário de chaveamento do indutor" title="Formulário de chaveamento do indutor"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Botão Estabilidade</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-reativa"></a>Potência reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base de potência do sistema). </p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/inductor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/capacitor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Capacitor</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/harmSource"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Fonte de Corrente Harmônica »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#indutor-no-psp-ufu" class="table-of-contents__link">Indutor no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-indutores" class="table-of-contents__link">Formulário de edição dos indutores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulations</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Indutor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Um reator destinado à conexão de derivação em uma rede para compensar a corrente capacitiva. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=421-01-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="indutor-no-psp-ufu"></a>Indutor no PSP-UFU<a class="hash-link" href="#indutor-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento indutor (ou reator) representa, geralmente, reator <em>shunt</em> no circuito do PSP-UFU.</p><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-indutores"></a>Formulário de edição dos indutores<a class="hash-link" href="#formulário-de-edição-dos-indutores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos indutores:</p><div><center><img src="/PSP/images/indForm.png" alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU"></center></div><p>Além do único contexto geral, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do indutor durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/indSw.png" alt="Formulário de chaveamento do indutor" title="Formulário de chaveamento do indutor"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-reativa"></a>Potência reativa<a class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base de potência do sistema). </p></div><div role="tabpanel" hidden=""><p>O botão &quot;Estabilidade&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/inductor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/capacitor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Capacitor</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/harmSource"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Fonte de Corrente Harmônica »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#indutor-no-psp-ufu" class="table-of-contents__link">Indutor no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-indutores" class="table-of-contents__link">Formulário de edição dos indutores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Instalação</h1></header><div class="markdown"><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="download"></a>Download<a aria-hidden="true" tabindex="-1" class="hash-link" href="#download" title="Direct link to heading">#</a></h2><p>O download do PSP-UFU pode ser realizado pelo <a href="https://github.com/Thales1330/PSP/releases/latest" target="_blank" rel="noopener noreferrer">&quot;<em>last release</em>&quot; do repositório do PSP-UFU no GitHub</a>.</p><p>Ao acessar ao link, role a página até encontrar os <em>Assets</em>:</p><div><center><img src="/PSP/images/download.png" alt="Assets download" title="Informações da Figura"></center></div><p>Como o PSP-UFU é um <em>software</em> multiplataforma, você deve baixar o arquivo correto:</p><ul><li>Para o sistema operacional <a href="/PSP/docs/installation#windows">Windows</a> baixe os arquivos <em>.exe</em> (recomendado) ou <em>.zip</em>.</li><li>Para os sistemas <a href="/PSP/docs/installation#linux">Linux</a> baseados no Debian (por exemplo, Ubuntu, Mint) <em>64 bits</em> baixe o pacote de instalação <em>.deb</em>.</li></ul><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Se você está interessado em contribuir com o PSP-UFU construindo-o para outros sistemas operacionais (principalmente macOS!) entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><p>Na sequência são apresentados o passo-a-passo para instalação dos SOs <a href="/PSP/docs/installation#windows">Windows</a> e <a href="/PSP/docs/installation#linux">Linux</a>. Também são apresentadas as soluções para os erros mais recorrentes.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="windows"></a>Windows<a aria-hidden="true" tabindex="-1" class="hash-link" href="#windows" title="Direct link to heading">#</a></h2><p>Após baixar o arquivo <em>.exe</em> avance em cada passo do instalador. O executável irá solicitar automaticamente a instalação do <strong>Microsoft Visual C++ Redistribuível</strong> e, caso não esteja instalado no seu computador, realize os procedimentos do programa.</p><p>Caso o executável falhe em abrir, baixe o arquivo <em>.zip</em> e descompacte em alguma pasta do seu sistema. Nesse caso é necessário baixar e instalar manualmente o <a href="https://aka.ms/vs/16/release/vc_redist.x86.exe" target="_blank" rel="noopener noreferrer"><strong>Microsoft Visual C++ Redistribuível 32 bits</strong></a>.</p><p>Após a conclusão da instalação abra o PSP-UFU para verificar a correta instalação. O executável está presente na pasta &quot;<em>&lt;local de instalação&gt;/PSP-UFU/bin/PSP-UFU.exe</em>&quot;.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="problemas-recorrentes-no-windows"></a>Problemas recorrentes no Windows<a aria-hidden="true" tabindex="-1" class="hash-link" href="#problemas-recorrentes-no-windows" title="Direct link to heading">#</a></h3><ul><li>Ao baixar o navegador pode <strong>falsamente</strong> sinalizar que é um arquivo perigoso, basta clicar em &quot;Manter arquivo&quot;;</li><li>Ao abrir o instalador, novamente o Windows sinaliza que o arquivo é perigoso, basta clicar em &quot;Executar assim mesmo&quot;;</li><li>Ao abrir o programa e alterar o seu idioma, uma mensagem de erro é exibida. Para resolver esse problema basta executar o PSP-UFU como administrador (clicar com botão direito no atalho do programa e posteriormente em executar como administrador). Altere novamente o idioma e reinicie o programa;</li><li>Ao criar um novo projeto a tela de trabalho não é exibida ou é exibida uma tela preta. Vá nas opções gerais e altere o renderizador para &quot;Device Context&quot;. Caso apareça uma mensagem de erro, execute o programa como administrador e repita o processo. Reinicie o programa.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="linux"></a>Linux<a aria-hidden="true" tabindex="-1" class="hash-link" href="#linux" title="Direct link to heading">#</a></h2><p>A instalação no sistema Linux é facilmente realizado pelo pacote de instalação automático <em>.deb</em>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Note que essa versão para Linux está defasada e algumas funcionalidades do programa podem não estar presentes.</p><p>Caso queira contribuir com o projeto e construir a versão atual no sistema Linux baseado em Debian ou em outras distribuições entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="problemas-recorrentes-no-linux"></a>Problemas recorrentes no Linux<a aria-hidden="true" tabindex="-1" class="hash-link" href="#problemas-recorrentes-no-linux" title="Direct link to heading">#</a></h3><p>Pode ser necessário adicionar o <a href="https://wiki.wxwidgets.org/Compiling_and_getting_started#Running_wxWidgets_projects" target="_blank" rel="noopener noreferrer">diretório das bibliotecas do wxWidgets</a> nas variáveis de ambiente.
-Para isso, siga <strong>UM</strong> dos métodos apresentados abaixo:</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="método-i-recomendado"></a><em>Método I (recomendado)</em><a aria-hidden="true" tabindex="-1" class="hash-link" href="#método-i-recomendado" title="Direct link to heading">#</a></h4><p>Insira o seguinte comando no arquivo <strong>~/.bashrc</strong>:</p><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-shell codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">echo</span><span class="token plain"> </span><span class="token string" style="color:rgb(195, 232, 141)">&quot;export LD_LIBRARY_PATH=</span><span class="token string variable" style="color:rgb(191, 199, 213)">$LD_LIBRARY_PATH</span><span class="token string" style="color:rgb(195, 232, 141)">:/usr/local/lib&quot;</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;&gt;</span><span class="token plain"> ~/.bashrc</span></div></div></div></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="método-ii-não-recomendado"></a><em>Método II (não recomendado)</em><a aria-hidden="true" tabindex="-1" class="hash-link" href="#método-ii-não-recomendado" title="Direct link to heading">#</a></h4><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Utilizando esse método os passos abaixo devem ser executados <strong>todas as vezes</strong> que você quiser abrir o PSP-UFU.</p></div></div><ol><li>Abra o terminal e acesse o diretório onde está instalado o PSP-UFU:</li></ol><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-shell codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">cd</span><span class="token plain"> /usr/local/bin</span></div></div></div></div></div><ol start="2"><li>Insira as bibliotecas compartilhadas do wxWidgets nas variáveis de ambiente:</li></ol><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-shell codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">export</span><span class="token plain"> </span><span class="token assign-left variable" style="color:rgb(191, 199, 213)">LD_LIBRARY_PATH</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token variable" style="color:rgb(191, 199, 213)">$LD_LIBRARY_PATH</span><span class="token plain">:/usr/local/lib</span></div></div></div></div></div><ol start="3"><li>Então, execute o PSP-UFU na <strong>mesma seção</strong> do terminal:</li></ol><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-shell codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">./PSP-UFU</span></div></div></div></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Verifique se o computador possui as unidades de vídeo instaladas corretamente, caso contrário, o programa não exibirá a área de trabalho para criar e editar diagramas de linha única ou diagramas de blocos de controle.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/installation.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Installation</h1></header><div class="markdown"><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="download"></a>Download<a class="hash-link" href="#download" title="Direct link to heading">#</a></h2><p>The PSP-UFU can be downloaded from the <a href="https://github.com/Thales1330/PSP/releases/latest" target="_blank" rel="noopener noreferrer"><em>last release</em> of the PSP-UFU repository on GitHub</a>.</p><p>When accessing the link, scroll down the page until you find the <em> Assets </em>:</p><div><center><img src="/PSP/images/download.png" alt="Assets download" title="Informações da Figura"></center></div><p>As the PSP-UFU is a multiplatform software, you must download the correct file:</p><ul><li>For the operating system <a href="/PSP/docs/installation#windows">Windows</a> download the files <em>.exe</em> (recommended) or <em>.zip</em>.</li><li>For <a href="/PSP/docs/installation#linux">Linux</a> systems based on Debian (for example, Ubuntu, Mint) <em>64 bits</em> download the installation package <em>.deb</em>.</li></ul><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Tip</h5></div><div class="admonition-content"><p>If you are interested in contributing to PSP-UFU by building it for other operating systems (mainly macOS!) Contact us at <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><p>Na sequência são apresentados o passo-a-passo para instalação dos SOs <a href="/PSP/docs/installation#windows">Windows</a> e <a href="/PSP/docs/installation#linux">Linux</a>. Também são apresentadas as soluções para os erros mais recorrentes.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="windows"></a>Windows<a class="hash-link" href="#windows" title="Direct link to heading">#</a></h2><p>Após baixar o arquivo <em>.exe</em> avance em cada passo do instalador. O executável irá solicitar automaticamente a instalação do <strong>Microsoft Visual C++ Redistribuível</strong> e, caso não esteja instalado no seu computador, realize os procedimentos do programa.</p><p>Caso o executável falhe em abrir, baixe o arquivo <em>.zip</em> e descompacte em alguma pasta do seu sistema. Nesse caso é necessário baixar e instalar manualmente o <a href="https://aka.ms/vs/16/release/vc_redist.x86.exe" target="_blank" rel="noopener noreferrer"><strong>Microsoft Visual C++ Redistribuível 32 bits</strong></a>.</p><p>Após a conclusão da instalação abra o PSP-UFU para verificar a correta instalação. O executável está presente na pasta &quot;<em>&lt;local de instalação&gt;/PSP-UFU/bin/PSP-UFU.exe</em>&quot;.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="problemas-recorrentes-no-windows"></a>Problemas recorrentes no Windows<a class="hash-link" href="#problemas-recorrentes-no-windows" title="Direct link to heading">#</a></h3><ul><li>Ao baixar o navegador pode <strong>falsamente</strong> sinalizar que é um arquivo perigoso, basta clicar em &quot;Manter arquivo&quot;;</li><li>Ao abrir o instalador, novamente o Windows sinaliza que o arquivo é perigoso, basta clicar em &quot;Executar assim mesmo&quot;;</li><li>Ao abrir o programa e alterar o seu idioma, uma mensagem de erro é exibida. Para resolver esse problema basta executar o PSP-UFU como administrador (clicar com botão direito no atalho do programa e posteriormente em executar como administrador). Altere novamente o idioma e reinicie o programa;</li><li>Ao criar um novo projeto a tela de trabalho não é exibida ou é exibida uma tela preta. Vá nas opções gerais e altere o renderizador para &quot;Device Context&quot;. Caso apareça uma mensagem de erro, execute o programa como administrador e repita o processo. Reinicie o programa.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="linux"></a>Linux<a class="hash-link" href="#linux" title="Direct link to heading">#</a></h2><p>A instalação no sistema Linux é facilmente realizado pelo pacote de instalação automático <em>.deb</em>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Note que essa versão para Linux está defasada e algumas funcionalidades do programa podem não estar presentes.</p><p>Caso queira contribuir com o projeto e construir a versão atual no sistema Linux baseado em Debian ou em outras distribuições entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="problemas-recorrentes-no-linux"></a>Problemas recorrentes no Linux<a class="hash-link" href="#problemas-recorrentes-no-linux" title="Direct link to heading">#</a></h3><p>Pode ser necessário adicionar o <a href="https://wiki.wxwidgets.org/Compiling_and_getting_started#Running_wxWidgets_projects" target="_blank" rel="noopener noreferrer">diretório das bibliotecas do wxWidgets</a> nas variáveis de ambiente.
+Para isso, siga <strong>UM</strong> dos métodos apresentados abaixo:</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="método-i-recomendado"></a><em>Método I (recomendado)</em><a class="hash-link" href="#método-i-recomendado" title="Direct link to heading">#</a></h4><p>Insira o seguinte comando no arquivo <strong>~/.bashrc</strong>:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly shell"><div tabindex="0" class="prism-code language-shell codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">echo</span><span class="token plain"> </span><span class="token string" style="color:rgb(195, 232, 141)">&quot;export LD_LIBRARY_PATH=</span><span class="token string variable" style="color:rgb(191, 199, 213)">$LD_LIBRARY_PATH</span><span class="token string" style="color:rgb(195, 232, 141)">:/usr/local/lib&quot;</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;&gt;</span><span class="token plain"> ~/.bashrc</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="método-ii-não-recomendado"></a><em>Método II (não recomendado)</em><a class="hash-link" href="#método-ii-não-recomendado" title="Direct link to heading">#</a></h4><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Utilizando esse método os passos abaixo devem ser executados <strong>todas as vezes</strong> que você quiser abrir o PSP-UFU.</p></div></div><ol><li>Abra o terminal e acesse o diretório onde está instalado o PSP-UFU:</li></ol><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly shell"><div tabindex="0" class="prism-code language-shell codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">cd</span><span class="token plain"> /usr/local/bin</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><ol start="2"><li>Insira as bibliotecas compartilhadas do wxWidgets nas variáveis de ambiente:</li></ol><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly shell"><div tabindex="0" class="prism-code language-shell codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">export</span><span class="token plain"> </span><span class="token assign-left variable" style="color:rgb(191, 199, 213)">LD_LIBRARY_PATH</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token variable" style="color:rgb(191, 199, 213)">$LD_LIBRARY_PATH</span><span class="token plain">:/usr/local/lib</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><ol start="3"><li>Então, execute o PSP-UFU na <strong>mesma seção</strong> do terminal:</li></ol><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly shell"><div tabindex="0" class="prism-code language-shell codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">./PSP-UFU</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Verifique se o computador possui as unidades de vídeo instaladas corretamente, caso contrário, o programa não exibirá a área de trabalho para criar e editar diagramas de linha única ou diagramas de blocos de controle.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/installation.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav 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class="docMainContainer_3EyW"><div class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Entrada / Saída</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>As entradas e saída são variáveis relacionadas às máquinas síncronas. Cada tipo de entrada e saída está relacionada a um regulador da máquina: <strong>Regulador de Tensão (AVR, do inglês <em>Automatic Voltage Regulator</em>)</strong> e <strong>Regulador de Velocidade (RV)</strong>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O sistema de controle deve ter <strong>ao menos uma entrada e uma saída</strong>.</p></div></div><p>As entradas e saídas do sistema de controle são definidas por esses blocos, os quais são distintos para cada tipo de escopo. A figura abaixo apresenta um exemplo de um formulário de edição das entradas e saídas de um AVR:</p><div><center><img src="/PSP/images/ioForm.png" alt="Formulário de edição de entradas e saídas de sistemas de controle de um AVR" title="Formulário de edição de entradas e saídas de sistemas de controle de um AVR"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="variáveis-de-entrada-e-saída"></a>Variáveis de entrada e saída<a aria-hidden="true" tabindex="-1" class="hash-link" href="#variáveis-de-entrada-e-saída" title="Direct link to heading">#</a></h2><p>O usuário deve sinalizar o tipo de bloco entre entrada e saída e, então, selecionar em uma lista a variável desejada. As seguintes variáveis de controle estão atualmente disponíveis no programa:</p><ul><li><p><strong>Tensão terminal</strong> (<em>entrada: AVR</em>): Módulo da tensão no barramento da máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, variável no tempo. Essa variável é normalmente utilizada no cálculo do erro da tensão de referência do AVR;</p></li><li><p><strong>Velocidade</strong> (<em>entrada: AVR e RV</em>): Velocidade da máquina síncrona, em rad/s, variável no tempo. Normalmente utilizada no cálculo do erro de velocidade nos reguladores de velocidade, além de entrada do PSS em AVRs;</p></li><li><p><strong>Potência ativa e reativa</strong> (<em>entrada: AVR</em>): Potência ativa fornecida pela máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, variável no tempo. Normalmente utilizada como entrada do PSS (potência ativa) e controle de sub e sobrecorrente de excitação nos AVRs;</p></li><li><p><strong>Tensão terminal inicial</strong> (<em>entrada: AVR</em>): Módulo da tensão no barramento da máquina síncrona prévia ao estudo dinâmico originado do fluxo de carga, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, fixo no tempo. Essa variável está normalmente associada à referência de tensão do AVR;</p></li><li><p><strong>Velocidade inicial</strong> (<em>entrada: AVR e RV</em>): Velocidade do sistema (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>π</mi><msub><mi>f</mi><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">2 \pi f_{ref}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03588em">π</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>), definida nas <a href="/PSP/docs/simulationConfig">opções de simulação</a>, em rad/s, fixa no tempo. Normalmente utiliza-se essa variável como referência de velocidade em RVs e normalização da velocidade;</p></li><li><p><strong>Potência mecânica inicial</strong> (<em>entrada: RV</em>): Potência mecânica inicial, calculada após a inicialização das máquinas síncronas com os dados originados do fluxo de carga, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, fixa no tempo. Normalmente é utilizada como referência de potência mecânica nos reguladores de velocidade;</p></li><li><p><strong>Variação de velocidade e potência ativa</strong> (<em>entrada: AVR</em>): Cálculo da variação dessas entradas entre os passos de integração normalizada pelo passo de integração, conforme a equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>−</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">\Delta x = \frac{x_n-x_{n-1}}{h}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.9463300000000001em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">h</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span>	é a variação da entrada (velocidade ou potência ativa);</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>	é a variável no passo atual e anterior, respectivamente;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	é o passo de integração.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A normalização é necessária para a correta utilização da razão de passo de controle, definida nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p></div></div></li><li><p><strong>Tensão de campo</strong> (<em>saída: AVR</em>): Define a tensão aplicada ao campo na máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> Utilizada como saída dos AVRs, controlando principalmente tensão no barramento conectado e/ou fator de potência da máquina;</p></li><li><p><strong>Potência mecânica</strong> (<em>saída: RV</em>): Define qual a potência mecânica aplicada no eixo da máquina síncrona. Utilizada como saída dos reguladores de velocidade, controlando principalmente a potência ativa injetada pela máquina e sua frequência.</p></li></ul></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/io.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/controlEditor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Editor de Controle</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/transferFunction"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Função Transferência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#variáveis-de-entrada-e-saída" class="table-of-contents__link">Variáveis de entrada e saída</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Entrada / Saída</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>As entradas e saída são variáveis relacionadas às máquinas síncronas. Cada tipo de entrada e saída está relacionada a um regulador da máquina: <strong>Regulador de Tensão (AVR, do inglês <em>Automatic Voltage Regulator</em>)</strong> e <strong>Regulador de Velocidade (RV)</strong>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O sistema de controle deve ter <strong>ao menos uma entrada e uma saída</strong>.</p></div></div><p>As entradas e saídas do sistema de controle são definidas por esses blocos, os quais são distintos para cada tipo de escopo. A figura abaixo apresenta um exemplo de um formulário de edição das entradas e saídas de um AVR:</p><div><center><img src="/PSP/images/ioForm.png" alt="Formulário de edição de entradas e saídas de sistemas de controle de um AVR" title="Formulário de edição de entradas e saídas de sistemas de controle de um AVR"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="variáveis-de-entrada-e-saída"></a>Variáveis de entrada e saída<a class="hash-link" href="#variáveis-de-entrada-e-saída" title="Direct link to heading">#</a></h2><p>O usuário deve sinalizar o tipo de bloco entre entrada e saída e, então, selecionar em uma lista a variável desejada. As seguintes variáveis de controle estão atualmente disponíveis no programa:</p><ul><li><p><strong>Tensão terminal</strong> (<em>entrada: AVR</em>): Módulo da tensão no barramento da máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, variável no tempo. Essa variável é normalmente utilizada no cálculo do erro da tensão de referência do AVR;</p></li><li><p><strong>Velocidade</strong> (<em>entrada: AVR e RV</em>): Velocidade da máquina síncrona, em rad/s, variável no tempo. Normalmente utilizada no cálculo do erro de velocidade nos reguladores de velocidade, além de entrada do PSS em AVRs;</p></li><li><p><strong>Potência ativa e reativa</strong> (<em>entrada: AVR</em>): Potência ativa fornecida pela máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, variável no tempo. Normalmente utilizada como entrada do PSS (potência ativa) e controle de sub e sobrecorrente de excitação nos AVRs;</p></li><li><p><strong>Tensão terminal inicial</strong> (<em>entrada: AVR</em>): Módulo da tensão no barramento da máquina síncrona prévia ao estudo dinâmico originado do fluxo de carga, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, fixo no tempo. Essa variável está normalmente associada à referência de tensão do AVR;</p></li><li><p><strong>Velocidade inicial</strong> (<em>entrada: AVR e RV</em>): Velocidade do sistema (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>π</mi><msub><mi>f</mi><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">2 \pi f_{ref}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03588em">π</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>), definida nas <a href="/PSP/docs/simulationConfig">opções de simulação</a>, em rad/s, fixa no tempo. Normalmente utiliza-se essa variável como referência de velocidade em RVs e normalização da velocidade;</p></li><li><p><strong>Potência mecânica inicial</strong> (<em>entrada: RV</em>): Potência mecânica inicial, calculada após a inicialização das máquinas síncronas com os dados originados do fluxo de carga, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, fixa no tempo. Normalmente é utilizada como referência de potência mecânica nos reguladores de velocidade;</p></li><li><p><strong>Variação de velocidade e potência ativa</strong> (<em>entrada: AVR</em>): Cálculo da variação dessas entradas entre os passos de integração normalizada pelo passo de integração, conforme a equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>−</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">\Delta x = \frac{x_n-x_{n-1}}{h}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.9463300000000001em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">h</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span>	é a variação da entrada (velocidade ou potência ativa);</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>	é a variável no passo atual e anterior, respectivamente;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	é o passo de integração.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A normalização é necessária para a correta utilização da razão de passo de controle, definida nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p></div></div></li><li><p><strong>Tensão de campo</strong> (<em>saída: AVR</em>): Define a tensão aplicada ao campo na máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> Utilizada como saída dos AVRs, controlando principalmente tensão no barramento conectado e/ou fator de potência da máquina;</p></li><li><p><strong>Potência mecânica</strong> (<em>saída: RV</em>): Define qual a potência mecânica aplicada no eixo da máquina síncrona. Utilizada como saída dos reguladores de velocidade, controlando principalmente a potência ativa injetada pela máquina e sua frequência.</p></li></ul></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/io.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/controlEditor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Editor de Controle</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/transferFunction"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Função Transferência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#variáveis-de-entrada-e-saída" class="table-of-contents__link">Variáveis de entrada e saída</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="docMainContainer_3EyW"><div class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Limitador absoluto</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Os blocos de limite absoluto cumprem a função de restringir os valores de entrada de acordo com os limites máximos e mínimos inseridos pelo usuário. Caso a entrada ultrapasse algum dos limites impostos, a saída será igual a esse valor limite até que a entrada retorne para a faixa permitida.</p><p>As restrições máximas e mínimas de valores dos sistemas de controle são modeladas utilizando um Limitador, cuja implementação é bastante simples e segue a seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>u</mi><mi>n</mi></msub><mo>&gt;</mo><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>u</mi><mi>n</mi></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub><mo>≤</mo><msub><mi>u</mi><mi>n</mi></msub><mo>≤</mo><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>u</mi><mi>n</mi></msub><mo>&lt;</mo><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">y_n = \begin{cases} L_{sup}{,} &amp; \text{se } u_n &gt; L_{sup} \\ u_n{,} &amp; \text{se } L_{inf} \le u_n \le L_{sup}\\ L_{inf}{,} &amp; \text{se } u_n &lt; L_{inf} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span 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style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	Éé o valor de saída atual do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{sup}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{inf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>	são os limites superior e inferior, respectivamente</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor de entrada atual do bloco</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-do-bloco-limitador"></a>Formulário de edição do bloco Limitador<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-do-bloco-limitador" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco limitador.</p><div><center><img src="/PSP/images/limiterForm.png" alt="Formulário de edição de dados do bloco limitador no PSP-UFU" title="Formulário de edição de dados do bloco limitador no PSP-UFU"></center></div><p>Esse bloco não linear é definido pelo <strong>limite superior e inferior</strong>, inseridos pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tais blocos são bastante encontrados nos sistemas de controle para limitar a atuação do controlador, além de representar limites físicos de modelos implementados, como, por exemplo, valores máximos de corrente de excitação ou potência mecânica fornecida pelas turbinas.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/limiter.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/exponential"><div 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Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Limitador absoluto</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Os blocos de limite absoluto cumprem a função de restringir os valores de entrada de acordo com os limites máximos e mínimos inseridos pelo usuário. Caso a entrada ultrapasse algum dos limites impostos, a saída será igual a esse valor limite até que a entrada retorne para a faixa permitida.</p><p>As restrições máximas e mínimas de valores dos sistemas de controle são modeladas utilizando um Limitador, cuja implementação é bastante simples e segue a seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>u</mi><mi>n</mi></msub><mo>&gt;</mo><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>u</mi><mi>n</mi></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub><mo>≤</mo><msub><mi>u</mi><mi>n</mi></msub><mo>≤</mo><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>u</mi><mi>n</mi></msub><mo>&lt;</mo><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">y_n = \begin{cases} L_{sup}{,} &amp; \text{se } u_n &gt; L_{sup} \\ u_n{,} &amp; \text{se } L_{inf} \le u_n \le L_{sup}\\ L_{inf}{,} &amp; \text{se } u_n &lt; L_{inf} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	Éé o valor de saída atual do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{sup}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{inf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>	são os limites superior e inferior, respectivamente</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor de entrada atual do bloco</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-do-bloco-limitador"></a>Formulário de edição do bloco Limitador<a class="hash-link" href="#formulário-de-edição-do-bloco-limitador" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco limitador.</p><div><center><img src="/PSP/images/limiterForm.png" alt="Formulário de edição de dados do bloco limitador no PSP-UFU" title="Formulário de edição de dados do bloco limitador no PSP-UFU"></center></div><p>Esse bloco não linear é definido pelo <strong>limite superior e inferior</strong>, inseridos pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tais blocos são bastante encontrados nos sistemas de controle para limitar a atuação do controlador, além de representar limites físicos de modelos implementados, como, por exemplo, valores máximos de corrente de excitação ou potência mecânica fornecida pelas turbinas.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/limiter.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/exponential"><div 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absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3EyW"><div class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Linha</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Um meio de transmissão fabricado usado para transmitir energia eletromagnética entre dois pontos com um mínimo de radiação. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=704-02-02" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="linha-no-psp-ufu"></a>Linha no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#linha-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As linhas no PSP-UFU são modelos <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> equilibradas. Podem ser utilizadas como linhas de transmissão e distribuição de sistemas elétricos de potência.</p><p>A figura abaixo mostra o modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> implementado no PSP-UFU:</p><div><center><img src="/PSP/images/lineModel.svg" alt="Modelo PI da linha" title="Modelo PI da linha"></center></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mi mathvariant="bold">L</mi></msub></mrow><annotation encoding="application/x-tex">\bold{r_L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a resistência da linha;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">L</mi></msub></mrow><annotation encoding="application/x-tex">\bold{x_L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a reatância indutiva da linha;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">b</mi><mi mathvariant="bold">L</mi><mi mathvariant="bold">d</mi></msubsup></mrow><annotation encoding="application/x-tex">\bold{b_{L}^{d}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.126385em;vertical-align:-0.277277em"></span><span class="mord"><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.4227230000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.277277em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a susceptância capacitiva <em>shunt</em> da linha.</li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O modelo utilizado no PSP-UFU é <strong>equilibrado</strong> e <strong>não possui impedâncias mútuas</strong> entre as fases.</p></div></div><p>A linha pode ser inseridas com pontos de ancoragem, ou &quot;nós&quot;, para maior personalização gráfica do elemento, como apresentado nas <a href="/PSP/docs/cadTools#linha">Ferramentas CAD</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A linha deve ser inserida entre duas barras de <strong>mesma tensão nominal</strong>. Caso você tente inseri-la entre barras de tensão diferentes, uma mensagem de erro será exibida.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/docs/bus">barramento</a> e um <a href="/PSP/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-das-linhas"></a>Formulário de edição das linhas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-das-linhas" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das linhas de transmissão:</p><div><center><img src="/PSP/images/lineForm.png" alt="Formulário das linhas no PSP-UFU" title="Formulário das linhas no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais da linha e informações do fluxo de carga;</li><li><strong>Falta</strong>: local onde as impedâncias de sequência zero são inseridas.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os parâmetros necessários para construção da linha segundo seu modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> são inseridos na aba Geral, utilizados para construção da matriz admitância de sequência positiva e negativa.</p><p>Dados adicionais de impedâncias de sequência zero necessário para o cálculo de curtos-circuitos desbalanceados são editados na aba Falta, utilizados na construção da matriz admitância de sequência zero.</p></div></div><p>Além desses dois contextos, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da linha durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/lineSw.png" alt="Formulário de chaveamento da linha" title="Formulário de chaveamento da linha"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Falta</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Botão Estabilidade</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="tensão-nominal"></a>Tensão nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Campo de informação <em>não editável</em> que apresenta a tensão nominal da linha. Para alterar esse campo é necessário editar o campo correspondente do <a href="/PSP/docs/bus#geral">barramento</a> conectado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal da linha, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="parâmetros-do-modelo-pi"></a>Parâmetros do modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span><a aria-hidden="true" tabindex="-1" class="hash-link" href="#parâmetros-do-modelo-pi" title="Direct link to heading">#</a></h4><p>Resistência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">r_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), Reatância indutiva (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">x_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) série e Susceptância capacitiva shunt <strong>total</strong> (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><msubsup><mi>b</mi><mi>L</mi><mi>d</mi></msubsup></mrow><annotation encoding="application/x-tex">2b_{L}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.124439em;vertical-align:-0.275331em"></span><span class="mord">2</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.424669em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em"><span></span></span></span></span></span></span></span></span></span></span>) presentes no modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> da linha.</p><p>Esses parâmetros são inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">\Omega/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">Ω</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">S/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>b</mi><mi>L</mi><mi>d</mi></msubsup></mrow><annotation encoding="application/x-tex">b_{L}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.124439em;vertical-align:-0.275331em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.424669em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em"><span></span></span></span></span></span></span></span></span></span></span>).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="comprimento-da-linha"></a>Comprimento da linha<a aria-hidden="true" tabindex="-1" class="hash-link" href="#comprimento-da-linha" title="Direct link to heading">#</a></h4><p>Utilizado para calcular o valor dos parâmetros da linha inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">\Omega/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">Ω</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> (ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">S/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span>).</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O comprimento da linha é ignorado caso não sejam utilizadas as unidades por quilometro.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal da linha como base para a conversão das unidades, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/line.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this 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href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulations</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Linha</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Um meio de transmissão fabricado usado para transmitir energia eletromagnética entre dois pontos com um mínimo de radiação. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=704-02-02" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="linha-no-psp-ufu"></a>Linha no PSP-UFU<a class="hash-link" href="#linha-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As linhas no PSP-UFU são modelos <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> equilibradas. Podem ser utilizadas como linhas de transmissão e distribuição de sistemas elétricos de potência.</p><p>A figura abaixo mostra o modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> implementado no PSP-UFU:</p><div><center><img src="/PSP/images/lineModel.svg" alt="Modelo PI da linha" title="Modelo PI da linha"></center></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mi mathvariant="bold">L</mi></msub></mrow><annotation encoding="application/x-tex">\bold{r_L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a resistência da linha;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">L</mi></msub></mrow><annotation encoding="application/x-tex">\bold{x_L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a reatância indutiva da linha;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">b</mi><mi mathvariant="bold">L</mi><mi mathvariant="bold">d</mi></msubsup></mrow><annotation encoding="application/x-tex">\bold{b_{L}^{d}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.126385em;vertical-align:-0.277277em"></span><span class="mord"><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.4227230000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.277277em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a susceptância capacitiva <em>shunt</em> da linha.</li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O modelo utilizado no PSP-UFU é <strong>equilibrado</strong> e <strong>não possui impedâncias mútuas</strong> entre as fases.</p></div></div><p>A linha pode ser inseridas com pontos de ancoragem, ou &quot;nós&quot;, para maior personalização gráfica do elemento, como apresentado nas <a href="/PSP/docs/cadTools#linha">Ferramentas CAD</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A linha deve ser inserida entre duas barras de <strong>mesma tensão nominal</strong>. Caso você tente inseri-la entre barras de tensão diferentes, uma mensagem de erro será exibida.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/docs/bus">barramento</a> e um <a href="/PSP/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-das-linhas"></a>Formulário de edição das linhas<a class="hash-link" href="#formulário-de-edição-das-linhas" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das linhas de transmissão:</p><div><center><img src="/PSP/images/lineForm.png" alt="Formulário das linhas no PSP-UFU" title="Formulário das linhas no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais da linha e informações do fluxo de carga;</li><li><strong>Falta</strong>: local onde as impedâncias de sequência zero são inseridas.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os parâmetros necessários para construção da linha segundo seu modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> são inseridos na aba Geral, utilizados para construção da matriz admitância de sequência positiva e negativa.</p><p>Dados adicionais de impedâncias de sequência zero necessário para o cálculo de curtos-circuitos desbalanceados são editados na aba Falta, utilizados na construção da matriz admitância de sequência zero.</p></div></div><p>Além desses dois contextos, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da linha durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/lineSw.png" alt="Formulário de chaveamento da linha" title="Formulário de chaveamento da linha"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-nominal"></a>Tensão nominal<a class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Campo de informação <em>não editável</em> que apresenta a tensão nominal da linha. Para alterar esse campo é necessário editar o campo correspondente do <a href="/PSP/docs/bus#geral">barramento</a> conectado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal da linha, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="parâmetros-do-modelo-pi"></a>Parâmetros do modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span><a class="hash-link" href="#parâmetros-do-modelo-pi" title="Direct link to heading">#</a></h4><p>Resistência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">r_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), Reatância indutiva (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">x_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) série e Susceptância capacitiva shunt <strong>total</strong> (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><msubsup><mi>b</mi><mi>L</mi><mi>d</mi></msubsup></mrow><annotation encoding="application/x-tex">2b_{L}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.124439em;vertical-align:-0.275331em"></span><span class="mord">2</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.424669em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em"><span></span></span></span></span></span></span></span></span></span></span>) presentes no modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> da linha.</p><p>Esses parâmetros são inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">\Omega/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">Ω</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">S/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>b</mi><mi>L</mi><mi>d</mi></msubsup></mrow><annotation encoding="application/x-tex">b_{L}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.124439em;vertical-align:-0.275331em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.424669em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em"><span></span></span></span></span></span></span></span></span></span></span>).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="comprimento-da-linha"></a>Comprimento da linha<a class="hash-link" href="#comprimento-da-linha" title="Direct link to heading">#</a></h4><p>Utilizado para calcular o valor dos parâmetros da linha inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">\Omega/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">Ω</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> (ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">S/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span>).</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O comprimento da linha é ignorado caso não sejam utilizadas as unidades por quilometro.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal da linha como base para a conversão das unidades, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="parâmetros-do-modelo-pi-de-sequência-zero"></a>Parâmetros do modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> de sequência zero<a class="hash-link" href="#parâmetros-do-modelo-pi-de-sequência-zero" title="Direct link to heading">#</a></h4><p>Esses parâmetros são necessários para o correto cálculo das correntes de <a href="/PSP/docs/fault">falta desbalanceadas</a> (fase-fase, fase-fase-terra e fase-terra) e devem ser inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A não inserção desses dados acarretarão em resultados incorretos para <a href="/PSP/docs/fault">faltas desbalanceadas</a>.</p></div></div></div><div role="tabpanel" hidden=""><p>O botão &quot;Estabilidade&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção da linha durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/line.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/bus"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Barramento</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/transformer"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Transformador »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#linha-no-psp-ufu" class="table-of-contents__link">Linha no PSP-UFU</a></li><li><a href="#formulário-de-edição-das-linhas" class="table-of-contents__link">Formulário de edição das linhas</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Carga</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><ol><li>a potência ativa, reativa ou aparente gerada, transmitida ou distribuída dentro de um sistema;</li><li>a potência demandada por um grupo de consumidores classificados de acordo com suas particularidades e características, por exemplo, carga de aquecimento, carga reativa diurna, etc. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=601-01-15" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</li></ol></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="carga-no-psp-ufu"></a>Carga no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As cargas são os elementos de potência consumidores de potência genéricos. Nos estudos de fluxo de carga é possível modelar a carga como <strong>potência ou impedância constante</strong> e em estabilidade pode-se compor a carga na proporção desejada em três tipos (<strong>carga ZIP</strong>): potência constante, impedância constante e corrente constante.</p><p>As cargas de potência constante nos estudos de fluxo de carga são inseridos no vetor de potências e permanecem fixos durante toda a simulação.</p><p>Os elementos shunt, como reatores, banco de capacitores e cargas de impedância constante, são representados simplesmente por suas admitâncias em derivação. Como os parâmetros de entrada geralmente se constituem de suas potências ativa (P) e reativa (Q) nominais (no caso de reatores e capacitores, P=0), deve-se, portanto, encontrar sua admitância com a equação abaixo para inseri-la na matriz admitância:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mfrac><mrow><mi>P</mi><mo>−</mo><mi>j</mi><mi>Q</mi></mrow><msup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\overline{y} = \frac{P-jQ}{\dot{V}^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.825em;vertical-align:-0.19444em"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.17052em;vertical-align:-0.8101900000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em"><span style="top:-2.18981em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord mathdefault">Q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8101900000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Para os estudos de <strong>curto-circuito</strong> e <strong>harmônicos</strong> as cargas são modeladas como impedância constante, independentemente da sinalização indicada em seu formulário de edição de dados.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="carga-no-estudo-de-estabilidade"></a>Carga no estudo de estabilidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#carga-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A modelagem das cargas do sistema de forma exata se torna impraticável em um problemas de estabilidade, visto à grande quantidade e variedade dos componentes envolvidos, além de apresentar modificações em suas composições causado por vários fatores, como tempo, condições climáticas e economia. Portanto várias aproximações devem ser utilizadas na formulação de um modelo de carga nos estudos aqui postos.</p><p>Uma forma interessante de se representar cargas dependentes da tensão são as cargas ZIP, as quais são funções quadráticas, possuindo três parcelas: impedância constante (Z), corrente constante (I) e potência constante (P). As potências ativa e reativa são obtidas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>L</mi></msub><mo>=</mo><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><mrow><msub><mi>V</mi><mi>i</mi></msub><mn>0</mn></mrow></mfrac><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_L=P_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+P_{I0} \left(\frac{V_i}{V_i0} \right)+P_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.604038em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.6540080000000001em"><span style="top:-3.9029000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>L</mi></msub><mo>=</mo><msub><mi>Q</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>Q</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><mrow><msub><mi>V</mi><mi>i</mi></msub><mn>0</mn></mrow></mfrac><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>Q</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_L=Q_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+Q_{I0} \left(\frac{V_i}{V_i0} \right)+Q_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.604038em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.6540080000000001em"><span style="top:-3.9029000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">P_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">Q_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	são as potências ativa e reativa da carga ZIP, respectivamente, conectadas em uma barra genérica i;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão atual de uma barra genérica i;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">V_{i0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão inicial de uma barra genérica i, obtida do estudo de fluxo de carga;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{Z0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{Z0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de impedância constante para potências ativa e reativa, respectivamente;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{I0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{I0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de corrente constante para potências ativa e reativa, respectivamente;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de potência constante para potências ativa e reativa, respectivamente.</p><p>Os valores das parcelas de potência da carga ZIP são obtidos utilizando as potências da carga após a convergência do cálculo do fluxo de carga (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{L0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, para a potência ativa e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{L0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, para a potência reativa), utilizando as seguintes expressões:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>Z</mi></mrow></msub><mn>100</mn></mfrac><mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub><msubsup><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow><mn>2</mn></msubsup></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>I</mi></mrow></msub><mn>100</mn></mfrac><mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>P</mi></mrow></msub><mn>100</mn></mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mstyle></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} P_{Z0}=\displaystyle \frac{k_{PZ}}{100} \frac{P_{L0}}{V_{i0}^2}\\ P_{I0}=\displaystyle \frac{k_{PI}}{100} \frac{P_{L0}}{V_{i0}}\\ P_{P0}=\displaystyle \frac{k_{PP}}{100} P_{L0} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.60004em;vertical-align:-3.05002em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5500200000000004em"><span style="top:-0.9999899999999999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-0.99499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.2899900000000002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.5849900000000003em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.8799900000000005em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.1749900000000006em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.2049900000000004em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.1500100000000004em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.29501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.59001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.885010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.180010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.475010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.50501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.80002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.05002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5495919999999996em"><span style="top:-5.549592em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9628639999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.215288em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.007848em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.0495919999999996em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PZ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PP}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> representam a composição em impedância, corrente e potência constantes, respectivamente, da parcela ativa da carga, os quais são inseridos pelo usuário na forma de porcentagem, sendo a soma desses três valores necessariamente igual a 100%. Os parâmetros <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QZ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QP}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> podem ser interpretados de forma análoga, porém compõe a parcela reativa da carga.</p><p>O comportamento das potências, retiradas de simulações no PSP-UFU, de cada parcela que compõe a carga ZIP são evidenciadas na figura abaixo.</p><div><center><img src="/PSP/images/loadModel.svg" alt="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente" title="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente"></center></div><p>Cargas de corrente e potência constantes possuem problemas em tensões muito baixas. À medida que a tensão diminui as correntes dessas cargas não reduz, como pode ser observado na figura acima, resultando em perda de precisão e problemas na convergência de processos iterativos. Para contornar esse problema utiliza-se uma tensão pré-definida pelo usuário (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>l</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow><annotation encoding="application/x-tex">V_{low}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02691em">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), a qual as cargas (ou parcelas) de corrente e potência constantes são modeladas como impedância constante, resultando no comportamento de potência e corrente apresentados na figura acima.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-das-cargas"></a>Formulário de edição das cargas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-das-cargas" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das cargas:</p><div><center><img src="/PSP/images/loadForm.png" alt="Formulário das cargas no PSP-UFU" title="Formulário das cargas no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais das cargas, informações e o tipo de carga no fluxo de carga;</li><li><strong>Estabilidade</strong>: contendo opções de visualização de dados da carga em gráficos no tempo e opções de parametrização da carga ZIP.</li></ul><p>Além desses dois contextos, pode ser observado o botão &quot;Chaveamento&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da carga durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/loadSw.png" alt="Formulário de chaveamento da carga" title="Formulário de chaveamento da carga"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Estabilidade</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Botão Chaveamento</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-ativa"></a>Potência ativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-ativa" title="Direct link to heading">#</a></h4><p>Parcela de potência ativa da carga. Pode ser inserido em MW, kW W ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base do sistema).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-reativa"></a>Potência reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Parcela de potência reativa da carga. Pode ser inserido em Mvar, kvar var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base do sistema).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="tipo-de-carga-fluxo-de-carga"></a>Tipo de carga (fluxo de carga)<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tipo-de-carga-fluxo-de-carga" title="Direct link to heading">#</a></h4><p>Tipo da carga para o estudo de fluxo de carga, podendo ser selecionado dois tipos: Potência constante e Impedância constante. As cargas de potência constantes são inseridas nos vetores de potência e permanecem invariantes no cálculo, enquanto que para as cargas de impedância constante determina-se o valor da impedância utilizando a potência e tensão nominal, a qual é inserida na matriz admitância.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/load.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/transformer"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Transformador</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/capacitor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Capacitor »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#carga-no-psp-ufu" class="table-of-contents__link">Carga no PSP-UFU</a><ul><li><a href="#carga-no-estudo-de-estabilidade" class="table-of-contents__link">Carga no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-das-cargas" class="table-of-contents__link">Formulário de edição das cargas</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Carga</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><ol><li>a potência ativa, reativa ou aparente gerada, transmitida ou distribuída dentro de um sistema;</li><li>a potência demandada por um grupo de consumidores classificados de acordo com suas particularidades e características, por exemplo, carga de aquecimento, carga reativa diurna, etc. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=601-01-15" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</li></ol></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="carga-no-psp-ufu"></a>Carga no PSP-UFU<a class="hash-link" href="#carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As cargas são os elementos de potência consumidores de potência genéricos. Nos estudos de fluxo de carga é possível modelar a carga como <strong>potência ou impedância constante</strong> e em estabilidade pode-se compor a carga na proporção desejada em três tipos (<strong>carga ZIP</strong>): potência constante, impedância constante e corrente constante.</p><p>As cargas de potência constante nos estudos de fluxo de carga são inseridos no vetor de potências e permanecem fixos durante toda a simulação.</p><p>Os elementos shunt, como reatores, banco de capacitores e cargas de impedância constante, são representados simplesmente por suas admitâncias em derivação. Como os parâmetros de entrada geralmente se constituem de suas potências ativa (P) e reativa (Q) nominais (no caso de reatores e capacitores, P=0), deve-se, portanto, encontrar sua admitância com a equação abaixo para inseri-la na matriz admitância:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mfrac><mrow><mi>P</mi><mo>−</mo><mi>j</mi><mi>Q</mi></mrow><msup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\overline{y} = \frac{P-jQ}{\dot{V}^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.825em;vertical-align:-0.19444em"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.17052em;vertical-align:-0.8101900000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em"><span style="top:-2.18981em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord mathdefault">Q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8101900000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Para os estudos de <strong>curto-circuito</strong> e <strong>harmônicos</strong> as cargas são modeladas como impedância constante, independentemente da sinalização indicada em seu formulário de edição de dados.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="carga-no-estudo-de-estabilidade"></a>Carga no estudo de estabilidade<a class="hash-link" href="#carga-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A modelagem das cargas do sistema de forma exata se torna impraticável em um problemas de estabilidade, visto à grande quantidade e variedade dos componentes envolvidos, além de apresentar modificações em suas composições causado por vários fatores, como tempo, condições climáticas e economia. Portanto várias aproximações devem ser utilizadas na formulação de um modelo de carga nos estudos aqui postos.</p><p>Uma forma interessante de se representar cargas dependentes da tensão são as cargas ZIP, as quais são funções quadráticas, possuindo três parcelas: impedância constante (Z), corrente constante (I) e potência constante (P). As potências ativa e reativa são obtidas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>L</mi></msub><mo>=</mo><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><mrow><msub><mi>V</mi><mi>i</mi></msub><mn>0</mn></mrow></mfrac><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_L=P_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+P_{I0} \left(\frac{V_i}{V_i0} \right)+P_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.604038em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.6540080000000001em"><span style="top:-3.9029000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>L</mi></msub><mo>=</mo><msub><mi>Q</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>Q</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><mrow><msub><mi>V</mi><mi>i</mi></msub><mn>0</mn></mrow></mfrac><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>Q</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_L=Q_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+Q_{I0} \left(\frac{V_i}{V_i0} \right)+Q_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.604038em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.6540080000000001em"><span style="top:-3.9029000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">P_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">Q_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	são as potências ativa e reativa da carga ZIP, respectivamente, conectadas em uma barra genérica i;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão atual de uma barra genérica i;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">V_{i0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão inicial de uma barra genérica i, obtida do estudo de fluxo de carga;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{Z0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{Z0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de impedância constante para potências ativa e reativa, respectivamente;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{I0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{I0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de corrente constante para potências ativa e reativa, respectivamente;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de potência constante para potências ativa e reativa, respectivamente.</p><p>Os valores das parcelas de potência da carga ZIP são obtidos utilizando as potências da carga após a convergência do cálculo do fluxo de carga (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{L0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, para a potência ativa e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{L0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, para a potência reativa), utilizando as seguintes expressões:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>Z</mi></mrow></msub><mn>100</mn></mfrac><mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub><msubsup><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow><mn>2</mn></msubsup></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>I</mi></mrow></msub><mn>100</mn></mfrac><mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>P</mi></mrow></msub><mn>100</mn></mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mstyle></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} P_{Z0}=\displaystyle \frac{k_{PZ}}{100} \frac{P_{L0}}{V_{i0}^2}\\ P_{I0}=\displaystyle \frac{k_{PI}}{100} \frac{P_{L0}}{V_{i0}}\\ P_{P0}=\displaystyle \frac{k_{PP}}{100} P_{L0} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.60004em;vertical-align:-3.05002em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5500200000000004em"><span style="top:-0.9999899999999999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-0.99499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.2899900000000002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.5849900000000003em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.8799900000000005em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.1749900000000006em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.2049900000000004em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.1500100000000004em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.29501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.59001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.885010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.180010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.475010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.50501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.80002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.05002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5495919999999996em"><span style="top:-5.549592em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9628639999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.215288em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.007848em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.0495919999999996em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PZ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PP}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> representam a composição em impedância, corrente e potência constantes, respectivamente, da parcela ativa da carga, os quais são inseridos pelo usuário na forma de porcentagem, sendo a soma desses três valores necessariamente igual a 100%. Os parâmetros <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QZ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QP}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> podem ser interpretados de forma análoga, porém compõe a parcela reativa da carga.</p><p>O comportamento das potências, retiradas de simulações no PSP-UFU, de cada parcela que compõe a carga ZIP são evidenciadas na figura abaixo.</p><div><center><img src="/PSP/images/loadModel.svg" alt="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente" title="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente"></center></div><p>Cargas de corrente e potência constantes possuem problemas em tensões muito baixas. À medida que a tensão diminui as correntes dessas cargas não reduz, como pode ser observado na figura acima, resultando em perda de precisão e problemas na convergência de processos iterativos. Para contornar esse problema utiliza-se uma tensão pré-definida pelo usuário (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>l</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow><annotation encoding="application/x-tex">V_{low}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02691em">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), a qual as cargas (ou parcelas) de corrente e potência constantes são modeladas como impedância constante, resultando no comportamento de potência e corrente apresentados na figura acima.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-das-cargas"></a>Formulário de edição das cargas<a class="hash-link" href="#formulário-de-edição-das-cargas" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das cargas:</p><div><center><img src="/PSP/images/loadForm.png" alt="Formulário das cargas no PSP-UFU" title="Formulário das cargas no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais das cargas, informações e o tipo de carga no fluxo de carga;</li><li><strong>Estabilidade</strong>: contendo opções de visualização de dados da carga em gráficos no tempo e opções de parametrização da carga ZIP.</li></ul><p>Além desses dois contextos, pode ser observado o botão &quot;Chaveamento&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da carga durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/loadSw.png" alt="Formulário de chaveamento da carga" title="Formulário de chaveamento da carga"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Chaveamento</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-ativa"></a>Potência ativa<a class="hash-link" href="#potência-ativa" title="Direct link to heading">#</a></h4><p>Parcela de potência ativa da carga. Pode ser inserido em MW, kW W ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base do sistema).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-reativa"></a>Potência reativa<a class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Parcela de potência reativa da carga. Pode ser inserido em Mvar, kvar var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base do sistema).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tipo-de-carga-fluxo-de-carga"></a>Tipo de carga (fluxo de carga)<a class="hash-link" href="#tipo-de-carga-fluxo-de-carga" title="Direct link to heading">#</a></h4><p>Tipo da carga para o estudo de fluxo de carga, podendo ser selecionado dois tipos: Potência constante e Impedância constante. As cargas de potência constantes são inseridas nos vetores de potência e permanecem invariantes no cálculo, enquanto que para as cargas de impedância constante determina-se o valor da impedância utilizando a potência e tensão nominal, a qual é inserida na matriz admitância.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-dados"></a>Imprimir dados<a class="hash-link" href="#imprimir-dados" title="Direct link to heading">#</a></h4><p>Exibe os dados da carga nos gráficos no tempo.
+Os seguintes dados são exibidos:</p><ul><li>módulo da tensão</li><li>corrente</li><li>potências ativa e reativa.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-composição-de-carga-zip"></a>Utilizar composição de carga ZIP<a class="hash-link" href="#utilizar-composição-de-carga-zip" title="Direct link to heading">#</a></h4><p>Permite compor a carga ZIP para estudos de estabilidade. Caso essa opção não esteja selecionada é usada a composição global definida nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>Por padrão, todas as cargas no PSP-UFU são definidas como impedância constante no estudo de estabilidade.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso todas as barras possuírem o mesmo comportamento de carga ZIP, a parametrização poderá ser realizada nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> para todas as cargas, sem a necessidade de parametrização individual.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="composição-de-potência-ativa-e-reativa"></a>Composição de potência ativa e reativa<a class="hash-link" href="#composição-de-potência-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Valores da composição da carga ZIP, inseridas para potência ativa e reativa separadamente, em porcentagem. A soma da composição deve ser igual a 100%, caso contrário será exibida uma mensagem de erro ao usuário.</p></div><div role="tabpanel" hidden=""><p>O botão &quot;Chaveamento&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/load.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/transformer"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Transformador</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/capacitor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Capacitor »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#carga-no-psp-ufu" class="table-of-contents__link">Carga no PSP-UFU</a><ul><li><a href="#carga-no-estudo-de-estabilidade" class="table-of-contents__link">Carga no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-das-cargas" class="table-of-contents__link">Formulário de edição das cargas</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Tela Principal</h1></header><div class="markdown"><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="organização-da-tela-principal"></a>Organização da tela principal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#organização-da-tela-principal" title="Direct link to heading">#</a></h2><p>A tela principal do PSP-UFU é dividida em três seções: <em><a href="/PSP/docs/mainScreen#menu-ribbon">Menu &quot;Ribbon&quot;</a></em>, <em><a href="/PSP/docs/mainScreen#%C3%A1rea-de-trabalho">Área de Trabalho</a></em> e <em><a href="/PSP/docs/mainScreen#barra-de-status">Barra de Status</a></em>.</p><div><center><img src="/PSP/images/mainScreen.svg" alt="Tela principal" title="Tela principal do PSP-UFU no Windows"></center></div><p>Resumidamente, no <strong>Menu Ribbon</strong> são acessadas as ferramentas do PSP-UFU; na <strong>Área de Trabalho</strong> são criados os diagramas unifilares de potência; e na <strong>Barra de Status</strong> são exibidas importantes informações acerca do estado da Área de Trabalho.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="menu-ribbon"></a>Menu Ribbon<a aria-hidden="true" tabindex="-1" class="hash-link" href="#menu-ribbon" title="Direct link to heading">#</a></h3><blockquote><p>Em informática o ribbon (faixa, em inglês) é um formato de apresentação de interface baseada na GUI onde a barra de ferramentas é mostrada através de uma barra mais larga com icones maiores possibilitando o uso dos aplicativos por dispositivos <em>touch screen</em>.
-<a href="https://pt.wikipedia.org/wiki/Ribbon_(inform%C3%A1tica)" target="_blank" rel="noopener noreferrer">Wikipedia</a></p></blockquote><p>O menu Ribbon é composto de três submenus: “Arquivo”, “Ferramentas” e “Simulação”.</p><div><center><img src="/PSP/images/menuFile.png" alt="Menu Arquivo" title="Menu Arquivo"></center></div><p>Em <strong>Arquivo</strong> estão as ferramentas gerais do programa, as quais permitem ao usuário criar e salvar novos projetos, além daquelas que possibilitam carregar os projetos já existentes e importar arquivos de outros programas. Esse submenu também possui opções gerais do programa e acesso a informações sobre o programa.</p><div><center><img src="/PSP/images/menuTools.png" alt="Menu Ferramentas" title="Menu Ferramentas"></center></div><p>Em <strong>Ferramentas</strong> estão presentes os itens específicos, os quais estão relacionados à criação e manipulação da rede elétrica de potência, relatórios tabulares de saída de dados e acesso ao formulário de criação de gráficos no tempo. Algumas configurações específicas do projeto ativo são acessadas por esse menu.</p><div><center><img src="/PSP/images/menuSimulation.png" alt="Menu Simulação" title="Menu Simulação"></center></div><p>Finalmente, o submenu <strong>Simulação</strong> proporciona o acesso do usuário aos cálculos realizados pelo <em>software</em> e suas configurações, além da ferramenta de “solução contínua”, a qual resolve os cálculos estáticos (fluxo de potência, curto-circuito e harmônicos) após quaisquer mudanças na rede de potência, como remoção de componente ou alteração de seus dados. Todas as configurações de simulação do projeto ativo são acessadas por esse menu.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="área-de-trabalho"></a>Área de Trabalho<a aria-hidden="true" tabindex="-1" class="hash-link" href="#área-de-trabalho" title="Direct link to heading">#</a></h3><p>Na área de trabalho estão presentes os projetos em execução, os quais são divididos por abas identificadas pelo nome do projeto (caso já esteja gravado no disco). É nessa área que são inseridos e excluídos os elementos elétricos, cujas ferramentas são acessadas por meio do <a href="/PSP/docs/mainScreen#menu-ribbon">Menu Ribbon</a>.</p><p>As edições gráficas dos elementos, como posição, conexão e tamanho dos barramentos, são realizadas por meio da ação <em>drag-and-drop</em> (arrastar e soltar) com a utilização do mouse. A edição dos dados elétricos é feita por meio de formulários de dados acessados clicando duas vezes sobre os elementos.</p><div><center><img src="/PSP/images/workspace.png" alt="Área de Trabalho" title="Área de Trabalho"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="barra-de-status"></a>Barra de Status<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barra-de-status" title="Direct link to heading">#</a></h3><p>A barra de status é responsável por informações interessantes sobre as circunstâncias atuais do programa: modo de operação do mouse (editar, mover, arrastar), zoom aplicado e posição do mouse na área de trabalho, além de dados sobre as ações dos usuários, como: inserção de elementos, informação sobre copiar e colar, etc.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p><strong>Sempre fique atendo à barra de status</strong>, pois são fornecidas informações importantes acerca da operação do programa. Caso tenha alguma dúvida siga as instruções apresentadas nesse componente.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="configurações-gerais"></a>Configurações gerais<a aria-hidden="true" tabindex="-1" class="hash-link" href="#configurações-gerais" title="Direct link to heading">#</a></h2><p>As configurações gerais do programa são acessadas no submenu Ribbon <strong>Arquivo</strong>. Essas configurações são aplicadas para todos os projetos e permanecem gravadas no disco.</p><div><center><img src="/PSP/images/generalSettings.png" alt="Configurações gerais" title="Configurações gerais"></center></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado</h5></div><div class="admonition-content"><p>Algumas configurações de segurança do seu computador (principalmente em sistemas Windows) podem gerar uma mensagem de erro ao confirmar as alterações das configurações gerais.</p><p>Para resolver esse problema basta <strong>executar o PSP-UFU como administrador</strong> (clicar com botão direito no atalho do programa e posteriormente em executar como administrador). Altere novamente as configurações e reinicie o programa.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="idioma"></a>Idioma<a aria-hidden="true" tabindex="-1" class="hash-link" href="#idioma" title="Direct link to heading">#</a></h3><p>Atualmente os seguintes idiomas estão disponíveis no PSP-UFU:</p><ul><li>Inglês</li><li>Português</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O programa deve ser reiniciado para surtir efeito da alteração do idioma.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="renderização"></a>Renderização<a aria-hidden="true" tabindex="-1" class="hash-link" href="#renderização" title="Direct link to heading">#</a></h3><p>Define como os elementos gráficos do <a href="/PSP/docs/powerEditor">editor de potência</a> e do <a href="/PSP/docs/controlEditor">editor de controle</a> são desenhados na tela. Atualmente duas opções estão disponíveis:</p><ul><li><a href="https://www.opengl.org/about/" target="_blank" rel="noopener noreferrer">OpenGL</a></li><li><a href="https://docs.wxwidgets.org/3.0/classwx_graphics_context.html" target="_blank" rel="noopener noreferrer">Device Context</a></li></ul><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>o OpenGL pode não ser suportado pelo seu computador. Nesse caso, ao criar um novo projeto a tela de trabalho não é exibida ou é exibida uma tela preta. A solução é alterar o renderizador para &quot;Device Context&quot;. </p><p>Caso apareça uma mensagem de erro, execute o programa como administrador e repita o processo.</p></div></div><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O programa deve ser reiniciado para surtir efeito de alteração do renderizador.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="arquivos-de-projeto-do-psp-ufu"></a>Arquivos de projeto do PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#arquivos-de-projeto-do-psp-ufu" title="Direct link to heading">#</a></h2><p>As opções de criação, gravação e abertura de projetos no disco, assim como importação de arquivos de outros programas estão presentes no submenu Ribbon <strong>Arquivo</strong>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="novo-projeto"></a>Novo projeto<a aria-hidden="true" tabindex="-1" class="hash-link" href="#novo-projeto" title="Direct link to heading">#</a></h3><p>A criação de um novo projeto é realizada clicando no botão <strong>Novo projeto</strong> no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Arquivo</strong></a>. Essa ação cria um sistema em branco na área de trabalho, local onde é possível inserir os elementos elétricos por meio do <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Ferramentas</strong></a> ou pelas <a href="/PSP/docs/powerEditor#teclas-de-atalho">teclas de atalho</a>.</p><p>Esse sistema em branco pode ser ciado utilizando o <a href="/PSP/docs/powerEditor">Editor de Potência</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="salvar-e-salvar-como"></a>Salvar e Salvar como...<a aria-hidden="true" tabindex="-1" class="hash-link" href="#salvar-e-salvar-como" title="Direct link to heading">#</a></h3><p>A opção &quot;Salvar&quot; sobrepõe as alterações realizadas no projeto aberto e grava no disco. A opção &quot;Salvar como...&quot; cria um novo arquivo e grava o projeto no disco com o auxílio de uma janela de seleção de pasta (e definição do nome do arquivo).</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Para projetos que estão sendo gravados pela primeira vez a opção &quot;Salvar&quot; se comporta de forma idêntica à opção &quot;Salvar como...&quot;.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>O PSP-UFU grava os arquivos com a extensão <em>.psp</em>. Esses arquivos nada mais são que arquivos de texto utilizando a <a href="https://en.wikipedia.org/wiki/Markup_language" target="_blank" rel="noopener noreferrer">linguagem de marcação</a> XML (<a href="https://en.wikipedia.org/wiki/XML" target="_blank" rel="noopener noreferrer">eXtensible Markup Language</a>), que define uma série de regras de formatação dos dados de forma que eles são tanto legíveis por humanos quanto por máquinas.</p><p>Portanto, os dados elétricos contidos neles podem ser facilmente identificados e alterados, caso necessário.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="abrir-projeto"></a>Abrir projeto<a aria-hidden="true" tabindex="-1" class="hash-link" href="#abrir-projeto" title="Direct link to heading">#</a></h3><p>Essa opção abre os projetos gravados no disco por meio com o auxílio de uma janela de seleção de arquivos. </p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="importar-projeto"></a>Importar projeto<a aria-hidden="true" tabindex="-1" class="hash-link" href="#importar-projeto" title="Direct link to heading">#</a></h3><p>O PSP-UFU permite a importação de arquivos dos seguintes programas:</p><ul><li><a href="http://www.cepel.br/pt_br/produtos/programas-computacionais-por-categoria/menu/anarede-analise-de-redes-eletricas.htm" target="_blank" rel="noopener noreferrer">ANAREDE</a></li><li><a href="https://matpower.org/" target="_blank" rel="noopener noreferrer">Matpower</a></li></ul><p>Para importação dos arquivos do <strong>ANAREDE</strong> são utilizados tanto o arquivo de dados elétricos (<em>.pwf</em>) quanto de dados gráficos dos elementos (<em>.lst</em>).</p><p>Uma vez que o arquivo do <strong>Matpower</strong> (<em>.m</em>) não possui dados gráficos dos elementos elétricos, o diagrama unifilar é automaticamente gerado pelo PSP-UFU ao importá-lo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="geração-automática-do-layout-de-diagramas-unifilares"></a>Geração automática do <em>layout</em> de diagramas unifilares<a aria-hidden="true" tabindex="-1" class="hash-link" href="#geração-automática-do-layout-de-diagramas-unifilares" title="Direct link to heading">#</a></h4><p>Para criar o <em>layout</em> automático, utilizou-se a teoria dos grafos baseada no posicionamento direcionado à força e aplicação de grafos com arestas ponderadas. De acordo com a teoria dos grafos, o sistema é modelado como vértices e arestas. Nesse contexto, os barramentos são os vértices e os ramos do sistema (linhas e transformadores) são as arestas. As localizações de elementos de derivação, assim como os nós dos elementos, são automaticamente controladas pelo PSP-UFU.</p><p>A ferramenta de <em>layout</em> automático é composta por um processo iterativo e os resultados são mais refinados com um número maior de iterações. Os melhores resultados são obtidos utilizando o número de iterações igual ou superior a cinco vezes o número de barramentos do sistema.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O <em>layout</em> automático ainda está em desenvolvimento e algumas melhorias podem ser feitas para evitar cruzamentos e sobreposições, porém, em seu estado atual, é totalmente funcional e satisfatório.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Uma vez que o Matpower pode converter os formatos CDF (<em>Common Data Format</em>) e PSS/E RAW para arquivos <em>.m</em>, estes podem ser importados de forma indireta para o PSP-UFU.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a 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tabindex="-1" href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulations</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Tela Principal</h1></header><div class="markdown"><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="organização-da-tela-principal"></a>Organização da tela principal<a class="hash-link" href="#organização-da-tela-principal" title="Direct link to heading">#</a></h2><p>A tela principal do PSP-UFU é dividida em três seções: <em><a href="/PSP/docs/mainScreen#menu-ribbon">Menu &quot;Ribbon&quot;</a></em>, <em><a href="/PSP/docs/mainScreen#%C3%A1rea-de-trabalho">Área de Trabalho</a></em> e <em><a href="/PSP/docs/mainScreen#barra-de-status">Barra de Status</a></em>.</p><div><center><img src="/PSP/images/mainScreen.svg" alt="Tela principal" title="Tela principal do PSP-UFU no Windows"></center></div><p>Resumidamente, no <strong>Menu Ribbon</strong> são acessadas as ferramentas do PSP-UFU; na <strong>Área de Trabalho</strong> são criados os diagramas unifilares de potência; e na <strong>Barra de Status</strong> são exibidas importantes informações acerca do estado da Área de Trabalho.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="menu-ribbon"></a>Menu Ribbon<a class="hash-link" href="#menu-ribbon" title="Direct link to heading">#</a></h3><blockquote><p>Em informática o ribbon (faixa, em inglês) é um formato de apresentação de interface baseada na GUI onde a barra de ferramentas é mostrada através de uma barra mais larga com icones maiores possibilitando o uso dos aplicativos por dispositivos <em>touch screen</em>.
+<a href="https://pt.wikipedia.org/wiki/Ribbon_(inform%C3%A1tica)" target="_blank" rel="noopener noreferrer">Wikipedia</a></p></blockquote><p>O menu Ribbon é composto de três submenus: “Arquivo”, “Ferramentas” e “Simulação”.</p><div><center><img src="/PSP/images/menuFile.png" alt="Menu Arquivo" title="Menu Arquivo"></center></div><p>Em <strong>Arquivo</strong> estão as ferramentas gerais do programa, as quais permitem ao usuário criar e salvar novos projetos, além daquelas que possibilitam carregar os projetos já existentes e importar arquivos de outros programas. Esse submenu também possui opções gerais do programa e acesso a informações sobre o programa.</p><div><center><img src="/PSP/images/menuTools.png" alt="Menu Ferramentas" title="Menu Ferramentas"></center></div><p>Em <strong>Ferramentas</strong> estão presentes os itens específicos, os quais estão relacionados à criação e manipulação da rede elétrica de potência, relatórios tabulares de saída de dados e acesso ao formulário de criação de gráficos no tempo. Algumas configurações específicas do projeto ativo são acessadas por esse menu.</p><div><center><img src="/PSP/images/menuSimulation.png" alt="Menu Simulação" title="Menu Simulação"></center></div><p>Finalmente, o submenu <strong>Simulação</strong> proporciona o acesso do usuário aos cálculos realizados pelo <em>software</em> e suas configurações, além da ferramenta de “solução contínua”, a qual resolve os cálculos estáticos (fluxo de potência, curto-circuito e harmônicos) após quaisquer mudanças na rede de potência, como remoção de componente ou alteração de seus dados. Todas as configurações de simulação do projeto ativo são acessadas por esse menu.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="área-de-trabalho"></a>Área de Trabalho<a class="hash-link" href="#área-de-trabalho" title="Direct link to heading">#</a></h3><p>Na área de trabalho estão presentes os projetos em execução, os quais são divididos por abas identificadas pelo nome do projeto (caso já esteja gravado no disco). É nessa área que são inseridos e excluídos os elementos elétricos, cujas ferramentas são acessadas por meio do <a href="/PSP/docs/mainScreen#menu-ribbon">Menu Ribbon</a>.</p><p>As edições gráficas dos elementos, como posição, conexão e tamanho dos barramentos, são realizadas por meio da ação <em>drag-and-drop</em> (arrastar e soltar) com a utilização do mouse. A edição dos dados elétricos é feita por meio de formulários de dados acessados clicando duas vezes sobre os elementos.</p><div><center><img src="/PSP/images/workspace.png" alt="Área de Trabalho" title="Área de Trabalho"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barra-de-status"></a>Barra de Status<a class="hash-link" href="#barra-de-status" title="Direct link to heading">#</a></h3><p>A barra de status é responsável por informações interessantes sobre as circunstâncias atuais do programa: modo de operação do mouse (editar, mover, arrastar), zoom aplicado e posição do mouse na área de trabalho, além de dados sobre as ações dos usuários, como: inserção de elementos, informação sobre copiar e colar, etc.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p><strong>Sempre fique atendo à barra de status</strong>, pois são fornecidas informações importantes acerca da operação do programa. Caso tenha alguma dúvida siga as instruções apresentadas nesse componente.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="configurações-gerais"></a>Configurações gerais<a class="hash-link" href="#configurações-gerais" title="Direct link to heading">#</a></h2><p>As configurações gerais do programa são acessadas no submenu Ribbon <strong>Arquivo</strong>. Essas configurações são aplicadas para todos os projetos e permanecem gravadas no disco.</p><div><center><img src="/PSP/images/generalSettings.png" alt="Configurações gerais" title="Configurações gerais"></center></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado</h5></div><div class="admonition-content"><p>Algumas configurações de segurança do seu computador (principalmente em sistemas Windows) podem gerar uma mensagem de erro ao confirmar as alterações das configurações gerais.</p><p>Para resolver esse problema basta <strong>executar o PSP-UFU como administrador</strong> (clicar com botão direito no atalho do programa e posteriormente em executar como administrador). Altere novamente as configurações e reinicie o programa.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="idioma"></a>Idioma<a class="hash-link" href="#idioma" title="Direct link to heading">#</a></h3><p>Atualmente os seguintes idiomas estão disponíveis no PSP-UFU:</p><ul><li>Inglês</li><li>Português</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O programa deve ser reiniciado para surtir efeito da alteração do idioma.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="renderização"></a>Renderização<a class="hash-link" href="#renderização" title="Direct link to heading">#</a></h3><p>Define como os elementos gráficos do <a href="/PSP/docs/powerEditor">editor de potência</a> e do <a href="/PSP/docs/controlEditor">editor de controle</a> são desenhados na tela. Atualmente duas opções estão disponíveis:</p><ul><li><a href="https://www.opengl.org/about/" target="_blank" rel="noopener noreferrer">OpenGL</a></li><li><a href="https://docs.wxwidgets.org/3.0/classwx_graphics_context.html" target="_blank" rel="noopener noreferrer">Device Context</a></li></ul><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>o OpenGL pode não ser suportado pelo seu computador. Nesse caso, ao criar um novo projeto a tela de trabalho não é exibida ou é exibida uma tela preta. A solução é alterar o renderizador para &quot;Device Context&quot;. </p><p>Caso apareça uma mensagem de erro, execute o programa como administrador e repita o processo.</p></div></div><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O programa deve ser reiniciado para surtir efeito de alteração do renderizador.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="arquivos-de-projeto-do-psp-ufu"></a>Arquivos de projeto do PSP-UFU<a class="hash-link" href="#arquivos-de-projeto-do-psp-ufu" title="Direct link to heading">#</a></h2><p>As opções de criação, gravação e abertura de projetos no disco, assim como importação de arquivos de outros programas estão presentes no submenu Ribbon <strong>Arquivo</strong>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="novo-projeto"></a>Novo projeto<a class="hash-link" href="#novo-projeto" title="Direct link to heading">#</a></h3><p>A criação de um novo projeto é realizada clicando no botão <strong>Novo projeto</strong> no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Arquivo</strong></a>. Essa ação cria um sistema em branco na área de trabalho, local onde é possível inserir os elementos elétricos por meio do <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Ferramentas</strong></a> ou pelas <a href="/PSP/docs/powerEditor#teclas-de-atalho">teclas de atalho</a>.</p><p>Esse sistema em branco pode ser ciado utilizando o <a href="/PSP/docs/powerEditor">Editor de Potência</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="salvar-e-salvar-como"></a>Salvar e Salvar como...<a class="hash-link" href="#salvar-e-salvar-como" title="Direct link to heading">#</a></h3><p>A opção &quot;Salvar&quot; sobrepõe as alterações realizadas no projeto aberto e grava no disco. A opção &quot;Salvar como...&quot; cria um novo arquivo e grava o projeto no disco com o auxílio de uma janela de seleção de pasta (e definição do nome do arquivo).</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Para projetos que estão sendo gravados pela primeira vez a opção &quot;Salvar&quot; se comporta de forma idêntica à opção &quot;Salvar como...&quot;.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>O PSP-UFU grava os arquivos com a extensão <em>.psp</em>. Esses arquivos nada mais são que arquivos de texto utilizando a <a href="https://en.wikipedia.org/wiki/Markup_language" target="_blank" rel="noopener noreferrer">linguagem de marcação</a> XML (<a href="https://en.wikipedia.org/wiki/XML" target="_blank" rel="noopener noreferrer">eXtensible Markup Language</a>), que define uma série de regras de formatação dos dados de forma que eles são tanto legíveis por humanos quanto por máquinas.</p><p>Portanto, os dados elétricos contidos neles podem ser facilmente identificados e alterados, caso necessário.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="abrir-projeto"></a>Abrir projeto<a class="hash-link" href="#abrir-projeto" title="Direct link to heading">#</a></h3><p>Essa opção abre os projetos gravados no disco por meio com o auxílio de uma janela de seleção de arquivos. </p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="importar-projeto"></a>Importar projeto<a class="hash-link" href="#importar-projeto" title="Direct link to heading">#</a></h3><p>O PSP-UFU permite a importação de arquivos dos seguintes programas:</p><ul><li><a href="http://www.cepel.br/pt_br/produtos/programas-computacionais-por-categoria/menu/anarede-analise-de-redes-eletricas.htm" target="_blank" rel="noopener noreferrer">ANAREDE</a></li><li><a href="https://matpower.org/" target="_blank" rel="noopener noreferrer">Matpower</a></li></ul><p>Para importação dos arquivos do <strong>ANAREDE</strong> são utilizados tanto o arquivo de dados elétricos (<em>.pwf</em>) quanto de dados gráficos dos elementos (<em>.lst</em>).</p><p>Uma vez que o arquivo do <strong>Matpower</strong> (<em>.m</em>) não possui dados gráficos dos elementos elétricos, o diagrama unifilar é automaticamente gerado pelo PSP-UFU ao importá-lo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="geração-automática-do-layout-de-diagramas-unifilares"></a>Geração automática do <em>layout</em> de diagramas unifilares<a class="hash-link" href="#geração-automática-do-layout-de-diagramas-unifilares" title="Direct link to heading">#</a></h4><p>Para criar o <em>layout</em> automático, utilizou-se a teoria dos grafos baseada no posicionamento direcionado à força e aplicação de grafos com arestas ponderadas. De acordo com a teoria dos grafos, o sistema é modelado como vértices e arestas. Nesse contexto, os barramentos são os vértices e os ramos do sistema (linhas e transformadores) são as arestas. As localizações de elementos de derivação, assim como os nós dos elementos, são automaticamente controladas pelo PSP-UFU.</p><p>A ferramenta de <em>layout</em> automático é composta por um processo iterativo e os resultados são mais refinados com um número maior de iterações. Os melhores resultados são obtidos utilizando o número de iterações igual ou superior a cinco vezes o número de barramentos do sistema.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O <em>layout</em> automático ainda está em desenvolvimento e algumas melhorias podem ser feitas para evitar cruzamentos e sobreposições, porém, em seu estado atual, é totalmente funcional e satisfatório.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Uma vez que o Matpower pode converter os formatos CDF (<em>Common Data Format</em>) e PSS/E RAW para arquivos <em>.m</em>, estes podem ser importados de forma indireta para o PSP-UFU.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a 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class="table-of-contents__link">Novo projeto</a></li><li><a href="#salvar-e-salvar-como" class="table-of-contents__link">Salvar e Salvar como...</a></li><li><a href="#abrir-projeto" class="table-of-contents__link">Abrir projeto</a></li><li><a href="#importar-projeto" class="table-of-contents__link">Importar projeto</a></li></ul></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a 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docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Expressão Matemática</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O bloco de expressão matemática utiliza a biblioteca externa <a href="http://warp.povusers.org/FunctionParser/" target="_blank" rel="noopener noreferrer">fparser</a> e permite ao usuário inserir expressões matemáticas genéricas de qualquer complexidade combinadas a estruturas condicionais em um diagrama de controle, aumentando significativamente a capacidade de generalização do programa.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="sintaxe-das-expressões-matemáticas"></a>Sintaxe das expressões matemáticas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#sintaxe-das-expressões-matemáticas" title="Direct link to heading">#</a></h2><p>As sintaxes das operações (para expressões <code>A</code> e <code>B</code>) são apresentadas na tabela abaixo:</p><table><thead><tr><th>Operador</th><th>Expressão</th></tr></thead><tbody><tr><td><code>( )</code></td><td>Expressão em parênteses primeiro</td></tr><tr><td><code>A unit</code></td><td>Um multiplicador de unidade (se um estiver sido adicionado)</td></tr><tr><td><code>A^B</code></td><td>Exponenciação (<code>A</code> elevado à potência <code>B</code>)</td></tr><tr><td><code>-A</code></td><td>Operação unária de oposto</td></tr><tr><td><code>!A</code></td><td>Operação lógica unária de negação (resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> se <code>int(A)</code> é <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, senão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>)</td></tr><tr><td><code>A*B</code> <code>A/B</code> <code>A%B</code></td><td>Multiplicação, divisão e módulo</td></tr><tr><td><code>A+B</code> <code>A-B</code></td><td>Adição e subtração</td></tr><tr><td><code>A=B</code> <code>A&lt;B</code> <code>A&lt;=B</code> <code>A!=B</code> <code>A&gt;B</code> <code>A&gt;=B</code></td><td>Comparação entre <code>A</code> e <code>B</code> (resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>)</td></tr><tr><td><code>A&amp;B</code></td><td>Resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> se <code>int(A)</code> e <code>int(B)</code> forem diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, senão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></td></tr><tr><td><code>A</code>|<code>B</code></td><td>Resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> se <code>int(A)</code> ou <code>int(B)</code> forem diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, senão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></td></tr></tbody></table><p>As <strong>funções matemáticas</strong> suportadas pela biblioteca, as quais podem ser utilizadas na construção de diagrama de blocos, são descritas na tabela abaixo:</p><table><thead><tr><th>Expressão</th><th>Descrição</th></tr></thead><tbody><tr><td><code>abs(A)</code></td><td>Valor absoluto de <code>A</code>. Para números reais, se <code>A</code> é negativo, retorna <code>–A</code>, senão retorna <code>A</code>. Para números complexos a expressão é equivalente a <code>hypot(real(x),imag(x))</code>.</td></tr><tr><td><code>acos(A)</code></td><td>Arco cosseno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>acosh(A)</code></td><td>O mesmo que <code>acos()</code> mas para cosseno hiperbólico</td></tr><tr><td><code>arg(A)</code></td><td>Ângulo de fase de um número complexo <code>A</code>.</td></tr><tr><td><code>asin(A)</code></td><td>Arco seno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>asinh(A)</code></td><td>Mesmo que <code>asin()</code>, mas para seno hiperbólico</td></tr><tr><td><code>atan(A)</code></td><td>Arco tangente de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>atan2(A,B)</code></td><td>Arco tangente de <code>A/B</code>, a qual os sinais dos dois argumentos para determinar o quadrante do resultado. Retorna a solução de do seguinte sistema: <code>hypot(A,B)*sin(x)=A, hypot(A,B)*cos(x)=B</code>. O valor retornado varia de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>π</mi></mrow><annotation encoding="application/x-tex">-\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span>.</td></tr><tr><td><code>atanh(A)</code></td><td>Mesmo que <code>atan()</code>, mas para tangente hiperbólica.</td></tr><tr><td><code>cbrt(A)</code></td><td>Raiz cúbica de <code>A</code>.</td></tr><tr><td><code>conj(A)</code></td><td>Conjugado complexo de <code>A</code>.</td></tr><tr><td><code>ceil(A)</code></td><td>Teto de <code>A</code>. Arredonda para o próximo maior inteiro.</td></tr><tr><td><code>cos(A)</code></td><td>Cosseno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>cosh(A)</code></td><td>Mesmo que <code>cos()</code>, mas para cosseno hiperbólico</td></tr><tr><td><code>cot(A)</code></td><td>Cotangente de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>csc(A)</code></td><td>Cossecante de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>exp(A)</code></td><td>Exponencial de <code>A</code>. Retorna o valor de e elevado a potência A.</td></tr><tr><td><code>exp2(A)</code></td><td>Exponencial de <code>A</code> na base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span>.</td></tr><tr><td><code>floor(A)</code></td><td>Piso de <code>A</code>. Arredonda para o próximo menor inteiro.</td></tr><tr><td><code>hypot(A,B)</code></td><td>Função de distância Euclidiana.</td></tr><tr><td><code>if(A,B,C)</code></td><td>Se <code>int(A)</code> é diferente de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, retorna o valor de <code>B</code>, senão retorna <code>C</code>.</td></tr><tr><td><code>imag(A)</code></td><td>Retorna a parte imaginária do número complexo <code>A</code>.</td></tr><tr><td><code>int(A)</code></td><td>Arredonda <code>A</code> para o inteiro mais próximo.</td></tr><tr><td><code>log(A)</code></td><td>Logaritmo natural (base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">e</span></span></span></span></span>) de <code>A</code>.</td></tr><tr><td><code>log2(A)</code></td><td>Logaritmo na base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span> de <code>A</code>.</td></tr><tr><td><code>log10(A)</code></td><td>Logaritmo na base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">0</span></span></span></span></span> de A.</td></tr><tr><td><code>max(A,B)</code></td><td>Se <code>A&gt;B</code>, o resultado é <code>A</code>, senão é <code>B</code>.</td></tr><tr><td><code>min(A,B)</code></td><td>Se <code>A&lt;B</code>, o resultado é <code>A</code>, senão é <code>B</code>.</td></tr><tr><td><code>polar(A,B)</code></td><td>Retorna o número complexo de magnitude <code>A</code> e ângulo de fase <code>B</code> (em radianos).</td></tr><tr><td><code>pow(A,B)</code></td><td>Exponenciação (<code>A</code> elevado à potência <code>B</code>)</td></tr><tr><td><code>real(A,B)</code></td><td>Retorna a parte real do número complexo <code>A</code>.</td></tr><tr><td><code>sec(A)</code></td><td>Secante de <code>A</code>.</td></tr><tr><td><code>sin(A)</code></td><td>Seno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>sinh(A)</code></td><td>Mesmo que <code>sin()</code>, mas para seno hiperbólico.</td></tr><tr><td><code>sqrt(A)</code></td><td>Raiz quadrada de <code>A</code>.</td></tr><tr><td><code>tan(A)</code></td><td>Tangente de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>tanh(A)</code></td><td>Mesmo que <code>tan()</code>, mas para tangente hiperbólica.</td></tr><tr><td><code>trunc(A)</code></td><td>Valor truncado de <code>A</code>. Retorna o número inteiro de <code>A</code> sem a parcela fracionada.</td></tr></tbody></table><p>É possível atribuir novas variáveis utilizando a seguinte sintaxe: </p><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token operator" style="color:rgb(137, 221, 255)">&lt;</span><span class="token plain">nome da variável</span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">&lt;</span><span class="token plain">expressão</span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">&lt;</span><span class="token plain">função</span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;</span></div></div></div></div></div><p>Por exemplo:</p><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">comprimento </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain">y</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">y</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">comprimento</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token function" style="color:rgb(130, 170, 255)">sin</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">comprimento</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span></div></div></div></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>O caractere de espaço e de nova linha são ignorados na interpretação da expressão, portanto para maior organização, o código anterior pode ser escrito da seguinte forma:</p><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">comprimento </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain">y</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">y</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain"></span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">comprimento</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token function" style="color:rgb(130, 170, 255)">sin</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">comprimento</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span></div></div></div></div></div></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Note que a expressão que determina o valor de saída do bloco <strong>não possui</strong> o caractere <code>;</code> em seu final.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-de-dados-do-bloco-de-expressão-matemática"></a>Formulário de edição de dados do bloco de Expressão Matemática<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-do-bloco-de-expressão-matemática" title="Direct link to heading">#</a></h2><p>O formulário de inserção e edição das entradas do bloco e da expressão matemática genérica, assim como ferramentas de auxílio de sua construção é apresentado na figura abaixo.</p><div><center><img src="/PSP/images/mathExpressionForm.png" alt="Formulário de edição de dados do bloco de expressão matemática no PSP-UFU" title="Formulário de edição de dados do bloco de expressão matemática no PSP-UFU"></center></div><p>No campo de “Variáveis de entrada” é inserida uma lista com os nomes das entradas separados por espaços. Um número qualquer de entrada pode ser definido nessa lista esses nomes são apresentados no ícone gráfico presente no editor de controle, além de serem destacados na expressão inserida pelo usuário. O número de entradas e saídas se comporta de forma semelhante aos <a href="/PSP/docs/sum">blocos somadores</a>.</p><p>Abaixo do campo das variáveis de entrada está presente o local para inserção da expressão matemática. A sintaxe da expressão inserida pelo usuário possui realce (por meio de diferentes formas e cores da fonte) para números, operadores, variáveis de entrada, funções e constantes, facilitando a criação, manipulação e identificação de erros de digitação e lógica.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Como ferramenta de auxílio ao usuário foi desenvolvida uma verificação da expressão inserida.</p></div></div><p>Tal ferramenta irá encontrar erros e indicará ao usuário qual o tipo do erro, além da sua localização, destacando-o. A figura anterior exemplifica a identificação automática do erro pelo PSP-UFU, assim como sua posição na expressão inserida.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="exemplo-de-expressão-matemática"></a>Exemplo de Expressão Matemática<a aria-hidden="true" tabindex="-1" class="hash-link" href="#exemplo-de-expressão-matemática" title="Direct link to heading">#</a></h2><p>A corrente de campo em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> pode ser estimada de forma aproximada utilizando as potências ativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span>) e reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span>), assim como as reatâncias transitórias de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">x_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">x_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) e o módulo da tensão terminal (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>) da máquina:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>i</mi><mi>f</mi></msub><mo>≈</mo><msqrt><mrow><msup><mrow><mo fence="true">(</mo><mi>V</mi><mo>+</mo><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>P</mi><mn>2</mn></msup></mrow></msqrt><mo>+</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>x</mi><mi>d</mi></msub><msub><mi>x</mi><mi>q</mi></msub></mfrac><mo>+</mo><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mo fence="true">)</mo></mrow><mo>×</mo><mfrac><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mrow><mo fence="true">(</mo><mi>V</mi><mo>+</mo><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mo>+</mo><msup><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mn>2</mn></msup></mrow><msqrt><mrow><msup><mrow><mo fence="true">(</mo><mi>V</mi><mo>+</mo><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>P</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mrow><annotation encoding="application/x-tex">i_f \approx \sqrt{ \left( V + Q&#x27; \right)^2 + P^2} + \left(\frac{x_d}{x_q} +1{,}0 \right) \times \frac{Q&#x27; \left( V + Q&#x27; \right) + {P&#x27;}^2}{\sqrt{\left( V + Q&#x27;\right)^2 + P^2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9456279999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.454121em"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.385879em"><span class="svg-align" style="top:-3.8em"><span class="pstrut" style="height:3.8em"></span><span class="mord" style="padding-left:1em"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6778919999999999em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954008em"><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.345879em"><span class="pstrut" style="height:3.8em"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em"><svg width="400em" height="1.8800000000000001em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Expressão Matemática</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O bloco de expressão matemática utiliza a biblioteca externa <a href="http://warp.povusers.org/FunctionParser/" target="_blank" rel="noopener noreferrer">fparser</a> e permite ao usuário inserir expressões matemáticas genéricas de qualquer complexidade combinadas a estruturas condicionais em um diagrama de controle, aumentando significativamente a capacidade de generalização do programa.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="sintaxe-das-expressões-matemáticas"></a>Sintaxe das expressões matemáticas<a class="hash-link" href="#sintaxe-das-expressões-matemáticas" title="Direct link to heading">#</a></h2><p>As sintaxes das operações (para expressões <code>A</code> e <code>B</code>) são apresentadas na tabela abaixo:</p><table><thead><tr><th>Operador</th><th>Expressão</th></tr></thead><tbody><tr><td><code>( )</code></td><td>Expressão em parênteses primeiro</td></tr><tr><td><code>A unit</code></td><td>Um multiplicador de unidade (se um estiver sido adicionado)</td></tr><tr><td><code>A^B</code></td><td>Exponenciação (<code>A</code> elevado à potência <code>B</code>)</td></tr><tr><td><code>-A</code></td><td>Operação unária de oposto</td></tr><tr><td><code>!A</code></td><td>Operação lógica unária de negação (resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> se <code>int(A)</code> é <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, senão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>)</td></tr><tr><td><code>A*B</code> <code>A/B</code> <code>A%B</code></td><td>Multiplicação, divisão e módulo</td></tr><tr><td><code>A+B</code> <code>A-B</code></td><td>Adição e subtração</td></tr><tr><td><code>A=B</code> <code>A&lt;B</code> <code>A&lt;=B</code> <code>A!=B</code> <code>A&gt;B</code> <code>A&gt;=B</code></td><td>Comparação entre <code>A</code> e <code>B</code> (resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>)</td></tr><tr><td><code>A&amp;B</code></td><td>Resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> se <code>int(A)</code> e <code>int(B)</code> forem diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, senão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></td></tr><tr><td><code>A</code>|<code>B</code></td><td>Resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> se <code>int(A)</code> ou <code>int(B)</code> forem diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, senão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></td></tr></tbody></table><p>As <strong>funções matemáticas</strong> suportadas pela biblioteca, as quais podem ser utilizadas na construção de diagrama de blocos, são descritas na tabela abaixo:</p><table><thead><tr><th>Expressão</th><th>Descrição</th></tr></thead><tbody><tr><td><code>abs(A)</code></td><td>Valor absoluto de <code>A</code>. Para números reais, se <code>A</code> é negativo, retorna <code>–A</code>, senão retorna <code>A</code>. Para números complexos a expressão é equivalente a <code>hypot(real(x),imag(x))</code>.</td></tr><tr><td><code>acos(A)</code></td><td>Arco cosseno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>acosh(A)</code></td><td>O mesmo que <code>acos()</code> mas para cosseno hiperbólico</td></tr><tr><td><code>arg(A)</code></td><td>Ângulo de fase de um número complexo <code>A</code>.</td></tr><tr><td><code>asin(A)</code></td><td>Arco seno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>asinh(A)</code></td><td>Mesmo que <code>asin()</code>, mas para seno hiperbólico</td></tr><tr><td><code>atan(A)</code></td><td>Arco tangente de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>atan2(A,B)</code></td><td>Arco tangente de <code>A/B</code>, a qual os sinais dos dois argumentos para determinar o quadrante do resultado. Retorna a solução de do seguinte sistema: <code>hypot(A,B)*sin(x)=A, hypot(A,B)*cos(x)=B</code>. O valor retornado varia de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>π</mi></mrow><annotation encoding="application/x-tex">-\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span>.</td></tr><tr><td><code>atanh(A)</code></td><td>Mesmo que <code>atan()</code>, mas para tangente hiperbólica.</td></tr><tr><td><code>cbrt(A)</code></td><td>Raiz cúbica de <code>A</code>.</td></tr><tr><td><code>conj(A)</code></td><td>Conjugado complexo de <code>A</code>.</td></tr><tr><td><code>ceil(A)</code></td><td>Teto de <code>A</code>. Arredonda para o próximo maior inteiro.</td></tr><tr><td><code>cos(A)</code></td><td>Cosseno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>cosh(A)</code></td><td>Mesmo que <code>cos()</code>, mas para cosseno hiperbólico</td></tr><tr><td><code>cot(A)</code></td><td>Cotangente de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>csc(A)</code></td><td>Cossecante de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>exp(A)</code></td><td>Exponencial de <code>A</code>. Retorna o valor de e elevado a potência A.</td></tr><tr><td><code>exp2(A)</code></td><td>Exponencial de <code>A</code> na base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span>.</td></tr><tr><td><code>floor(A)</code></td><td>Piso de <code>A</code>. Arredonda para o próximo menor inteiro.</td></tr><tr><td><code>hypot(A,B)</code></td><td>Função de distância Euclidiana.</td></tr><tr><td><code>if(A,B,C)</code></td><td>Se <code>int(A)</code> é diferente de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, retorna o valor de <code>B</code>, senão retorna <code>C</code>.</td></tr><tr><td><code>imag(A)</code></td><td>Retorna a parte imaginária do número complexo <code>A</code>.</td></tr><tr><td><code>int(A)</code></td><td>Arredonda <code>A</code> para o inteiro mais próximo.</td></tr><tr><td><code>log(A)</code></td><td>Logaritmo natural (base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">e</span></span></span></span></span>) de <code>A</code>.</td></tr><tr><td><code>log2(A)</code></td><td>Logaritmo na base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span> de <code>A</code>.</td></tr><tr><td><code>log10(A)</code></td><td>Logaritmo na base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">0</span></span></span></span></span> de A.</td></tr><tr><td><code>max(A,B)</code></td><td>Se <code>A&gt;B</code>, o resultado é <code>A</code>, senão é <code>B</code>.</td></tr><tr><td><code>min(A,B)</code></td><td>Se <code>A&lt;B</code>, o resultado é <code>A</code>, senão é <code>B</code>.</td></tr><tr><td><code>polar(A,B)</code></td><td>Retorna o número complexo de magnitude <code>A</code> e ângulo de fase <code>B</code> (em radianos).</td></tr><tr><td><code>pow(A,B)</code></td><td>Exponenciação (<code>A</code> elevado à potência <code>B</code>)</td></tr><tr><td><code>real(A,B)</code></td><td>Retorna a parte real do número complexo <code>A</code>.</td></tr><tr><td><code>sec(A)</code></td><td>Secante de <code>A</code>.</td></tr><tr><td><code>sin(A)</code></td><td>Seno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>sinh(A)</code></td><td>Mesmo que <code>sin()</code>, mas para seno hiperbólico.</td></tr><tr><td><code>sqrt(A)</code></td><td>Raiz quadrada de <code>A</code>.</td></tr><tr><td><code>tan(A)</code></td><td>Tangente de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>tanh(A)</code></td><td>Mesmo que <code>tan()</code>, mas para tangente hiperbólica.</td></tr><tr><td><code>trunc(A)</code></td><td>Valor truncado de <code>A</code>. Retorna o número inteiro de <code>A</code> sem a parcela fracionada.</td></tr></tbody></table><p>É possível atribuir novas variáveis utilizando a seguinte sintaxe: </p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token operator" style="color:rgb(137, 221, 255)">&lt;</span><span class="token plain">nome da variável</span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">&lt;</span><span class="token plain">expressão</span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">&lt;</span><span class="token plain">função</span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Por exemplo:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">comprimento </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain">y</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">y</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">comprimento</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token function" style="color:rgb(130, 170, 255)">sin</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">comprimento</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>O caractere de espaço e de nova linha são ignorados na interpretação da expressão, portanto para maior organização, o código anterior pode ser escrito da seguinte forma:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">comprimento </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain">y</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">y</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain"></span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">comprimento</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token function" style="color:rgb(130, 170, 255)">sin</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">comprimento</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Note que a expressão que determina o valor de saída do bloco <strong>não possui</strong> o caractere <code>;</code> em seu final.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-bloco-de-expressão-matemática"></a>Formulário de edição de dados do bloco de Expressão Matemática<a class="hash-link" href="#formulário-de-edição-de-dados-do-bloco-de-expressão-matemática" title="Direct link to heading">#</a></h2><p>O formulário de inserção e edição das entradas do bloco e da expressão matemática genérica, assim como ferramentas de auxílio de sua construção é apresentado na figura abaixo.</p><div><center><img src="/PSP/images/mathExpressionForm.png" alt="Formulário de edição de dados do bloco de expressão matemática no PSP-UFU" title="Formulário de edição de dados do bloco de expressão matemática no PSP-UFU"></center></div><p>No campo de “Variáveis de entrada” é inserida uma lista com os nomes das entradas separados por espaços. Um número qualquer de entrada pode ser definido nessa lista esses nomes são apresentados no ícone gráfico presente no editor de controle, além de serem destacados na expressão inserida pelo usuário. O número de entradas e saídas se comporta de forma semelhante aos <a href="/PSP/docs/sum">blocos somadores</a>.</p><p>Abaixo do campo das variáveis de entrada está presente o local para inserção da expressão matemática. A sintaxe da expressão inserida pelo usuário possui realce (por meio de diferentes formas e cores da fonte) para números, operadores, variáveis de entrada, funções e constantes, facilitando a criação, manipulação e identificação de erros de digitação e lógica.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Como ferramenta de auxílio ao usuário foi desenvolvida uma verificação da expressão inserida.</p></div></div><p>Tal ferramenta irá encontrar erros e indicará ao usuário qual o tipo do erro, além da sua localização, destacando-o. A figura anterior exemplifica a identificação automática do erro pelo PSP-UFU, assim como sua posição na expressão inserida.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="exemplo-de-expressão-matemática"></a>Exemplo de Expressão Matemática<a class="hash-link" href="#exemplo-de-expressão-matemática" title="Direct link to heading">#</a></h2><p>A corrente de campo em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> pode ser estimada de forma aproximada utilizando as potências ativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span>) e reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span>), assim como as reatâncias transitórias de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">x_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">x_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) e o módulo da tensão terminal (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>) da máquina:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>i</mi><mi>f</mi></msub><mo>≈</mo><msqrt><mrow><msup><mrow><mo fence="true">(</mo><mi>V</mi><mo>+</mo><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>P</mi><mn>2</mn></msup></mrow></msqrt><mo>+</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>x</mi><mi>d</mi></msub><msub><mi>x</mi><mi>q</mi></msub></mfrac><mo>+</mo><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mo fence="true">)</mo></mrow><mo>×</mo><mfrac><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mrow><mo fence="true">(</mo><mi>V</mi><mo>+</mo><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mo>+</mo><msup><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mn>2</mn></msup></mrow><msqrt><mrow><msup><mrow><mo fence="true">(</mo><mi>V</mi><mo>+</mo><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>P</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mrow><annotation encoding="application/x-tex">i_f \approx \sqrt{ \left( V + Q&#x27; \right)^2 + P^2} + \left(\frac{x_d}{x_q} +1{,}0 \right) \times \frac{Q&#x27; \left( V + Q&#x27; \right) + {P&#x27;}^2}{\sqrt{\left( V + Q&#x27;\right)^2 + P^2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9456279999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.454121em"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.385879em"><span class="svg-align" style="top:-3.8em"><span class="pstrut" style="height:3.8em"></span><span class="mord" style="padding-left:1em"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6778919999999999em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954008em"><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.345879em"><span class="pstrut" style="height:3.8em"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em"><svg width="400em" height="1.8800000000000001em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7
@@ -36,16 +38,16 @@ c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30
 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722
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-M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.502996em"><span></span></span></span></span></span></span></span><span style="top:-3.5670040000000003em"><span class="pstrut" style="height:3.3370040000000003em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-4.014004em"><span class="pstrut" style="height:3.3370040000000003em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9559em"><span style="top:-3.2047920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.73em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mi>q</mi></msub><mo>×</mo><mfrac><mi>P</mi><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">P&#x27; = x_q \times \frac{P}{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.217331em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.872331em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mi>q</mi></msub><mo>×</mo><mfrac><mi>Q</mi><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">Q&#x27; = x_q \times \frac{Q}{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.269439em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.924439em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.446108em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li></ul><p>Primeiramente, deve-se inserir no campo &quot;variáveis de entrada&quot; as grandezas fornecidas pelo PSP-UFU. Nesse caso, como pode ser observado no bloco de <a href="/PSP/docs/io">entrada / saída</a>, todas as variáveis necessárias são fornecidas: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>. Essas variáveis devem ser inseridas separadas por espaço: <code>p q v</code>.</p><p>Utilizando três blocos de <a href="/PSP/docs/io">entrada / saída</a> pode-se fornecer tais dados ao conectá-los ao bloco de expressão matemática.</p><p>As reatâncias transitórias de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">x_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">x_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) podem ser definidas diretamente na expressão matemática:</p><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.146</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">xq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.0969</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div></div></div><p>Os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">P&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">Q&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> podem ser também calculados:</p><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">yp </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> p </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> q </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div></div></div><p>Com todos os dados necessários pode-se calcular a corrente de campo:</p><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">1.0</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">       </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yp</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div></div></div><p>Finalmente, pode-se definir a saída do bloco (não inserindo <code>;</code>):</p><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f</span></div></div></div></div></div><p>Portanto, para as entradas definidas como <code>p q v</code>, a expressão matemática total para cálculo da corrente de campo será:</p><div class="mdxCodeBlock_1XEh"><div class="codeBlockContent_1u-d"><button type="button" aria-label="Copy code to clipboard" class="copyButton_10dd">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_3iAC"><div class="codeBlockLines_b7E3" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.146</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">xq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.0969</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yp </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> p </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> q </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">1.0</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">       </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yp</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f</span></div></div></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ul><li>NIEMINEN, J.; YLILUOMA, J. Function Parser for C++ v4.5.2. Disponível em: <a href="http://warp.povusers.org/FunctionParser/" target="_blank" rel="noopener noreferrer">http://warp.povusers.org/FunctionParser/</a>. </li></ul></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/mathExpression.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/gain"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Ganho</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/powerFlow"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Fluxo de Potência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#sintaxe-das-expressões-matemáticas" class="table-of-contents__link">Sintaxe das expressões matemáticas</a></li><li><a href="#formulário-de-edição-de-dados-do-bloco-de-expressão-matemática" class="table-of-contents__link">Formulário de edição de dados do bloco de Expressão Matemática</a></li><li><a href="#exemplo-de-expressão-matemática" class="table-of-contents__link">Exemplo de Expressão Matemática</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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+M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.502996em"><span></span></span></span></span></span></span></span><span style="top:-3.5670040000000003em"><span class="pstrut" style="height:3.3370040000000003em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-4.014004em"><span class="pstrut" style="height:3.3370040000000003em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9559em"><span style="top:-3.2047920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.73em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mi>q</mi></msub><mo>×</mo><mfrac><mi>P</mi><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">P&#x27; = x_q \times \frac{P}{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.217331em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.872331em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mi>q</mi></msub><mo>×</mo><mfrac><mi>Q</mi><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">Q&#x27; = x_q \times \frac{Q}{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.269439em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.924439em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.446108em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li></ul><p>Primeiramente, deve-se inserir no campo &quot;variáveis de entrada&quot; as grandezas fornecidas pelo PSP-UFU. Nesse caso, como pode ser observado no bloco de <a href="/PSP/docs/io">entrada / saída</a>, todas as variáveis necessárias são fornecidas: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>. Essas variáveis devem ser inseridas separadas por espaço: <code>p q v</code>.</p><p>Utilizando três blocos de <a href="/PSP/docs/io">entrada / saída</a> pode-se fornecer tais dados ao conectá-los ao bloco de expressão matemática.</p><p>As reatâncias transitórias de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">x_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">x_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) podem ser definidas diretamente na expressão matemática:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.146</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">xq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.0969</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">P&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">Q&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> podem ser também calculados:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">yp </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> p </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> q </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Com todos os dados necessários pode-se calcular a corrente de campo:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">1.0</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">       </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yp</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Finalmente, pode-se definir a saída do bloco (não inserindo <code>;</code>):</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Portanto, para as entradas definidas como <code>p q v</code>, a expressão matemática total para cálculo da corrente de campo será:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.146</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">xq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.0969</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yp </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> p </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> q </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">1.0</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">       </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yp</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ul><li>NIEMINEN, J.; YLILUOMA, J. Function Parser for C++ v4.5.2. Disponível em: <a href="http://warp.povusers.org/FunctionParser/" target="_blank" rel="noopener noreferrer">http://warp.povusers.org/FunctionParser/</a>. </li></ul></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/mathExpression.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/gain"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Ganho</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/powerFlow"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Fluxo de Potência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#sintaxe-das-expressões-matemáticas" class="table-of-contents__link">Sintaxe das expressões matemáticas</a></li><li><a href="#formulário-de-edição-de-dados-do-bloco-de-expressão-matemática" class="table-of-contents__link">Formulário de edição de dados do bloco de Expressão Matemática</a></li><li><a href="#exemplo-de-expressão-matemática" class="table-of-contents__link">Exemplo de Expressão Matemática</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Multiplicador</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Esse elemento é bastante simples e cumpre com a função matemáticas de seu nome.</p><p>Possui duas entradas de valores reais e sua saída é computada na sequência do nó superior pelo inferior, ou seja, a saída será o valor presente no nó 1 multiplicado pelo nó 2.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Nós não conectados no multiplicador são assumidos com o valor unitário e para o divisor é considerado valor nulo.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/multiplier.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Editor de Potência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O Editor de Elementos de Potência no PSP-UFU permite a inserção, manipulação e edição de elementos elétricos de potência por meio de um diagrama unifilar.</p><blockquote><p>(<strong>Diagrama unifilar</strong>) Um diagrama do sistema no qual as linhas polifásicas são representados por sua única linha equivalente. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=601-02-04" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Todos os elementos inseridos pode ser manipulados e editados utilizando as <a href="/PSP/docs/cadTools">Ferramentas CAD</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="novo-projeto"></a>Novo projeto<a aria-hidden="true" tabindex="-1" class="hash-link" href="#novo-projeto" title="Direct link to heading">#</a></h2><p>A criação de um novo projeto é realizada clicando no item <strong>Novo projeto</strong> no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Arquivo</strong></a>. Essa ação cria um sistema em branco na área de trabalho, local onde é possível inserir os elementos elétricos por meio do <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Ferramentas</strong></a> ou pelas <a href="/PSP/docs/powerEditor#teclas-de-atalho">teclas de atalho</a>.</p><p>Abaixo são apresentados os elementos elétricos de potência que podem ser inseridos no PSP-UFU:</p><ul><li><a href="/PSP/docs/bus">Barramento</a></li><li>Elementos de ramo<ul><li><a href="/PSP/docs/line">Linha</a></li><li><a href="/PSP/docs/transformer">Transformador</a></li></ul></li><li>Elementos Shunt<ul><li><a href="/PSP/docs/load">Carga</a></li><li><a href="/PSP/docs/capacitor">Capacitor</a></li><li><a href="/PSP/docs/inductor">Indutor</a></li><li><a href="/PSP/docs/harmSource">Fonte de Corrente Harmônica</a></li></ul></li><li>Máquinas<ul><li><a href="/PSP/docs/syncGenerator">Gerador Síncrono</a></li><li><a href="/PSP/docs/syncMotor">Compensador Síncrono</a></li><li><a href="/PSP/docs/indMotor">Motor de Indução</a></li></ul></li></ul><div><center><img src="/PSP/images/powerElements.svg" alt="Elementos de potência" title="Elementos de potência"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="inserindo-elementos"></a>Inserindo elementos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#inserindo-elementos" title="Direct link to heading">#</a></h2><p>Por meio do submenu <strong>Ferramentas</strong> pode-se inserir os elementos de potência, como é apresentado na imagem abaixo.</p><div><center><img src="/PSP/images/addPowerElements.png" alt="Adicionar elementos de potência" title="Adicionar elementos de potência"></center></div><p>No caso de inserção de um barramento, o usuário deve apenas clicar na posição desejada, já para os outros componentes será solicitado que o usuário selecione em uma ou duas barras, o qual o novo componente será conectado.</p><p>Sempre que são inseridos novos elementos, um formulário de dados é exibido e pode-se editar os dados elétricos e acessar formulários adicionais, como dados de estabilidade.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O primeiro elemento que a ser inserido no projeto <strong>deve</strong> ser um barramento. Os demais elementos elétricos devem ser conectados em uma ou mais barras.</p></div></div><p>Além dos elementos de potência pode-se inserir os elementos de <em><a href="/PSP/docs/text">texto vinculado</a></em>, os quais estão associados a uma grandeza que se deseja vizualizar diretamente na <a href="/PSP/docs/mainScreen#%C3%A1rea-de-trabalho">Área de Trabalho</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao inserir um barramento, a mesma irá acompanhar o ponteiro do mouse até que você clique na posição desejada. Para os demais elementos <strong>deve-se clicar em uma mais barras</strong> para exibir o elemento inserido.</p><p><strong>Sempre fique atento às instruções da <a href="/PSP/docs/mainScreen#barra-de-status">Barra de Status</a>.</strong></p></div></div><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/timelapseBuild.mp4" type="video/mp4"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="teclas-de-atalho"></a>Teclas de atalho<a aria-hidden="true" tabindex="-1" class="hash-link" href="#teclas-de-atalho" title="Direct link to heading">#</a></h3><p>É possível inserir todos os elementos de potência por meio de teclas de atalho, facilitando e agilizando a criação dos diagramas unifilares no PSP-UFU. A tabela abaixo apresenta tais atalhos:</p><table><thead><tr><th>Elemento</th><th align="center">Atalho</th><th>Elemento</th><th align="center">Atalho</th></tr></thead><tbody><tr><td>Barramento</td><td align="center">B</td><td>Carga</td><td align="center">Shift + L</td></tr><tr><td>Linha</td><td align="center">L</td><td>Capacitor</td><td align="center">Shift + C</td></tr><tr><td>Transformador</td><td align="center">T</td><td>Indutor</td><td align="center">Shift + I</td></tr><tr><td>Gerador</td><td align="center">G</td><td>Corrente harmônica</td><td align="center">Shift + H</td></tr><tr><td>Motor de indução</td><td align="center">I</td><td>Texto</td><td align="center">A</td></tr><tr><td>Compensador síncrono</td><td align="center">K</td><td></td><td align="center"></td></tr></tbody></table><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="editando-dados-elétricos"></a>Editando dados elétricos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#editando-dados-elétricos" title="Direct link to heading">#</a></h2><p>Ao inserir um elemento, seu respectivo formulário de edição é exibido para inserção dos dados elétricos d elemento. Todos os dados podem ser <strong>alterados</strong> ao <strong>clicar duas vezes sobre o elemento</strong> ou por opção do <strong>menu de contexto</strong> exibido ao clicar com o botão direito sobre o elemento.</p><p>Cada formulário de edição de dados elétricos dos elementos possuem suas particularidades e são descritos em documentação específica: <a href="/PSP/docs/bus">Barramento</a>, <a href="/PSP/docs/line">Linha</a>, <a href="/PSP/docs/transformer">Transformador</a>, <a href="/PSP/docs/load">Carga</a>, <a href="/PSP/docs/capacitor">Capacitor</a>, <a href="/PSP/docs/inductor">Indutor</a>, <a href="/PSP/docs/harmSource">Fonte de Corrente Harmônica</a>, <a href="/PSP/docs/syncGenerator">Gerador Síncrono</a>, <a href="/PSP/docs/syncMotor">Compensador Síncrono</a>, <a href="/PSP/docs/indMotor">Motor de Indução</a>.</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/powerEditor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/cadTools"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Ferramentas CAD</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/bus"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Barramento »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#novo-projeto" class="table-of-contents__link">Novo projeto</a></li><li><a href="#inserindo-elementos" class="table-of-contents__link">Inserindo elementos</a><ul><li><a href="#teclas-de-atalho" class="table-of-contents__link">Teclas de atalho</a></li></ul></li><li><a href="#editando-dados-elétricos" class="table-of-contents__link">Editando dados elétricos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/docs/installation">Installation</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mainScreen">Tela Principal</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/cadTools">Ferramentas CAD</a></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Power Element Editor</a><ul class="menu__list"><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/docs/powerEditor">Editor de Potência</a></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!" tabindex="0">Power Elements</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/bus">Barramento</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/line">Linha</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/transformer">Transformador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/load">Carga</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/capacitor">Capacitor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/inductor">Indutor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmSource">Fonte de Corrente Harmônica</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/syncGenerator">Gerador Síncrono</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/syncMotor">Compensador Síncrono</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/indMotor">Motor de Indução</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Control Element Editor</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/controlEditor">Editor de Controle</a></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!" tabindex="-1">Control Elements</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/io">Entrada / Saída</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/transferFunction">Função Transferência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/sum">Somador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/multiplier">Multiplicador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulations</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Editor de Potência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O Editor de Elementos de Potência no PSP-UFU permite a inserção, manipulação e edição de elementos elétricos de potência por meio de um diagrama unifilar.</p><blockquote><p>(<strong>Diagrama unifilar</strong>) Um diagrama do sistema no qual as linhas polifásicas são representados por sua única linha equivalente. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=601-02-04" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Todos os elementos inseridos pode ser manipulados e editados utilizando as <a href="/PSP/docs/cadTools">Ferramentas CAD</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="novo-projeto"></a>Novo projeto<a class="hash-link" href="#novo-projeto" title="Direct link to heading">#</a></h2><p>A criação de um novo projeto é realizada clicando no item <strong>Novo projeto</strong> no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Arquivo</strong></a>. Essa ação cria um sistema em branco na área de trabalho, local onde é possível inserir os elementos elétricos por meio do <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Ferramentas</strong></a> ou pelas <a href="/PSP/docs/powerEditor#teclas-de-atalho">teclas de atalho</a>.</p><p>Abaixo são apresentados os elementos elétricos de potência que podem ser inseridos no PSP-UFU:</p><ul><li><a href="/PSP/docs/bus">Barramento</a></li><li>Elementos de ramo<ul><li><a href="/PSP/docs/line">Linha</a></li><li><a href="/PSP/docs/transformer">Transformador</a></li></ul></li><li>Elementos Shunt<ul><li><a href="/PSP/docs/load">Carga</a></li><li><a href="/PSP/docs/capacitor">Capacitor</a></li><li><a href="/PSP/docs/inductor">Indutor</a></li><li><a href="/PSP/docs/harmSource">Fonte de Corrente Harmônica</a></li></ul></li><li>Máquinas<ul><li><a href="/PSP/docs/syncGenerator">Gerador Síncrono</a></li><li><a href="/PSP/docs/syncMotor">Compensador Síncrono</a></li><li><a href="/PSP/docs/indMotor">Motor de Indução</a></li></ul></li></ul><div><center><img src="/PSP/images/powerElements.svg" alt="Elementos de potência" title="Elementos de potência"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserindo-elementos"></a>Inserindo elementos<a class="hash-link" href="#inserindo-elementos" title="Direct link to heading">#</a></h2><p>Por meio do submenu <strong>Ferramentas</strong> pode-se inserir os elementos de potência, como é apresentado na imagem abaixo.</p><div><center><img src="/PSP/images/addPowerElements.png" alt="Adicionar elementos de potência" title="Adicionar elementos de potência"></center></div><p>No caso de inserção de um barramento, o usuário deve apenas clicar na posição desejada, já para os outros componentes será solicitado que o usuário selecione em uma ou duas barras, o qual o novo componente será conectado.</p><p>Sempre que são inseridos novos elementos, um formulário de dados é exibido e pode-se editar os dados elétricos e acessar formulários adicionais, como dados de estabilidade.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O primeiro elemento que a ser inserido no projeto <strong>deve</strong> ser um barramento. Os demais elementos elétricos devem ser conectados em uma ou mais barras.</p></div></div><p>Além dos elementos de potência pode-se inserir os elementos de <em><a href="/PSP/docs/text">texto vinculado</a></em>, os quais estão associados a uma grandeza que se deseja vizualizar diretamente na <a href="/PSP/docs/mainScreen#%C3%A1rea-de-trabalho">Área de Trabalho</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao inserir um barramento, a mesma irá acompanhar o ponteiro do mouse até que você clique na posição desejada. Para os demais elementos <strong>deve-se clicar em uma mais barras</strong> para exibir o elemento inserido.</p><p><strong>Sempre fique atento às instruções da <a href="/PSP/docs/mainScreen#barra-de-status">Barra de Status</a>.</strong></p></div></div><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/timelapseBuild.mp4" type="video/mp4"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="teclas-de-atalho"></a>Teclas de atalho<a class="hash-link" href="#teclas-de-atalho" title="Direct link to heading">#</a></h3><p>É possível inserir todos os elementos de potência por meio de teclas de atalho, facilitando e agilizando a criação dos diagramas unifilares no PSP-UFU. A tabela abaixo apresenta tais atalhos:</p><table><thead><tr><th>Elemento</th><th align="center">Atalho</th><th>Elemento</th><th align="center">Atalho</th></tr></thead><tbody><tr><td>Barramento</td><td align="center">B</td><td>Carga</td><td align="center">Shift + L</td></tr><tr><td>Linha</td><td align="center">L</td><td>Capacitor</td><td align="center">Shift + C</td></tr><tr><td>Transformador</td><td align="center">T</td><td>Indutor</td><td align="center">Shift + I</td></tr><tr><td>Gerador</td><td align="center">G</td><td>Corrente harmônica</td><td align="center">Shift + H</td></tr><tr><td>Motor de indução</td><td align="center">I</td><td>Texto</td><td align="center">A</td></tr><tr><td>Compensador síncrono</td><td align="center">K</td><td></td><td align="center"></td></tr></tbody></table><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="editando-dados-elétricos"></a>Editando dados elétricos<a class="hash-link" href="#editando-dados-elétricos" title="Direct link to heading">#</a></h2><p>Ao inserir um elemento, seu respectivo formulário de edição é exibido para inserção dos dados elétricos d elemento. Todos os dados podem ser <strong>alterados</strong> ao <strong>clicar duas vezes sobre o elemento</strong> ou por opção do <strong>menu de contexto</strong> exibido ao clicar com o botão direito sobre o elemento.</p><p>Cada formulário de edição de dados elétricos dos elementos possuem suas particularidades e são descritos em documentação específica: <a href="/PSP/docs/bus">Barramento</a>, <a href="/PSP/docs/line">Linha</a>, <a href="/PSP/docs/transformer">Transformador</a>, <a href="/PSP/docs/load">Carga</a>, <a href="/PSP/docs/capacitor">Capacitor</a>, <a href="/PSP/docs/inductor">Indutor</a>, <a href="/PSP/docs/harmSource">Fonte de Corrente Harmônica</a>, <a href="/PSP/docs/syncGenerator">Gerador Síncrono</a>, <a href="/PSP/docs/syncMotor">Compensador Síncrono</a>, <a href="/PSP/docs/indMotor">Motor de Indução</a>.</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/powerEditor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/cadTools"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Ferramentas CAD</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/bus"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Barramento »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#novo-projeto" class="table-of-contents__link">Novo projeto</a></li><li><a href="#inserindo-elementos" class="table-of-contents__link">Inserindo elementos</a><ul><li><a href="#teclas-de-atalho" class="table-of-contents__link">Teclas de atalho</a></li></ul></li><li><a href="#editando-dados-elétricos" class="table-of-contents__link">Editando dados elétricos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Fluxo de Potência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Um estudo fundamental no planejamento da expansão e operação de um sistema elétrico é o <strong>fluxo de potência (ou fluxo de carga)</strong> uma vez que a operação satisfatória desse sistema depende do conhecimento dos efeitos da interligação, de novas cargas, de novas centrais geradoras e de novas linhas antes que elas sejam instaladas. Esse estudo tem como objetivo encontrar o fluxo de potência nos ramos e as tensões nodais do Sistema Elétrico de Potência (SEP) impostas pela geração e a carga.</p><p>As equações formuladas a partir dos modelos dos elementos presentes no estudo de fluxo de carga são não lineares e não possuem solução analítica explícita. Portanto é necessária a utilização de métodos numéricos iterativos para solução do problema de fluxo de carga.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulação-do-problema-de-fluxo-de-carga"></a>Formulação do problema de fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulação-do-problema-de-fluxo-de-carga" title="Direct link to heading">#</a></h2><p>O problema de fluxo de carga pode ser representado por um sistema de equações e inequações algébricas não-lineares que correspondem às leis de Kirchhoff e a um conjunto de restrições de operação impostos pelos componentes de uma rede elétrica.
-Na formulação do problema a cada barra da rede são associadas quatro variáveis, sendo que duas delas surgem como dados e duas como incógnitas (em uma barra de índice <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>):</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a magnitude da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é o ângulo da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a injeção líquida de potência ativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a injeção líquida de potência reativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</li></ul><p>Relativo às variáveis que são incógnitas e os dados do sistema, podem ser definidos <strong>três tipos de barras</strong>:</p><ul><li><strong>Barra PQ</strong>:	<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra PV</strong>:	<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra de Referência</strong>	<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As <strong>barras PQ</strong> geralmente são representadas pelos barramentos de carga, os quais não há geração e controle de tensão.</p><p>As <strong>barras PV</strong> se caracterizam pelo controle de tensão mediante a injeção ou absorção de potência reativa por meio do controle da excitação de uma máquina síncrona.</p><p>A <strong>barra de Referência</strong> (ou de folga, de oscilação) tem como função, assim como o próprio nome diz, servir de referência de tensão e ângulo do sistema. Essa barra é necessariamente geradora, uma vez que ela é responsável pelo equilíbrio do balanço de potência do sistema.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O tipo de barra deve ser definido no <a href="/PSP/docs/bus">elemento barramento</a>.</p><p>Note que <strong>o sistema deve possuir somente uma barra de referência</strong>.</p></div></div><p>Como mencionado anteriormente, as equações são não-lineares e a solução analítica não é prática. As soluções dessas equações seguem <strong><a href="https://en.wikipedia.org/wiki/Iterative_method" target="_blank" rel="noopener noreferrer">processos iterativos</a></strong>, em que são atribuídos valores estimados (ou iniciais) para as barras com tensões desconhecidas e, baseado na potência ativa e reativa e módulo da tensão especificados, calcula-se por meio das equações previamente apresentadas as novas tensões complexas em cada nó do sistema.</p><p>Na sequência, esse conjunto de valores para as tensões em cada barra é utilizado para novamente calcular outro grupo de tensões. Cada cálculo de um novo conjunto de tensões é chamado iteração. O processo iterativo é repetido até que as mudanças em todas as barras sejam menores do que um valor pré-estipulado, obtendo assim a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="execução-do-fluxo-de-carga-no-psp-ufu"></a>Execução do fluxo de carga no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#execução-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do fluxo de carga é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Fluxo de carga</strong>.</p><div><center><img src="/PSP/images/menuSimulationPF.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso o fluxo de carga tenha sido executado com sucesso, as setas de potência serão exibidas, a barra de status indicará sucesso na operação e os <a href="/PSP/docs/text">elementos de texto</a> serão atualizados.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Evite construir o circuito com o cálculo contínuo habilitado, uma vez que configurações temporárias podem levar a erros de execução da simulação.</p><p>Para desabilitar o cálculo contínuo clique no botão <strong>Desabilitar solução</strong>.</p></div></div><p>Os resultados do fluxo de carga são exibidos nos <a href="/PSP/docs/text">elementos de texto vinculado</a>, ao posicionar o mouse sobre os elementos e em <a href="/PSP/docs/tabularReport">relatórios tabulares</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="erros-comuns-na-execução-do-fluxo-de-carga"></a>Erros comuns na execução do fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#erros-comuns-na-execução-do-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao fluxo de carga.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado"></a>A seguinte mensagem de erro é exibida: &quot;O número máximo de iterações foi alcançado&quot;<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando o método de solução numérica selecionado nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> atinge o número máximo de iterações inserido. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li><li><strong>O número máximo de iterações está muito baixo</strong>. Alguns circuitos exigem um número maior de iterações, portanto altere o valor do máximo de iterações nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>. Também tente alternar entre os métodos numéricos de solução disponíveis.</li><li><strong>Os parâmetros de simulação estão inadequados</strong>. Caso um parâmetro do método de solução esteja inadequado, como fator de aceleração ou tolerância, o cálculo pode não alcançar a convergência. Altere esses parâmetros nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="os-dados-de-saída-são-exibidos-como-nan-ou-nan"></a>Os dados de saída são exibidos como &quot;NaN&quot; ou &quot;nan&quot;<a aria-hidden="true" tabindex="-1" class="hash-link" href="#os-dados-de-saída-são-exibidos-como-nan-ou-nan" title="Direct link to heading">#</a></h4><p>Isso ocorre devido a erros de operações matemáticas nos cálculos de fluxo de carga. &quot;NaN&quot; significa <em>Not a Number</em>.</p><ul><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu"></a>Métodos de solução numérica do fluxo de carga no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Os métodos implementados no programa para solução do problema de fluxo de carga no PSP-UFU são <strong>Gauss-Seidel (GS)</strong> e <strong>Newton-Raphson (NR)</strong>. Além desses métodos clássicos, um método híbrido pode ser utilizado (definido nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>), em que é utilizado inicialmente o GS e na sequência o NR, aumentando a chance de convergência do NR.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="gauss-seidel"></a>Gauss-Seidel<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gauss-seidel" title="Direct link to heading">#</a></h3><p>O método de Gauss-Seidel tem sido bastante utilizado nas últimas décadas para solução do problema de fluxo de carga, uma vez que não há a necessidade de fatorar a matrizes, reduzindo o esforço computacional. Atualmente, restrições computacionais são menos problemáticas e outros métodos são normalmente escolhidos, porém o Gauss-Seidel ainda possui valor didático e, visto que o PSP-UFU também possui fins educacionais, optou-se pela implementação desse método.</p><p>Para iniciar as iterações do método são necessários valores iniciais para as tensões (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_i^0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) que devem ser calculadas. Normalmente para barras do tipo PQ tem-se <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=1{,}0+j0{,}0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> e para barras do tipo PV <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=V_i^{esp}+j0{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> [p.u.], em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">V_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> é o módulo da tensão especificada para a barra PV. A barra de referência tem o módulo e ângulo de tensão fixos e não participam no processo iterativo.</p><p>O método de Gauss-Seidel mostra um <strong>número excessivo de iterações</strong> e, com o intuito reduzi-los, multiplicam-se as correções de tensões por uma constante. Essa operação amplia o valor da correção, trazendo a tensão para mais perto do valor do valor final. Os multiplicadores que realizam essa convergência melhorada são chamados de fatores de aceleração. Para qualquer sistema existem valores ótimos para os fatores de aceleração e uma escolha inadequada pode resultar em uma convergência mais lenta ou torná-la impossível. Normalmente é utilizado um fator de aceleração igual a 1,6, valor definido como padrão no programa. A utilização do fator de aceleração é realizada por meio da equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mi>α</mi><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow><mo>+</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)} = \alpha \left( \dot{V}_{i}^{(v + 1)}- \dot{V}_{i_{AC}}^{(v)} \right) + \dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.27686399999999994em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração anterior com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span> é o fator de aceleração</li></ul><p>O fluxograma abaixo demonstra como o método de Gauss-Seidel foi implementado no PSP-UFU:</p><div><center><img src="/PSP/images/gaussSeidel.svg" alt="Método numérico de Gauss-Seidel para fluxo de carga" title="Método numérico de Gauss-Seidel para fluxo de carga"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="newton-raphson"></a>Newton-Raphson<a aria-hidden="true" tabindex="-1" class="hash-link" href="#newton-raphson" title="Direct link to heading">#</a></h3><p>O método de Newton-Raphson (também conhecido como método de Newton ou Newton-Fourier) para solução do fluxo de carga é descrito em vários livros e artigos. Atualmente é o algoritmo mais utilizado para solução do fluxo de carga. Para casos bem condicionados, esse método geralmente converge em 4 a 5 iterações, porém existe a possibilidade da técnica contornar o ponto da solução sem nunca atingi-la, o que também justifica implementação do método de Gauss-Seidel no <em>software</em>.</p><p>A expansão da <a href="https://en.wikipedia.org/wiki/Taylor_series" target="_blank" rel="noopener noreferrer">série de Taylor</a> para uma função de duas ou mais variáveis é a base do método de Newton-Raphson para resolver o problema do fluxo de carga. Fazendo a expansão em série de Taylor, para duas equações e duas incógnitas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_1\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_2\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span>), sem listar as derivadas parciais maiores que 1 na forma matricial, obtêm-se:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>2</mn></msub><mo>−</mo><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} K_1 - f_1\left( x_{1}^{(0)}, x_{2}^{(0)} \right)\\ K_2 - f_2\left( x_{1}^{(0)}, x_{2}^{(0)} \right) \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x_1} &amp; \displaystyle\frac{\partial f_1}{\partial x_2}\\ \displaystyle\frac{\partial f_2}{\partial x_1} &amp; \displaystyle\frac{\partial f_2}{\partial x_2} \end{bmatrix} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6010299999999997em;vertical-align:-1.55002em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em"><span style="top:-4.05002em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.41488em;vertical-align:-1.95744em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span> é o resultado da equação <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> é a estimativa iniciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span> é o valor acrescido de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> que resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>, ou seja: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow><mo>=</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right) = f_1\left( x_{1}^{(0)} + \Delta x_1, x_{2}^{(0)} + \Delta x_2 \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span></span></li></ul><p>Essa expressão pode ser resumida em:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>J</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \Delta K_1^{(0)}\\ \Delta K_2^{(0)} \end{bmatrix} = \begin{bmatrix} J \end{bmatrix}^{(0)} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.09618em">J</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1279000000000001em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>J</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[J]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.09618em">J</span><span class="mclose">]</span></span></span></span></span> é a <strong>Matriz Jacobiana</strong></p><p>Com a equação acima é possível calcular os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>. Entretanto, esses valores somados às estimativas iniciais não determinam a solução correta, sendo necessário repetir o processo de determinação das constantes, formação da matriz jacobiana e solução da equação acima, o qual será refeito determinando novas estimativas <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>Esse processo é repetido até que as correções se tornem tão pequenas que satisfaçam uma precisão escolhida.</p><p>Para aplicar o método de Newton-Raphson à solução das equações do fluxo de carga, utiliza-se as equações que representam as potências ativa e reativa injetadas em uma barra.</p><p>Assim como no método de Gauss-Seidel, a barra de referência é omitida da solução iterativa para determinar as tensões, pois a tensão complexa dessa barra é especificada. Como é conhecido o valor da potência ativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras do tipo PQ e PV, além da potência reativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras PQ pode-se definir <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> como os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span>. Os valores estimados do módulo e ângulo da tensão correspondem aos valores estimados para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>O jacobiano consiste nas derivadas parciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> em relação a cada uma das variáveis das equações injeção de potência líquida na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>. A matriz coluna formada por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> corresponde às correções de ângulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta \theta_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e módulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) das tensões de barra.</p><p>Com isso pode-se escrever a equação matricial de um sistema de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">n</span></span></span></span></span> barras, em que a barra número 1 corresponde à barra de referência e as barras de número 2 a n são barras do tipo PQ:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtable rowspacing="0.15999999999999992em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mo>+</mo><mtext>NPV</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{array}{r} \text{NPQ} + \text{NPV} \begin{cases} ~\\ ~ \end{cases}\\ ~\\ \text{NPQ} \begin{cases} ~\\ ~ \end{cases} \end{array} \begin{bmatrix} \Delta P_2\\ \vdots\\ \Delta P_n\\ \Delta Q_2\\ \vdots\\ \Delta Q_n\\ \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial P_2}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial P_2}{\partial \theta_n} &amp; \displaystyle\frac{\partial P_2}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial P_2}{\partial V_n}\\ \vdots &amp; \ddots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ \displaystyle\frac{\partial P_n}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial P_n}{\partial \theta_n} &amp; \displaystyle\frac{\partial P_n}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial P_n}{\partial V_n}\\ \displaystyle\frac{\partial Q_2}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_2}{\partial \theta_n} &amp; \displaystyle\frac{\partial Q_2}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_2}{\partial V_n}\\ \vdots &amp; \ddots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ \displaystyle\frac{\partial Q_n}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_n}{\partial \theta_n} &amp; \displaystyle\frac{\partial Q_n}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_n}{\partial V_n} \end{bmatrix} \begin{bmatrix} \Delta \theta_2\\ \vdots\\ \Delta \theta_2\\ \Delta V_2\\ \vdots\\ \Delta V_n\\ \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.520000000000003em;vertical-align:-4.010000000000001em"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.8500299999999994em"><span style="top:-5.850029999999999em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord text"><span class="mord">NPV</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.7599999999999993em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-1.6499999999999997em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.3500300000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:12.61603em;vertical-align:-6.050095em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPQ</mtext></mrow><annotation encoding="application/x-tex">\text{NPQ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord text"><span class="mord">NPQ</span></span></span></span></span></span> é o número de barras PQ</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPV</mtext></mrow><annotation encoding="application/x-tex">\text{NPV}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord text"><span class="mord">NPV</span></span></span></span></span></span> é o número de barras PV</li></ul><p>O processo iterativo se inicia calculando as potências ativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ e PV e as potências reativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ, ambas as equações utilizando as estimativas iniciais das tensões complexas. Calcula-se, então, as correções de potência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">\Delta P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">\Delta Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span></span></span></span></span>):</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi><mo>=</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup><mspace linebreak="newline"></mspace><mi mathvariant="normal">Δ</mi><mi>Q</mi><mo>=</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta P = P_i^{esp} - P_i^{calc}\\ \Delta Q = Q_i^{esp} - Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O passo seguinte é a formação da matriz jacobiana. Com isso é possível calcular as correções de módulo e ângulo das tensões de todas as barras (com exceção da barra de referência). Para tanto, no PSP-UFU utiliza-se o método de <a href="https://en.wikipedia.org/wiki/Gaussian_elimination" target="_blank" rel="noopener noreferrer">Eliminação Gaussiana</a> e em sequência a substituição regressiva. Esse procedimento diminui o esforço computacional, uma vez que a inversão da matriz jacobiana em todas as iterações é evitada.</p><p>Com as correções de módulo e ângulo das tensões das barras calculados, aplicam-se as seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mspace linebreak="newline"></mspace><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\theta_i^{(v+1)}=\theta_i^{(v)}+\Delta \theta_i^{(v)}\\ V_i^{(v+1)}=V_i^{(v)}+\Delta V_i^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O processo é então reiniciado e será repetido até que se obtenha a convergência, quando as correções se tornam tão pequenas que satisfaçam uma tolerância pré-estipulada.</p><p>O fluxograma abaixo mostra o método de Newton-Raphson para solução do fluxo de carga implementado.</p><div><center><img src="/PSP/images/newtonRaphson.svg" alt="Método numérico de Newton-Raphson para fluxo de carga" title="Método numérico de Newton-Raphson para fluxo de carga"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="controles-e-limites-em-um-problema-de-fluxo-de-carga"></a>Controles e limites em um problema de fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" title="Direct link to heading">#</a></h2><p>Nas barras de <a href="/PSP/docs/syncGenerator">geração</a> e também naquelas em que se encontra um <a href="/PSP/docs/syncMotor">compensador síncrono</a> conectado, o controle da magnitude da tensão no barramento é realizado por meio do ajuste da corrente de campo das máquinas síncronas, as quais podem operar sobrexcitadas (injetando reativos) ou subexcitadas (absorvendo reativos). Os valores limites de potência reativa que podem ser injetadas ou absorvidas dependem da máquina síncrona em estudo. Esses limites são incluídos no fluxo de carga com a criação de dois novos parâmetros, a potência reativa máxima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e potência reativa mínima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>), sendo esses valores a soma dos limites individuais das máquinas em uma mesma barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</p><p>A manutenção da potência reativa dentro dos limites é realizada pela troca do tipo de barra, ou seja, as barras violadoras que controlam a tensão (PV), passam a ser barras de carga (PQ), cuja potência é fixada como o limite que seria ultrapassado (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e a tensão deixa de ser controlada partir de então.</p><p>A verificação de violação e troca de tipo de barra pode ser realizada a cada iteração ou ao final da convergência do cálculo. No PSP-UFU foi implementada a última estratégia, uma vez que separa os conceitos de cálculo e de verificação de limites, tornando mais fácil o desenvolvimento de novos métodos numéricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situação não violadora o cálculo iterativo deve ser retomado até que obtenha novamente a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MONTICELLI, A. J. Fluxo de Carga em Redes de Energia Elétrica. São Paulo: Edgar Blücher, 1983.</li><li>STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.</li><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton’s Method. IEEE Transaction on Power Apparatus and Systems, v. PAS-86, n. 11, nov. 1967. doi: <a href="https://doi.org/10.1109/TPAS.1967.291823" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPAS.1967.291823</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/powerFlow.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/mathExpression"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Expressão Matemática</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/fault"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Curto-Circuito »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#formulação-do-problema-de-fluxo-de-carga" class="table-of-contents__link">Formulação do problema de fluxo de carga</a></li><li><a href="#execução-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Execução do fluxo de carga no PSP-UFU</a><ul><li><a href="#erros-comuns-na-execução-do-fluxo-de-carga" class="table-of-contents__link">Erros comuns na execução do fluxo de carga</a></li></ul></li><li><a href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Métodos de solução numérica do fluxo de carga no PSP-UFU</a><ul><li><a href="#gauss-seidel" class="table-of-contents__link">Gauss-Seidel</a></li><li><a href="#newton-raphson" class="table-of-contents__link">Newton-Raphson</a></li></ul></li><li><a href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" class="table-of-contents__link">Controles e limites em um problema de fluxo de carga</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Fluxo de Potência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Um estudo fundamental no planejamento da expansão e operação de um sistema elétrico é o <strong>fluxo de potência (ou fluxo de carga)</strong> uma vez que a operação satisfatória desse sistema depende do conhecimento dos efeitos da interligação, de novas cargas, de novas centrais geradoras e de novas linhas antes que elas sejam instaladas. Esse estudo tem como objetivo encontrar o fluxo de potência nos ramos e as tensões nodais do Sistema Elétrico de Potência (SEP) impostas pela geração e a carga.</p><p>As equações formuladas a partir dos modelos dos elementos presentes no estudo de fluxo de carga são não lineares e não possuem solução analítica explícita. Portanto é necessária a utilização de métodos numéricos iterativos para solução do problema de fluxo de carga.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulação-do-problema-de-fluxo-de-carga"></a>Formulação do problema de fluxo de carga<a class="hash-link" href="#formulação-do-problema-de-fluxo-de-carga" title="Direct link to heading">#</a></h2><p>O problema de fluxo de carga pode ser representado por um sistema de equações e inequações algébricas não-lineares que correspondem às leis de Kirchhoff e a um conjunto de restrições de operação impostos pelos componentes de uma rede elétrica.
+Na formulação do problema a cada barra da rede são associadas quatro variáveis, sendo que duas delas surgem como dados e duas como incógnitas (em uma barra de índice <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>):</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a magnitude da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é o ângulo da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a injeção líquida de potência ativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a injeção líquida de potência reativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</li></ul><p>Relativo às variáveis que são incógnitas e os dados do sistema, podem ser definidos <strong>três tipos de barras</strong>:</p><ul><li><strong>Barra PQ</strong>:	<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra PV</strong>:	<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra de Referência</strong>	<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As <strong>barras PQ</strong> geralmente são representadas pelos barramentos de carga, os quais não há geração e controle de tensão.</p><p>As <strong>barras PV</strong> se caracterizam pelo controle de tensão mediante a injeção ou absorção de potência reativa por meio do controle da excitação de uma máquina síncrona.</p><p>A <strong>barra de Referência</strong> (ou de folga, de oscilação) tem como função, assim como o próprio nome diz, servir de referência de tensão e ângulo do sistema. Essa barra é necessariamente geradora, uma vez que ela é responsável pelo equilíbrio do balanço de potência do sistema.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O tipo de barra deve ser definido no <a href="/PSP/docs/bus">elemento barramento</a>.</p><p>Note que <strong>o sistema deve possuir somente uma barra de referência</strong>.</p></div></div><p>Como mencionado anteriormente, as equações são não-lineares e a solução analítica não é prática. As soluções dessas equações seguem <strong><a href="https://en.wikipedia.org/wiki/Iterative_method" target="_blank" rel="noopener noreferrer">processos iterativos</a></strong>, em que são atribuídos valores estimados (ou iniciais) para as barras com tensões desconhecidas e, baseado na potência ativa e reativa e módulo da tensão especificados, calcula-se por meio das equações previamente apresentadas as novas tensões complexas em cada nó do sistema.</p><p>Na sequência, esse conjunto de valores para as tensões em cada barra é utilizado para novamente calcular outro grupo de tensões. Cada cálculo de um novo conjunto de tensões é chamado iteração. O processo iterativo é repetido até que as mudanças em todas as barras sejam menores do que um valor pré-estipulado, obtendo assim a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="execução-do-fluxo-de-carga-no-psp-ufu"></a>Execução do fluxo de carga no PSP-UFU<a class="hash-link" href="#execução-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do fluxo de carga é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Fluxo de carga</strong>.</p><div><center><img src="/PSP/images/menuSimulationPF.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso o fluxo de carga tenha sido executado com sucesso, as setas de potência serão exibidas, a barra de status indicará sucesso na operação e os <a href="/PSP/docs/text">elementos de texto</a> serão atualizados.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Evite construir o circuito com o cálculo contínuo habilitado, uma vez que configurações temporárias podem levar a erros de execução da simulação.</p><p>Para desabilitar o cálculo contínuo clique no botão <strong>Desabilitar solução</strong>.</p></div></div><p>Os resultados do fluxo de carga são exibidos nos <a href="/PSP/docs/text">elementos de texto vinculado</a>, ao posicionar o mouse sobre os elementos e em <a href="/PSP/docs/tabularReport">relatórios tabulares</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="erros-comuns-na-execução-do-fluxo-de-carga"></a>Erros comuns na execução do fluxo de carga<a class="hash-link" href="#erros-comuns-na-execução-do-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao fluxo de carga.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado"></a>A seguinte mensagem de erro é exibida: &quot;O número máximo de iterações foi alcançado&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando o método de solução numérica selecionado nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> atinge o número máximo de iterações inserido. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li><li><strong>O número máximo de iterações está muito baixo</strong>. Alguns circuitos exigem um número maior de iterações, portanto altere o valor do máximo de iterações nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>. Também tente alternar entre os métodos numéricos de solução disponíveis.</li><li><strong>Os parâmetros de simulação estão inadequados</strong>. Caso um parâmetro do método de solução esteja inadequado, como fator de aceleração ou tolerância, o cálculo pode não alcançar a convergência. Altere esses parâmetros nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="os-dados-de-saída-são-exibidos-como-nan-ou-nan"></a>Os dados de saída são exibidos como &quot;NaN&quot; ou &quot;nan&quot;<a class="hash-link" href="#os-dados-de-saída-são-exibidos-como-nan-ou-nan" title="Direct link to heading">#</a></h4><p>Isso ocorre devido a erros de operações matemáticas nos cálculos de fluxo de carga. &quot;NaN&quot; significa <em>Not a Number</em>.</p><ul><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu"></a>Métodos de solução numérica do fluxo de carga no PSP-UFU<a class="hash-link" href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Os métodos implementados no programa para solução do problema de fluxo de carga no PSP-UFU são <strong>Gauss-Seidel (GS)</strong> e <strong>Newton-Raphson (NR)</strong>. Além desses métodos clássicos, um método híbrido pode ser utilizado (definido nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>), em que é utilizado inicialmente o GS e na sequência o NR, aumentando a chance de convergência do NR.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gauss-seidel"></a>Gauss-Seidel<a class="hash-link" href="#gauss-seidel" title="Direct link to heading">#</a></h3><p>O método de Gauss-Seidel tem sido bastante utilizado nas últimas décadas para solução do problema de fluxo de carga, uma vez que não há a necessidade de fatorar a matrizes, reduzindo o esforço computacional. Atualmente, restrições computacionais são menos problemáticas e outros métodos são normalmente escolhidos, porém o Gauss-Seidel ainda possui valor didático e, visto que o PSP-UFU também possui fins educacionais, optou-se pela implementação desse método.</p><p>Para iniciar as iterações do método são necessários valores iniciais para as tensões (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_i^0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) que devem ser calculadas. Normalmente para barras do tipo PQ tem-se <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=1{,}0+j0{,}0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> e para barras do tipo PV <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=V_i^{esp}+j0{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> [p.u.], em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">V_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> é o módulo da tensão especificada para a barra PV. A barra de referência tem o módulo e ângulo de tensão fixos e não participam no processo iterativo.</p><p>O método de Gauss-Seidel mostra um <strong>número excessivo de iterações</strong> e, com o intuito reduzi-los, multiplicam-se as correções de tensões por uma constante. Essa operação amplia o valor da correção, trazendo a tensão para mais perto do valor do valor final. Os multiplicadores que realizam essa convergência melhorada são chamados de fatores de aceleração. Para qualquer sistema existem valores ótimos para os fatores de aceleração e uma escolha inadequada pode resultar em uma convergência mais lenta ou torná-la impossível. Normalmente é utilizado um fator de aceleração igual a 1,6, valor definido como padrão no programa. A utilização do fator de aceleração é realizada por meio da equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mi>α</mi><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow><mo>+</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)} = \alpha \left( \dot{V}_{i}^{(v + 1)}- \dot{V}_{i_{AC}}^{(v)} \right) + \dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.27686399999999994em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração anterior com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span> é o fator de aceleração</li></ul><p>O fluxograma abaixo demonstra como o método de Gauss-Seidel foi implementado no PSP-UFU:</p><div><center><img src="/PSP/images/gaussSeidel.svg" alt="Método numérico de Gauss-Seidel para fluxo de carga" title="Método numérico de Gauss-Seidel para fluxo de carga"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="newton-raphson"></a>Newton-Raphson<a class="hash-link" href="#newton-raphson" title="Direct link to heading">#</a></h3><p>O método de Newton-Raphson (também conhecido como método de Newton ou Newton-Fourier) para solução do fluxo de carga é descrito em vários livros e artigos. Atualmente é o algoritmo mais utilizado para solução do fluxo de carga. Para casos bem condicionados, esse método geralmente converge em 4 a 5 iterações, porém existe a possibilidade da técnica contornar o ponto da solução sem nunca atingi-la, o que também justifica implementação do método de Gauss-Seidel no <em>software</em>.</p><p>A expansão da <a href="https://en.wikipedia.org/wiki/Taylor_series" target="_blank" rel="noopener noreferrer">série de Taylor</a> para uma função de duas ou mais variáveis é a base do método de Newton-Raphson para resolver o problema do fluxo de carga. Fazendo a expansão em série de Taylor, para duas equações e duas incógnitas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_1\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_2\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span>), sem listar as derivadas parciais maiores que 1 na forma matricial, obtêm-se:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>2</mn></msub><mo>−</mo><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} K_1 - f_1\left( x_{1}^{(0)}, x_{2}^{(0)} \right)\\ K_2 - f_2\left( x_{1}^{(0)}, x_{2}^{(0)} \right) \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x_1} &amp; \displaystyle\frac{\partial f_1}{\partial x_2}\\ \displaystyle\frac{\partial f_2}{\partial x_1} &amp; \displaystyle\frac{\partial f_2}{\partial x_2} \end{bmatrix} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6010299999999997em;vertical-align:-1.55002em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em"><span style="top:-4.05002em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.41488em;vertical-align:-1.95744em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span> é o resultado da equação <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> é a estimativa iniciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span> é o valor acrescido de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> que resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>, ou seja: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow><mo>=</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right) = f_1\left( x_{1}^{(0)} + \Delta x_1, x_{2}^{(0)} + \Delta x_2 \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span></span></li></ul><p>Essa expressão pode ser resumida em:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>J</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \Delta K_1^{(0)}\\ \Delta K_2^{(0)} \end{bmatrix} = \begin{bmatrix} J \end{bmatrix}^{(0)} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.09618em">J</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1279000000000001em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>J</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[J]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.09618em">J</span><span class="mclose">]</span></span></span></span></span> é a <strong>Matriz Jacobiana</strong></p><p>Com a equação acima é possível calcular os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>. Entretanto, esses valores somados às estimativas iniciais não determinam a solução correta, sendo necessário repetir o processo de determinação das constantes, formação da matriz jacobiana e solução da equação acima, o qual será refeito determinando novas estimativas <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>Esse processo é repetido até que as correções se tornem tão pequenas que satisfaçam uma precisão escolhida.</p><p>Para aplicar o método de Newton-Raphson à solução das equações do fluxo de carga, utiliza-se as equações que representam as potências ativa e reativa injetadas em uma barra.</p><p>Assim como no método de Gauss-Seidel, a barra de referência é omitida da solução iterativa para determinar as tensões, pois a tensão complexa dessa barra é especificada. Como é conhecido o valor da potência ativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras do tipo PQ e PV, além da potência reativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras PQ pode-se definir <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> como os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span>. Os valores estimados do módulo e ângulo da tensão correspondem aos valores estimados para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>O jacobiano consiste nas derivadas parciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> em relação a cada uma das variáveis das equações injeção de potência líquida na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>. A matriz coluna formada por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> corresponde às correções de ângulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta \theta_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e módulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) das tensões de barra.</p><p>Com isso pode-se escrever a equação matricial de um sistema de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">n</span></span></span></span></span> barras, em que a barra número 1 corresponde à barra de referência e as barras de número 2 a n são barras do tipo PQ:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtable rowspacing="0.15999999999999992em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mo>+</mo><mtext>NPV</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{array}{r} \text{NPQ} + \text{NPV} \begin{cases} ~\\ ~ \end{cases}\\ ~\\ \text{NPQ} \begin{cases} ~\\ ~ \end{cases} \end{array} \begin{bmatrix} \Delta P_2\\ \vdots\\ \Delta P_n\\ \Delta Q_2\\ \vdots\\ \Delta Q_n\\ \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial P_2}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial P_2}{\partial \theta_n} &amp; \displaystyle\frac{\partial P_2}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial P_2}{\partial V_n}\\ \vdots &amp; \ddots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ \displaystyle\frac{\partial P_n}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial P_n}{\partial \theta_n} &amp; \displaystyle\frac{\partial P_n}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial P_n}{\partial V_n}\\ \displaystyle\frac{\partial Q_2}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_2}{\partial \theta_n} &amp; \displaystyle\frac{\partial Q_2}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_2}{\partial V_n}\\ \vdots &amp; \ddots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ \displaystyle\frac{\partial Q_n}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_n}{\partial \theta_n} &amp; \displaystyle\frac{\partial Q_n}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_n}{\partial V_n} \end{bmatrix} \begin{bmatrix} \Delta \theta_2\\ \vdots\\ \Delta \theta_2\\ \Delta V_2\\ \vdots\\ \Delta V_n\\ \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.520000000000003em;vertical-align:-4.010000000000001em"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.8500299999999994em"><span style="top:-5.850029999999999em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord text"><span class="mord">NPV</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.7599999999999993em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-1.6499999999999997em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.3500300000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:12.61603em;vertical-align:-6.050095em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPQ</mtext></mrow><annotation encoding="application/x-tex">\text{NPQ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord text"><span class="mord">NPQ</span></span></span></span></span></span> é o número de barras PQ</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPV</mtext></mrow><annotation encoding="application/x-tex">\text{NPV}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord text"><span class="mord">NPV</span></span></span></span></span></span> é o número de barras PV</li></ul><p>O processo iterativo se inicia calculando as potências ativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ e PV e as potências reativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ, ambas as equações utilizando as estimativas iniciais das tensões complexas. Calcula-se, então, as correções de potência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">\Delta P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">\Delta Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span></span></span></span></span>):</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi><mo>=</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup><mspace linebreak="newline"></mspace><mi mathvariant="normal">Δ</mi><mi>Q</mi><mo>=</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta P = P_i^{esp} - P_i^{calc}\\ \Delta Q = Q_i^{esp} - Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O passo seguinte é a formação da matriz jacobiana. Com isso é possível calcular as correções de módulo e ângulo das tensões de todas as barras (com exceção da barra de referência). Para tanto, no PSP-UFU utiliza-se o método de <a href="https://en.wikipedia.org/wiki/Gaussian_elimination" target="_blank" rel="noopener noreferrer">Eliminação Gaussiana</a> e em sequência a substituição regressiva. Esse procedimento diminui o esforço computacional, uma vez que a inversão da matriz jacobiana em todas as iterações é evitada.</p><p>Com as correções de módulo e ângulo das tensões das barras calculados, aplicam-se as seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mspace linebreak="newline"></mspace><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\theta_i^{(v+1)}=\theta_i^{(v)}+\Delta \theta_i^{(v)}\\ V_i^{(v+1)}=V_i^{(v)}+\Delta V_i^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O processo é então reiniciado e será repetido até que se obtenha a convergência, quando as correções se tornam tão pequenas que satisfaçam uma tolerância pré-estipulada.</p><p>O fluxograma abaixo mostra o método de Newton-Raphson para solução do fluxo de carga implementado.</p><div><center><img src="/PSP/images/newtonRaphson.svg" alt="Método numérico de Newton-Raphson para fluxo de carga" title="Método numérico de Newton-Raphson para fluxo de carga"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="controles-e-limites-em-um-problema-de-fluxo-de-carga"></a>Controles e limites em um problema de fluxo de carga<a class="hash-link" href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" title="Direct link to heading">#</a></h2><p>Nas barras de <a href="/PSP/docs/syncGenerator">geração</a> e também naquelas em que se encontra um <a href="/PSP/docs/syncMotor">compensador síncrono</a> conectado, o controle da magnitude da tensão no barramento é realizado por meio do ajuste da corrente de campo das máquinas síncronas, as quais podem operar sobrexcitadas (injetando reativos) ou subexcitadas (absorvendo reativos). Os valores limites de potência reativa que podem ser injetadas ou absorvidas dependem da máquina síncrona em estudo. Esses limites são incluídos no fluxo de carga com a criação de dois novos parâmetros, a potência reativa máxima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e potência reativa mínima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>), sendo esses valores a soma dos limites individuais das máquinas em uma mesma barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</p><p>A manutenção da potência reativa dentro dos limites é realizada pela troca do tipo de barra, ou seja, as barras violadoras que controlam a tensão (PV), passam a ser barras de carga (PQ), cuja potência é fixada como o limite que seria ultrapassado (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e a tensão deixa de ser controlada partir de então.</p><p>A verificação de violação e troca de tipo de barra pode ser realizada a cada iteração ou ao final da convergência do cálculo. No PSP-UFU foi implementada a última estratégia, uma vez que separa os conceitos de cálculo e de verificação de limites, tornando mais fácil o desenvolvimento de novos métodos numéricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situação não violadora o cálculo iterativo deve ser retomado até que obtenha novamente a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MONTICELLI, A. J. Fluxo de Carga em Redes de Energia Elétrica. São Paulo: Edgar Blücher, 1983.</li><li>STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.</li><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton’s Method. IEEE Transaction on Power Apparatus and Systems, v. PAS-86, n. 11, nov. 1967. doi: <a href="https://doi.org/10.1109/TPAS.1967.291823" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPAS.1967.291823</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/powerFlow.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/mathExpression"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Expressão Matemática</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/fault"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Curto-Circuito »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#formulação-do-problema-de-fluxo-de-carga" class="table-of-contents__link">Formulação do problema de fluxo de carga</a></li><li><a href="#execução-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Execução do fluxo de carga no PSP-UFU</a><ul><li><a href="#erros-comuns-na-execução-do-fluxo-de-carga" class="table-of-contents__link">Erros comuns na execução do fluxo de carga</a></li></ul></li><li><a href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Métodos de solução numérica do fluxo de carga no PSP-UFU</a><ul><li><a href="#gauss-seidel" class="table-of-contents__link">Gauss-Seidel</a></li><li><a href="#newton-raphson" class="table-of-contents__link">Newton-Raphson</a></li></ul></li><li><a href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" class="table-of-contents__link">Controles e limites em um problema de fluxo de carga</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Limitador de taxa</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Os blocos de limite de taxa têm como objetivo conter a taxa de crescimento ou decrescimento da sua entrada a valores inseridos pelo usuário. A taxa de variação é calculada a partir dos valores de entrada atual e passado e, caso seja violada, será definida uma saída que respeite essas taxas (de crescimento ou decrescimento).</p><p>A taxa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>) é calculada de acordo com a equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mi>n</mi></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">r = \frac{u_n - y_{n-1}}{h}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.9463300000000001em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">h</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a entrada atual do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor de saída anterior do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	é o passo de integração</li></ul><p>Obtida a taxa, o bloco irá resultar em um valor calculado pelas seguintes condições:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>h</mi><mo>×</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><mi>r</mi><mo>&gt;</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>u</mi><mi>n</mi></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub><mo>≤</mo><mi>r</mi><mo>≤</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>h</mi><mo>×</mo><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><mi>r</mi><mo>&lt;</mo><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} h \times R_{cre}{,} &amp; \text{se } r &gt; R_{cre} \\ u_n{,} &amp; \text{se } R_{dec} \le r \le R_{cre}\\ h \times R_{dec}{,} &amp; \text{se } r &lt; R_{dec} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{cre}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{dec}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	são as taxas de crescimento e decaimento, respectivamente</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-de-dados-do-limitador-de-taxa"></a>Formulário de edição de dados do Limitador de taxa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-do-limitador-de-taxa" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco limitador de taxa.</p><div><center><img src="/PSP/images/rateLimitForm.png" alt="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU" title="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU"></center></div><p>Esse bloco é definido pelo <strong>limite superior (taxa de crescimento máxima) e inferior (taxa de decrescimento máxima)</strong>, inseridos pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Esses blocos são úteis para limitar variações bruscas na entrada que possam comprometer o controle ou limites físicos de modelos de turbina, por exemplo.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div 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Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Limitador de taxa</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Os blocos de limite de taxa têm como objetivo conter a taxa de crescimento ou decrescimento da sua entrada a valores inseridos pelo usuário. A taxa de variação é calculada a partir dos valores de entrada atual e passado e, caso seja violada, será definida uma saída que respeite essas taxas (de crescimento ou decrescimento).</p><p>A taxa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>) é calculada de acordo com a equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mi>n</mi></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">r = \frac{u_n - y_{n-1}}{h}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.9463300000000001em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">h</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a entrada atual do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor de saída anterior do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	é o passo de integração</li></ul><p>Obtida a taxa, o bloco irá resultar em um valor calculado pelas seguintes condições:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>h</mi><mo>×</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><mi>r</mi><mo>&gt;</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>u</mi><mi>n</mi></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub><mo>≤</mo><mi>r</mi><mo>≤</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>h</mi><mo>×</mo><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><mi>r</mi><mo>&lt;</mo><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} h \times R_{cre}{,} &amp; \text{se } r &gt; R_{cre} \\ u_n{,} &amp; \text{se } R_{dec} \le r \le R_{cre}\\ h \times R_{dec}{,} &amp; \text{se } r &lt; R_{dec} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{cre}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{dec}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	são as taxas de crescimento e decaimento, respectivamente</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-limitador-de-taxa"></a>Formulário de edição de dados do Limitador de taxa<a class="hash-link" href="#formulário-de-edição-de-dados-do-limitador-de-taxa" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco limitador de taxa.</p><div><center><img src="/PSP/images/rateLimitForm.png" alt="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU" title="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU"></center></div><p>Esse bloco é definido pelo <strong>limite superior (taxa de crescimento máxima) e inferior (taxa de decrescimento máxima)</strong>, inseridos pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Esses blocos são úteis para limitar variações bruscas na entrada que possam comprometer o controle ou limites físicos de modelos de turbina, por exemplo.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a 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padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Configurações da Simulação</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>As parametrizações e configurações das simulações são acessadas no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no ícone do botão <strong>Configurações de simulação</strong>.</p><div><center><img src="/PSP/images/menuSimulationSettings.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-das-configurações-de-simulação"></a>Formulário de edição das configurações de simulação<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-das-configurações-de-simulação" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de edição das configurações de simulação:</p><div><center><img src="/PSP/images/simulationConfigForm.png" alt="Configurações de simulação" title="Configurações de simulação"></center></div><p>Esse formulário é subdividido em quatro contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais e comuns a várias simulações;</li><li><strong>Fluxo de carga</strong>: local onde é selecionado o <a href="/PSP/docs/powerFlow#m%C3%A9todos-de-solu%C3%A7%C3%A3o-num%C3%A9rica-do-fluxo-de-carga-no-psp-ufu">método de solução numérica</a>, assim como seus parâmetros;</li><li><strong>Estabilidade</strong>: contendo opções de simulação no tempo e parâmetros do <a href="/PSP/docs/stability#integra%C3%A7%C3%A3o-num%C3%A9rica">método de integração</a>;</li><li><strong>Carga ZIP</strong>: contém as opções gerais das <a href="/PSP/docs/load#carga-no-estudo-de-estabilidade">cargas do tipo ZIP</a> no estudo de <a href="/PSP/docs/stability">estabilidade</a>.</li></ul><hr><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Fluxo de carga</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Estabilidade</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Carga ZIP</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-base"></a>Potência base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-base" title="Direct link to heading">#</a></h4><p>Potência base do sistema utilizada para conversão dos dados reais em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> e vice-versa. Pode ser inserido em VA, kVA ou MVA.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>A potência base de cada elemento pode ser distinta da potência base do sistema. Para isso, basta marcar a opção &quot;Utilizar potência nominal [<em>do elemento</em>] como base&quot;, presente em seus respectivos <a href="/PSP/docs/powerEditor">formulários de edição da dados</a>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="frequência-do-sistema"></a>Frequência do sistema<a aria-hidden="true" tabindex="-1" class="hash-link" href="#frequência-do-sistema" title="Direct link to heading">#</a></h4><p>Define a frequência nominal do sistema.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao alterar a frequência nominal atente-se ao campo &quot;<a href="/PSP/docs/syncGenerator#frequ%C3%AAncia-de-circuito-aberto">Frequência de circuito aberto</a>&quot; das máquinas síncronas.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="cálculo-contínuo"></a>Cálculo contínuo<a aria-hidden="true" tabindex="-1" class="hash-link" href="#cálculo-contínuo" title="Direct link to heading">#</a></h4><p>Habilita ou desabilita o cálculo contínuo para os cálculos de <a href="/PSP/docs/fault">curto-circuito</a>, <a href="/PSP/docs/fault">nível de curto-circuito</a> e <a href="/PSP/docs/harmonics">distorções harmônicas</a>. O cálculo contínuo para o <a href="/PSP/docs/powerFlow">fluxo de carga</a> é sempre habilitado.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Para habilitar o cálculo contínuo aperte o botão <strong>Habilitar solução</strong> presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a>.</p><p>Com essa opção, os cálculos estáticos selecionados são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p></div></div></div></div></div><hr></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/simulationConfig.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/stability"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Estabilidade</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/text"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Texto Vinculado »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#formulário-de-edição-das-configurações-de-simulação" class="table-of-contents__link">Formulário de edição das configurações de simulação</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="#!">Control Element Editor</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/controlEditor">Editor de Controle</a></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!" tabindex="-1">Control Elements</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/io">Entrada / Saída</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/transferFunction">Função Transferência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/sum">Somador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/multiplier">Multiplicador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Simulations</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Configurações da Simulação</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>As parametrizações e configurações das simulações são acessadas no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no ícone do botão <strong>Configurações de simulação</strong>.</p><div><center><img src="/PSP/images/menuSimulationSettings.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-das-configurações-de-simulação"></a>Formulário de edição das configurações de simulação<a class="hash-link" href="#formulário-de-edição-das-configurações-de-simulação" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de edição das configurações de simulação:</p><div><center><img src="/PSP/images/simulationConfigForm.png" alt="Configurações de simulação" title="Configurações de simulação"></center></div><p>Esse formulário é subdividido em quatro contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais e comuns a várias simulações;</li><li><strong>Fluxo de carga</strong>: local onde é selecionado o <a href="/PSP/docs/powerFlow#m%C3%A9todos-de-solu%C3%A7%C3%A3o-num%C3%A9rica-do-fluxo-de-carga-no-psp-ufu">método de solução numérica</a>, assim como seus parâmetros;</li><li><strong>Estabilidade</strong>: contendo opções de simulação no tempo e parâmetros do <a href="/PSP/docs/stability#integra%C3%A7%C3%A3o-num%C3%A9rica">método de integração</a>;</li><li><strong>Carga ZIP</strong>: contém as opções gerais das <a href="/PSP/docs/load#carga-no-estudo-de-estabilidade">cargas do tipo ZIP</a> no estudo de <a href="/PSP/docs/stability">estabilidade</a>.</li></ul><hr><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Fluxo de carga</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Carga ZIP</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-base"></a>Potência base<a class="hash-link" href="#potência-base" title="Direct link to heading">#</a></h4><p>Potência base do sistema utilizada para conversão dos dados reais em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> e vice-versa. Pode ser inserido em VA, kVA ou MVA.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>A potência base de cada elemento pode ser distinta da potência base do sistema. Para isso, basta marcar a opção &quot;Utilizar potência nominal [<em>do elemento</em>] como base&quot;, presente em seus respectivos <a href="/PSP/docs/powerEditor">formulários de edição da dados</a>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="frequência-do-sistema"></a>Frequência do sistema<a class="hash-link" href="#frequência-do-sistema" title="Direct link to heading">#</a></h4><p>Define a frequência nominal do sistema.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao alterar a frequência nominal atente-se ao campo &quot;<a href="/PSP/docs/syncGenerator#frequ%C3%AAncia-de-circuito-aberto">Frequência de circuito aberto</a>&quot; das máquinas síncronas.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="cálculo-contínuo"></a>Cálculo contínuo<a class="hash-link" href="#cálculo-contínuo" title="Direct link to heading">#</a></h4><p>Habilita ou desabilita o cálculo contínuo para os cálculos de <a href="/PSP/docs/fault">curto-circuito</a>, <a href="/PSP/docs/fault">nível de curto-circuito</a> e <a href="/PSP/docs/harmonics">distorções harmônicas</a>. O cálculo contínuo para o <a href="/PSP/docs/powerFlow">fluxo de carga</a> é sempre habilitado.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Para habilitar o cálculo contínuo aperte o botão <strong>Habilitar solução</strong> presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a>.</p><p>Com essa opção, os cálculos estáticos selecionados são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p></div></div></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="método-de-solução"></a>Método de solução<a class="hash-link" href="#método-de-solução" title="Direct link to heading">#</a></h4><p>Define o método de solução numérica para o estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>. Na versão atual, está presente no PSP-UFU os seguintes métodos:</p><ul><li><a href="/PSP/docs/powerFlow#gauss-seidel">Gauss-Seidel</a></li><li><a href="/PSP/docs/powerFlow#newton-raphson">Newton-Raphson</a></li><li><a href="/PSP/docs/powerFlow#m%C3%A9todos-de-solu%C3%A7%C3%A3o-num%C3%A9rica-do-fluxo-de-carga-no-psp-ufu">Gauss-Newton híbrido</a></li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="fator-de-aceleração"></a>Fator de aceleração<a class="hash-link" href="#fator-de-aceleração" title="Direct link to heading">#</a></h4><p>Fator utilizado para acelerar a convergência nos métodos de Gauss. A utilização do fator de aceleração é realizada por meio da equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mi>α</mi><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow><mo>+</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)} = \alpha \left( \dot{V}_{i}^{(v + 1)}- \dot{V}_{i_{AC}}^{(v)} \right) + \dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.27686399999999994em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração anterior com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span> é o fator de aceleração</li></ul><p>Essa opção é habilitada somente para os métodos de <a href="/PSP/docs/powerFlow#gauss-seidel">Gauss-Seidel</a> e <a href="/PSP/docs/powerFlow#m%C3%A9todos-de-solu%C3%A7%C3%A3o-num%C3%A9rica-do-fluxo-de-carga-no-psp-ufu">Gauss-Newton híbrido</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tolerância"></a>Tolerância<a class="hash-link" href="#tolerância" title="Direct link to heading">#</a></h4><p>É o valor de erro entre as iterações (em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) no qual é obtida a convergência do método numérico.</p><p>Para o método de <a href="/PSP/docs/powerFlow#gauss-seidel">Gauss-Seidel</a> o erro é calculado pelo maior valor do módulo da diferença entre a tensão na iteração anterior e atual (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo>=</mo><mrow><mo fence="true">∣</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">∣</mo></mrow></mrow><annotation encoding="application/x-tex">e = \left| V_{i}^{(v + 1)} - V_{i}^{(v)} \right|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.8239899999999998em;vertical-align:-0.6500149999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.173975em"><span style="top:-1.955985em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-2.556985em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-2.572975em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-3.173975em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6500149999999999em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.173975em"><span style="top:-1.955985em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-2.556985em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-2.572975em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-3.173975em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6500149999999999em"><span></span></span></span></span></span></span></span></span></span></span></span>); enquanto que no método de <a href="/PSP/docs/powerFlow#newton-raphson">Newton-Raphson</a> esse erro é o módulo do maior valor de ajuste de potência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">Δ</mi><mi>P</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\Delta P|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">∣</span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mord">∣</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">Δ</mi><mi>Q</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\Delta Q|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">∣</span><span class="mord">Δ</span><span class="mord mathdefault">Q</span><span class="mord">∣</span></span></span></span></span>).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="iterações-máx"></a>Iterações máx<a class="hash-link" href="#iterações-máx" title="Direct link to heading">#</a></h4><p>É o número máximo de iterações que um método poderá alcançar. Caso esse valor seja alcançado, o cálculo de <a href="/PSP/docs/powerFlow">fluxo de carga</a> é interrompido com erro.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="ângulo-da-barra-de-referência"></a>Ângulo da barra de referência<a class="hash-link" href="#ângulo-da-barra-de-referência" title="Direct link to heading">#</a></h4><p>É o valor do ângulo da <a href="/PSP/docs/powerFlow">barra de referência</a>, em graus.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inércia-do-newton"></a>Inércia do Newton<a class="hash-link" href="#inércia-do-newton" title="Direct link to heading">#</a></h4><p>É o valor que multiplica as correções de potência. Valores maiores que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> incrementam a correção e menores que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> diminuem a correção.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Valores menores que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> <em>podem</em> auxiliar na convergência, porém aumentam o número de iterações e consequentemente o tempo de processamento do método.</p></div></div><p>Essa opção é habilitada somente para o método de <a href="/PSP/docs/powerFlow#newton-raphson">Newton-Raphson</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tolerância-do-gauss"></a>Tolerância do Gauss<a class="hash-link" href="#tolerância-do-gauss" title="Direct link to heading">#</a></h4><p>Define a tolerância do Gauss-Seidel para o método híbrido. Essa opção é habilitada somente para o método de <a href="/PSP/docs/powerFlow#m%C3%A9todos-de-solu%C3%A7%C3%A3o-num%C3%A9rica-do-fluxo-de-carga-no-psp-ufu">Gauss-Newton híbrido</a>.</p><p>Uma vez que o Newton-Raphson é consideravelmente sensível às condições iniciais, as primeiras iterações são calculadas utilizado o Gauss-Seidel até que o erro fique menor que a tolerância especificada no campo “Tolerância do Gauss”. A partir desse ponto o cálculo é realizado utilizando o método de Newton-Raphson até que se obtenha a convergência com um erro menor que a tolerância estipulada no campo “Tolerância”.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tal opção auxilia na convergência de sistemas impossíveis de resolver utilizando Newton-Raphson convencional.</p></div></div></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="passo-de-integração"></a>Passo de integração<a class="hash-link" href="#passo-de-integração" title="Direct link to heading">#</a></h4><p>Define o passo de integração para o método de integração <a href="/PSP/docs/stability#integra%C3%A7%C3%A3o-num%C3%A9rica">Trapezoidal Implícito</a>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Valores muito elevados poderão gerar erros de simulação, enquanto valores muito pequenos aumentarão significativamente o tempo de processamento do cálculo de estabilidade.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tempo-de-simulação"></a>Tempo de simulação<a class="hash-link" href="#tempo-de-simulação" title="Direct link to heading">#</a></h4><p>Define o tempo total de simulação, em segundos.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tolerância-1"></a>Tolerância<a class="hash-link" href="#tolerância-1" title="Direct link to heading">#</a></h4><p>Tolerância do processo iterativo (em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) do método de integração <a href="/PSP/docs/stability#integra%C3%A7%C3%A3o-num%C3%A9rica">Trapezoidal Implícito</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="iterações-máx-1"></a>Iterações máx<a class="hash-link" href="#iterações-máx-1" title="Direct link to heading">#</a></h4><p>Número máximo de iterações do processo iterativo do método de integração <a href="/PSP/docs/stability#integra%C3%A7%C3%A3o-num%C3%A9rica">Trapezoidal Implícito</a>. Caso o número de iterações ultrapasse esse valor, o cálculo de estabilidade será encerrado com erro.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="razão-de-passo-dos-controles"></a>Razão de passo dos controles<a class="hash-link" href="#razão-de-passo-dos-controles" title="Direct link to heading">#</a></h4><p>Define a quantidade passos de integração que o sistema de controle será submetido para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> passo de integração do sistema de potência.</p><p>Por exemplo, se o passo de integração for definido em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>01</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">0{,}01~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord">1</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span> e a razão de passo como <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">0</span></span></span></span></span>, o valor efetivo do passo de integração para os sistemas de controle será de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>001</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">0{,}001~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord">0</span><span class="mord">1</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span> (10 vezes menor).</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A utilização desse campo se justifica em situações de grandes diferenças nas constantes de tempo das máquinas síncronas (geralmente maiores) e dos sistemas de controle (geralmente menores).</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um valor de razão de passo de controles igual a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> não irá alterar o passo de integração do sistema de controle.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tempo-de-impressão"></a>Tempo de impressão<a class="hash-link" href="#tempo-de-impressão" title="Direct link to heading">#</a></h4><p>É o intervalo de tempo o qual o programa irá armazenar os dados para impressão em <a href="/PSP/docs/graphViewer">gráficos no tempo</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Valores muito pequenos (menores que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>001</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">0{,}001~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord">0</span><span class="mord">1</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span>) podem aumentar significativamente o tempo de processamento do cálculo de estabilidade.</p><p>O tempo de impressão <strong>deve ser maior ou igual</strong> ao passo de integração.</p></div></div></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-composição-geral-para-todas-as-cargas-do-sistema"></a>Utilizar composição geral para todas as cargas do sistema<a class="hash-link" href="#utilizar-composição-geral-para-todas-as-cargas-do-sistema" title="Direct link to heading">#</a></h4><p>Habilita a composição da <a href="/PSP/docs/load#carga-no-estudo-de-estabilidade">carga ZIP</a> para todas as cargas que <strong>não</strong> possuem parametrização individual.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="composição-da-carga-potências-ativa-e-reativa"></a>Composição da carga (potências ativa e reativa)<a class="hash-link" href="#composição-da-carga-potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Define a composição, em porcentagem, das parcelas de Impedância, Corrente e Potência constantes da carga. Essa composição pode ser implementada individualmente para a potência ativa e reativa.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A soma das parcelas da carga ZIP para uma potência deve ser 100%, caso contrário será exibida uma mensagem de erro.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="subtensão-a-qual-as-cargas-serão-modeladas-como-impedância-constante"></a>Subtensão a qual as cargas serão modeladas como impedância constante<a class="hash-link" href="#subtensão-a-qual-as-cargas-serão-modeladas-como-impedância-constante" title="Direct link to heading">#</a></h4><p>Define o valor percentual da tensão (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>l</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow><annotation encoding="application/x-tex">V_{low}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02691em">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) que, para valores inferiores a esse, a carga passa a ser considerada como impedância constante pura.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Cargas de corrente e potência constantes possuem problemas em tensões muito baixas. À medida que a tensão diminui as correntes dessas cargas não reduz, resultando em perda de precisão e problemas na convergência de processos iterativos. Para contornar esse problema utiliza-se uma tensão pré-definida (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>l</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow><annotation encoding="application/x-tex">V_{low}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02691em">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), a qual as cargas (ou parcelas) de corrente e potência constantes são modeladas como impedância constante.</p></div></div></div></div></div><hr></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/simulationConfig.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/stability"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Estabilidade</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/text"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Texto Vinculado »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#formulário-de-edição-das-configurações-de-simulação" class="table-of-contents__link">Formulário de edição das configurações de simulação</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Estabilidade</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Estabilidade de sistemas de potência é a habilidade de um Sistema Elétrico de Potência (SEP), para uma certa condição de operação, de reestabelecer um estado de operação de equilíbrio após ser submetido a um distúrbio físico. A estabilidade é reconhecida como um importante problema para a segurança da operação desde a década de 1920. Muitos <em>blackouts</em> de grande porte causados pela instabilidade do SEP têm ilustrado a importância desse fenômeno.</p><div><center><img src="/PSP/images/stability.svg" alt="Estabilidade" title="Estabilidade"></center></div><p>Normalmente são realizados dois tipos de estudos de estabilidade (uma classificação mais aprofundada pode ser observada <a href="https://doi.org/10.1109/TPWRS.2004.825981" target="_blank" rel="noopener noreferrer">nesse artigo</a>). A recuperação de uma rápida e brusca perturbação é chamada de <strong>estabilidade transitória</strong>, e sua solução é geralmente obtida no domínio do tempo. Outro estudo é denominado <strong>estabilidade dinâmica</strong>, utilizado para descrever a resposta do sistema frente a pequenas perturbações ao longo de muito tempo, o qual pode ser resolvido tanto no domínio da frequência quanto no domínio do tempo.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>No PSP-UFU, a estabilidade dinâmica é tratada como uma extensão da estabilidade transitória, em que os efeitos de pequenas constantes de tempo devem ser ignorados.</p></div></div><p>O modelo para representação do SEP em um estudo de estabilidade transitória é feito por meio do sistema de equações algébrico-diferenciais (EADs), em que o estado inicial é considerado estável e originado de um estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>. A solução das equações se dá por meio de um <a href="https://en.wikipedia.org/wiki/Numerical_integration" target="_blank" rel="noopener noreferrer">método de integração numérica</a> e, então, é observada a resposta do sistema de potência.</p><p>Caso a solução no domínio do tempo, após alguma perturbação, leve um ou mais parâmetros do sistema a valores fisicamente impossíveis ou, até mesmo, a solução das equações diferenciais não obtenha convergência em processos iterativos, o sistema é dito <strong>instável</strong>. Em contrapartida, caso as variáveis de estado dos elementos do sistema oscilem após uma perturbação e se estabeleçam em um novo ponto de operação, mesmo que distinto dos valores iniciais, o sistema é considerado <strong>estável</strong>.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="execução-do-estudo-de-estabilidade-no-psp-ufu"></a>Execução do estudo de estabilidade no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#execução-do-estudo-de-estabilidade-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, assim como os <a href="/PSP/docs/controlEditor">controles</a> das <a href="/PSP/docs/syncGenerator">máquinas síncronas</a>, a execução da estabilidade é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no ícone do botão <strong>Executar Estabilidade</strong>. Ao clicar na parte inferior do botão &quot;Executar Estabilidade&quot; será exibido um menu suspenso com a opção <strong>Lista de eventos de estabilidade</strong>.</p><div><center><img src="/PSP/images/menuSimulationStability.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><p>Ao executar o cálculo de estabilidade, uma caixa com o status do processo de simulação será exibida, indicando primeiramente a inicialização do estudo e posteriormente o tempo de simulação calculado.</p><p>Ao clicar na lista de eventos de estabilidade, será exibida uma janela com a descrição dos eventos de estabilidade inseridos.</p><div><center><img src="/PSP/images/stabList.png" alt="Lista de eventos de estabilidade" title="Lista de eventos de estabilidade"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="inserindo-um-evento-de-estabilidade"></a>Inserindo um evento de estabilidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#inserindo-um-evento-de-estabilidade" title="Direct link to heading">#</a></h3><p>Os distúrbios mais comuns aplicados em estudos de estabilidade transitória são faltas e chaveamentos. Tais operações são facilmente realizadas no PSP-UFU:</p><ul><li><p><strong>Faltas</strong>: Curtos-circuitos trifásicos podem ser inseridos nos <a href="/PSP/docs/bus">barramentos</a> do sistema por meio da inclusão da impedância de falta na matriz admitância de barras da mesma maneira realizada em um elemento <em>shunt</em>. Tal valor é definido pelo usuário e caso seja um curto-circuito franco, um valor não nulo, mas suficientemente próximo de zero, é aplicado, de forma que a tensão no barramento é levada a zero durante o distúrbio.</p></li><li><p><strong>Chaveamento de ramo</strong>: Da mesma forma que as faltas, o chaveamento de ramos é realizado por meio da alteração na matriz admitância, removendo ou inserindo os parâmetros do elemento a ser chaveado.
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href="#!">Control Element Editor</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/controlEditor">Editor de Controle</a></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!" tabindex="-1">Control Elements</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/io">Entrada / Saída</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/transferFunction">Função Transferência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/sum">Somador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/multiplier">Multiplicador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Simulations</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Estabilidade</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Estabilidade de sistemas de potência é a habilidade de um Sistema Elétrico de Potência (SEP), para uma certa condição de operação, de reestabelecer um estado de operação de equilíbrio após ser submetido a um distúrbio físico. A estabilidade é reconhecida como um importante problema para a segurança da operação desde a década de 1920. Muitos <em>blackouts</em> de grande porte causados pela instabilidade do SEP têm ilustrado a importância desse fenômeno.</p><div><center><img src="/PSP/images/stability.svg" alt="Estabilidade" title="Estabilidade"></center></div><p>Normalmente são realizados dois tipos de estudos de estabilidade (uma classificação mais aprofundada pode ser observada <a href="https://doi.org/10.1109/TPWRS.2004.825981" target="_blank" rel="noopener noreferrer">nesse artigo</a>). A recuperação de uma rápida e brusca perturbação é chamada de <strong>estabilidade transitória</strong>, e sua solução é geralmente obtida no domínio do tempo. Outro estudo é denominado <strong>estabilidade dinâmica</strong>, utilizado para descrever a resposta do sistema frente a pequenas perturbações ao longo de muito tempo, o qual pode ser resolvido tanto no domínio da frequência quanto no domínio do tempo.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>No PSP-UFU, a estabilidade dinâmica é tratada como uma extensão da estabilidade transitória, em que os efeitos de pequenas constantes de tempo devem ser ignorados.</p></div></div><p>O modelo para representação do SEP em um estudo de estabilidade transitória é feito por meio do sistema de equações algébrico-diferenciais (EADs), em que o estado inicial é considerado estável e originado de um estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>. A solução das equações se dá por meio de um <a href="https://en.wikipedia.org/wiki/Numerical_integration" target="_blank" rel="noopener noreferrer">método de integração numérica</a> e, então, é observada a resposta do sistema de potência.</p><p>Caso a solução no domínio do tempo, após alguma perturbação, leve um ou mais parâmetros do sistema a valores fisicamente impossíveis ou, até mesmo, a solução das equações diferenciais não obtenha convergência em processos iterativos, o sistema é dito <strong>instável</strong>. Em contrapartida, caso as variáveis de estado dos elementos do sistema oscilem após uma perturbação e se estabeleçam em um novo ponto de operação, mesmo que distinto dos valores iniciais, o sistema é considerado <strong>estável</strong>.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="execução-do-estudo-de-estabilidade-no-psp-ufu"></a>Execução do estudo de estabilidade no PSP-UFU<a class="hash-link" href="#execução-do-estudo-de-estabilidade-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, assim como os <a href="/PSP/docs/controlEditor">controles</a> das <a href="/PSP/docs/syncGenerator">máquinas síncronas</a>, a execução da estabilidade é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no ícone do botão <strong>Executar Estabilidade</strong>. Ao clicar na parte inferior do botão &quot;Executar Estabilidade&quot; será exibido um menu suspenso com a opção <strong>Lista de eventos de estabilidade</strong>.</p><div><center><img src="/PSP/images/menuSimulationStability.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><p>Ao executar o cálculo de estabilidade, uma caixa com o status do processo de simulação será exibida, indicando primeiramente a inicialização do estudo e posteriormente o tempo de simulação calculado.</p><p>Ao clicar na lista de eventos de estabilidade, será exibida uma janela com a descrição dos eventos de estabilidade inseridos.</p><div><center><img src="/PSP/images/stabList.png" alt="Lista de eventos de estabilidade" title="Lista de eventos de estabilidade"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserindo-um-evento-de-estabilidade"></a>Inserindo um evento de estabilidade<a class="hash-link" href="#inserindo-um-evento-de-estabilidade" title="Direct link to heading">#</a></h3><p>Os distúrbios mais comuns aplicados em estudos de estabilidade transitória são faltas e chaveamentos. Tais operações são facilmente realizadas no PSP-UFU:</p><ul><li><p><strong>Faltas</strong>: Curtos-circuitos trifásicos podem ser inseridos nos <a href="/PSP/docs/bus">barramentos</a> do sistema por meio da inclusão da impedância de falta na matriz admitância de barras da mesma maneira realizada em um elemento <em>shunt</em>. Tal valor é definido pelo usuário e caso seja um curto-circuito franco, um valor não nulo, mas suficientemente próximo de zero, é aplicado, de forma que a tensão no barramento é levada a zero durante o distúrbio.</p></li><li><p><strong>Chaveamento de ramo</strong>: Da mesma forma que as faltas, o chaveamento de ramos é realizado por meio da alteração na matriz admitância, removendo ou inserindo os parâmetros do elemento a ser chaveado.
 Cada <a href="/PSP/docs/powerEditor">elemento de potência</a> possui um botão de &quot;Chaveamento&quot; ou &quot;Estabilidade&quot;, em que pode ser inserido os tempos de remoção e/ou inserção do componente.</p></li><li><p><strong>Chaveamento de máquinas</strong>: A remoção de uma máquina síncrona é efetivada com a retirada de sua participação no vetor de correntes, além da remoção de sua admitância fictícia.
-Assim como os <a href="/PSP/docs/powerEditor">elementos de potência</a>, as <a href="/PSP/docs/syncGenerator">máquinas síncronas</a> possui um botão &quot;Chaveamento&quot;, em que pode ser inserido os tempos de remoção e/ou inserção do componente.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Mesmo que removida da barra, os parâmetros das <a href="/PSP/docs/syncGenerator">máquinas síncronas</a> continuam a ser calculados com a corrente do estator nula, podendo fornecer resultados em uma eventual reconexão.</p></div></div></li></ul><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Eventos nos <strong>sistemas de controle</strong> podem ser facilmente introduzidos com o bloco de <a href="/PSP/docs/mathExpression">expressão matemática</a>. Nesse caso, tais eventos <strong>não</strong> serão exibidos na lista de eventos de estabilidade.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="erros-comuns-na-execução-do-estudo-de-estabilidade"></a>Erros comuns na execução do estudo de estabilidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#erros-comuns-na-execução-do-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao estudo de estabilidade.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="a-seguinte-mensagem-de-erro-é-exibida-não-foi-possível-construir-a-matriz-admitância"></a>A seguinte mensagem de erro é exibida: &quot;Não foi possível construir a matriz admitância&quot;<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-não-foi-possível-construir-a-matriz-admitância" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível construir a matriz admitância de barras. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros do circuito estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-os-valores-de-saturação-do"></a>A seguinte mensagem de erro é exibida: &quot;Erro ao inicializar os valores de saturação do...&quot;<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-os-valores-de-saturação-do" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível calcular os fatores de saturação da <a href="/PSP/docs/syncGenerator">máquina síncrona</a>. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>O fator de saturação é menor que 1,2</strong>. Esse valor deve ser maior que 1,2, ou irá gerar erros na simulação. Caso não seja informado, a saturação da máquina não é considerada nos cálculos.</li><li><strong>Os parâmetros da máquina síncrona estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão das máquinas síncronas sejam inseridos, o cálculo dos fatores de saturação pode divergir. Verifique se os dados foram inseridos corretamente.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-avr--regulador-de-velocidade"></a>A seguinte mensagem de erro é exibida: &quot;Erro ao inicializar o AVR / regulador de velocidade...&quot;<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-avr--regulador-de-velocidade" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível <a href="/PSP/docs/controlEditor#inicializa%C3%A7%C3%A3o-do-sistema-de-controle">inicializar o sistema de controle</a> de uma <a href="/PSP/docs/syncGenerator">máquina síncrona</a>. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>O sistema de controle está vazio</strong>. Caso esteja habilitado o AVR e/ou o regulador de velocidade e o controle não foi inserido, esse erro pode ser acionado. Insira o controle da máquina ou desmarque a opção de utilização do AVR e/ou regulador de velocidade.</li><li><strong>O sistema de controle não possui ao menos uma entrada e uma saída</strong>. O sistema de controle deve ter ao menos uma <a href="/PSP/docs/io">entrada e uma saída</a>, caso contrário apresentará erro de execução.</li><li><strong>O passo de integração está muito pequeno</strong>. Caso o passo de integração esteja muito pequeno, os cálculos gerarão erros e irão divergir. Reduza o passo de integração nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="a-seguinte-mensagem-de-erro-é-exibida-impossível-resolver-as-máquinas-do-sistema"></a>A seguinte mensagem de erro é exibida: &quot;Impossível resolver as máquinas do sistema&quot;<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-impossível-resolver-as-máquinas-do-sistema" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível resolver as EADs das máquinas síncronas inseridas no sistemas. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros da máquina síncrona estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão das máquinas síncronas sejam inseridos, o cálculo das EADs pode se tornar impossível. Verifique se os dados foram inseridos corretamente.</li><li><strong>O passo de integração está muito pequeno</strong>. Caso o passo de integração esteja muito pequeno, os cálculos gerarão erros e irão divergir. Reduza o passo de integração nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</li><li><strong>Os parâmetros do circuito estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-escorregamento-do-motor"></a>A seguinte mensagem de erro é exibida: &quot;Erro ao inicializar o escorregamento do motor...&quot;<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-escorregamento-do-motor" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível calcular o valor de escorregamento inicial do motor de indução. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros de estabilidade do motor estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão dos motores de indução sejam inseridos, o cálculo do escorregamento pode se tornar impossível. Verifique se os dados foram inseridos corretamente.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="estrutura-da-ferramenta-de-estabilidade"></a>Estrutura da ferramenta de estabilidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#estrutura-da-ferramenta-de-estabilidade" title="Direct link to heading">#</a></h2><p>A estabilidade de um SEP é um problema dinâmico e necessita de modelos mais elaborados de elementos de potência comparados àqueles apresentados nos outros estudos. Esses modelos são descritos individualmente, com destaque às <a href="/PSP/docs/syncGenerator#gerador-s%C3%ADncrono-no-estudo-de-estabilidade">máquinas síncronas</a>, <a href="/PSP/docs/load#carga-no-estudo-de-estabilidade">cargas ZIP</a> e <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-estudo-de-estabilidade">motores de indução</a>.</p><p>A representação dos demais componentes do sistema elétrico: <a href="/PSP/docs/line">linhas de transmissão</a>, <a href="/PSP/docs/transformer">transformadores</a> e elementos <em>shunt</em> (com exceção de cargas ZIP), que formam a rede de transmissão ou distribuição balanceada, é realizada utilizando os mesmos modelos do <a href="/PSP/docs/powerFlow">fluxo de carga</a>.</p><p>Os modelos dinâmicos para a análise no domínio do tempo são na forma de um sistema de equações algébrico-diferenciais (EADs), descritas a seguir:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mi>g</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\dfrac{dx}{dt}= f(x,y,u)\\ g = f(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault">u</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.03588em">g</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>	são as variáveis de estado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span></span></span>	são as variáveis de algébricas</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">u</span></span></span></span></span>	são as variáveis de entrada</li></ul><p>As variáveis de entrada são inseridas pelo usuário e permanecem constantes durante todo o processo de cálculo. Para o problema de estabilidade, as variáveis algébricas iniciais correspondem às tensões e ângulos das barras calculadas no <a href="/PSP/docs/powerFlow">fluxo de carga</a>. As variáveis de estado iniciais são estimadas após a convergência do fluxo de potência (inicialização dos elementos dinâmicos).</p><p>Na sequência é apresentado, na forma de fluxogramas, a estrutura do módulo de estabilidade implementado no PSP-UFU. O fluxograma abaixo mostra a estrutura geral da ferramenta de estabilidade.</p><div><center><img src="/PSP/images/stabGeneral.svg" alt="Estrutura geral da ferramenta de estabilidade" title="Estrutura geral da ferramenta de estabilidade"></center></div><p>A inicialização da rede elétrica e dos elementos dinâmicos, assim como o precesso iterativo para solução das EADs são apresentados nos dois fluxogramas conseguintes.</p><div><center><img src="/PSP/images/stabInit.svg" alt="Inicialização da ferramenta de estabilidade" title="Inicialização da ferramenta de estabilidade"></center></div><div><center><img src="/PSP/images/stabIterative.svg" alt="Processo iterativo para cálculo dos elementos dinâmicos" title="Estrutura geral da ferramenta de estabilidade"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="integração-numérica"></a>Integração numérica<a aria-hidden="true" tabindex="-1" class="hash-link" href="#integração-numérica" title="Direct link to heading">#</a></h3><p>Para resolver as equações diferenciais da máquina é necessário um método de integração numérica. Tais métodos são classificados em dois grupos: métodos explícitos e métodos implícitos. Os métodos explícitos, devido à sua formulação, calculam diretamente o estado do sistema em um instante de tempo posterior, enquanto métodos implícitos envolvem estados atuais e posteriores em suas equações, exigindo, portanto, um processo iterativo.</p><p>As constantes de tempo presentes no estudo de estabilidade têm uma grande variação em seu valor (podem variar de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">10^{-3}~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">10~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span>). Isso torna o sistema de equações diferenciais da máquina síncrona como um sistema rígido (<em><a href="https://en.wikipedia.org/wiki/Stiff_equation" target="_blank" rel="noopener noreferrer">stiff equation</a></em>). Caso a análise da estabilidade numérica tanto das equações diferenciais rígidas quanto do método de integração obtenham o mesmo comportamento, o método é chamado de absolutamente estável, ou A-estável.</p><p>Métodos de integração numérica explícitos, como por exemplo o <a href="https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods" target="_blank" rel="noopener noreferrer">Runge-Kutta</a> de quarta ordem, não podem ser A-estáveis e, portanto, normalmente possuem comportamento ruim em problemas com equações diferenciais rígidas. Por outro lado, métodos implícitos podem ser A-estáveis. Um método implícito adequado para solução do comportamento dinâmico de sistemas elétricos é o <strong>Trapezoidal Implícito</strong>, por possuir as seguintes vantagens:</p><ul><li>É numericamente estável (A-estável);</li><li>É bastante rápida;</li><li>Possui boa precisão (dependendo somente do passo de integração utilizado).</li></ul><p>Tal método foi implementado no PSP-UFU tanto para solução das equações diferenciais da máquina síncrona quanto nas <a href="/PSP/docs/transferFunction">funções transferência</a> do <a href="/PSP/docs/controlEditor">sistema de controle</a>. Sua formulação é dada pela seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><mi>h</mi><mrow><mo fence="true">(</mo><msubsup><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><msubsup><mi>y</mi><mi>n</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y_{n+1}=y_n + 0{,}5 h \left(y_{n+1}&#x27; + y_{n}&#x27; \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	é o passo de integração</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor do estado no passo anterior (instante <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">t_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>)</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_{n+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor do estado no passo atual (instante <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">t_{n+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8234109999999999em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>)</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>y</mi><mi>n</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">y_{n}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.998892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span>	é a equação diferencial no passo anterior</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">y_{n+1}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.058331em;vertical-align:-0.30643899999999996em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.451892em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.30643899999999996em"><span></span></span></span></span></span></span></span></span></span></span>	é a equação diferencial no passo atual</li></ul><p>Nota-se nessa equação que o método é implícito devido ao termo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> aparecer em ambos os lados da equação, por isso é necessário um processo iterativo para sua solução.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>KUNDUR, P.; et al. Definition and classification of power system stability. IEEE Transactions Power Systems, v. 19, n. 2, mai 2004, p. 1387-1401. doi: <a href="https://doi.org/10.1109/TPWRS.2004.825981" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPWRS.2004.825981</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>PARK, R. H. Two-reactions Theory of Synchronous Machine. Generalized Method of Analysis – Part I. AIEE Transactions, v. 48, n. 3, jul 1929. doi: <a href="https://doi.org/10.1109/T-AIEE.1929.5055275" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/T-AIEE.1929.5055275</a></li><li>ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: <a href="https://doi.org/10.1109/9780470545577" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/9780470545577</a></li><li>SAUER, P. W.; PAI, M. A. Power System Dynamics and Stability. Pretience Hall, Upper Saddle River, 1998.</li><li>KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994.</li><li>DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: <a href="https://doi.org/10.1109/TPAS.1972.293341" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPAS.1972.293341</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/stability.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/harmonics"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Harmônicos</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/simulationConfig"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Configurações da Simulação »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#execução-do-estudo-de-estabilidade-no-psp-ufu" class="table-of-contents__link">Execução do estudo de estabilidade no PSP-UFU</a><ul><li><a href="#inserindo-um-evento-de-estabilidade" class="table-of-contents__link">Inserindo um evento de estabilidade</a></li><li><a href="#erros-comuns-na-execução-do-estudo-de-estabilidade" class="table-of-contents__link">Erros comuns na execução do estudo de estabilidade</a></li></ul></li><li><a href="#estrutura-da-ferramenta-de-estabilidade" class="table-of-contents__link">Estrutura da ferramenta de estabilidade</a><ul><li><a href="#integração-numérica" class="table-of-contents__link">Integração numérica</a></li></ul></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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+Assim como os <a href="/PSP/docs/powerEditor">elementos de potência</a>, as <a href="/PSP/docs/syncGenerator">máquinas síncronas</a> possui um botão &quot;Chaveamento&quot;, em que pode ser inserido os tempos de remoção e/ou inserção do componente.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Mesmo que removida da barra, os parâmetros das <a href="/PSP/docs/syncGenerator">máquinas síncronas</a> continuam a ser calculados com a corrente do estator nula, podendo fornecer resultados em uma eventual reconexão.</p></div></div></li></ul><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Eventos nos <strong>sistemas de controle</strong> podem ser facilmente introduzidos com o bloco de <a href="/PSP/docs/mathExpression">expressão matemática</a>. Nesse caso, tais eventos <strong>não</strong> serão exibidos na lista de eventos de estabilidade.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="erros-comuns-na-execução-do-estudo-de-estabilidade"></a>Erros comuns na execução do estudo de estabilidade<a class="hash-link" href="#erros-comuns-na-execução-do-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao estudo de estabilidade.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-não-foi-possível-construir-a-matriz-admitância"></a>A seguinte mensagem de erro é exibida: &quot;Não foi possível construir a matriz admitância&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-não-foi-possível-construir-a-matriz-admitância" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível construir a matriz admitância de barras. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros do circuito estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-os-valores-de-saturação-do"></a>A seguinte mensagem de erro é exibida: &quot;Erro ao inicializar os valores de saturação do...&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-os-valores-de-saturação-do" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível calcular os fatores de saturação da <a href="/PSP/docs/syncGenerator">máquina síncrona</a>. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>O fator de saturação é menor que 1,2</strong>. Esse valor deve ser maior que 1,2, ou irá gerar erros na simulação. Caso não seja informado, a saturação da máquina não é considerada nos cálculos.</li><li><strong>Os parâmetros da máquina síncrona estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão das máquinas síncronas sejam inseridos, o cálculo dos fatores de saturação pode divergir. Verifique se os dados foram inseridos corretamente.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-avr--regulador-de-velocidade"></a>A seguinte mensagem de erro é exibida: &quot;Erro ao inicializar o AVR / regulador de velocidade...&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-avr--regulador-de-velocidade" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível <a href="/PSP/docs/controlEditor#inicializa%C3%A7%C3%A3o-do-sistema-de-controle">inicializar o sistema de controle</a> de uma <a href="/PSP/docs/syncGenerator">máquina síncrona</a>. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>O sistema de controle está vazio</strong>. Caso esteja habilitado o AVR e/ou o regulador de velocidade e o controle não foi inserido, esse erro pode ser acionado. Insira o controle da máquina ou desmarque a opção de utilização do AVR e/ou regulador de velocidade.</li><li><strong>O sistema de controle não possui ao menos uma entrada e uma saída</strong>. O sistema de controle deve ter ao menos uma <a href="/PSP/docs/io">entrada e uma saída</a>, caso contrário apresentará erro de execução.</li><li><strong>O passo de integração está muito pequeno</strong>. Caso o passo de integração esteja muito pequeno, os cálculos gerarão erros e irão divergir. Reduza o passo de integração nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-impossível-resolver-as-máquinas-do-sistema"></a>A seguinte mensagem de erro é exibida: &quot;Impossível resolver as máquinas do sistema&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-impossível-resolver-as-máquinas-do-sistema" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível resolver as EADs das máquinas síncronas inseridas no sistemas. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros da máquina síncrona estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão das máquinas síncronas sejam inseridos, o cálculo das EADs pode se tornar impossível. Verifique se os dados foram inseridos corretamente.</li><li><strong>O passo de integração está muito pequeno</strong>. Caso o passo de integração esteja muito pequeno, os cálculos gerarão erros e irão divergir. Reduza o passo de integração nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</li><li><strong>Os parâmetros do circuito estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-escorregamento-do-motor"></a>A seguinte mensagem de erro é exibida: &quot;Erro ao inicializar o escorregamento do motor...&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-escorregamento-do-motor" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível calcular o valor de escorregamento inicial do motor de indução. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros de estabilidade do motor estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão dos motores de indução sejam inseridos, o cálculo do escorregamento pode se tornar impossível. Verifique se os dados foram inseridos corretamente.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="estrutura-da-ferramenta-de-estabilidade"></a>Estrutura da ferramenta de estabilidade<a class="hash-link" href="#estrutura-da-ferramenta-de-estabilidade" title="Direct link to heading">#</a></h2><p>A estabilidade de um SEP é um problema dinâmico e necessita de modelos mais elaborados de elementos de potência comparados àqueles apresentados nos outros estudos. Esses modelos são descritos individualmente, com destaque às <a href="/PSP/docs/syncGenerator#gerador-s%C3%ADncrono-no-estudo-de-estabilidade">máquinas síncronas</a>, <a href="/PSP/docs/load#carga-no-estudo-de-estabilidade">cargas ZIP</a> e <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-estudo-de-estabilidade">motores de indução</a>.</p><p>A representação dos demais componentes do sistema elétrico: <a href="/PSP/docs/line">linhas de transmissão</a>, <a href="/PSP/docs/transformer">transformadores</a> e elementos <em>shunt</em> (com exceção de cargas ZIP), que formam a rede de transmissão ou distribuição balanceada, é realizada utilizando os mesmos modelos do <a href="/PSP/docs/powerFlow">fluxo de carga</a>.</p><p>Os modelos dinâmicos para a análise no domínio do tempo são na forma de um sistema de equações algébrico-diferenciais (EADs), descritas a seguir:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mi>g</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\dfrac{dx}{dt}= f(x,y,u)\\ g = f(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault">u</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.03588em">g</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>	são as variáveis de estado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span></span></span>	são as variáveis de algébricas</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">u</span></span></span></span></span>	são as variáveis de entrada</li></ul><p>As variáveis de entrada são inseridas pelo usuário e permanecem constantes durante todo o processo de cálculo. Para o problema de estabilidade, as variáveis algébricas iniciais correspondem às tensões e ângulos das barras calculadas no <a href="/PSP/docs/powerFlow">fluxo de carga</a>. As variáveis de estado iniciais são estimadas após a convergência do fluxo de potência (inicialização dos elementos dinâmicos).</p><p>Na sequência é apresentado, na forma de fluxogramas, a estrutura do módulo de estabilidade implementado no PSP-UFU. O fluxograma abaixo mostra a estrutura geral da ferramenta de estabilidade.</p><div><center><img src="/PSP/images/stabGeneral.svg" alt="Estrutura geral da ferramenta de estabilidade" title="Estrutura geral da ferramenta de estabilidade"></center></div><p>A inicialização da rede elétrica e dos elementos dinâmicos, assim como o precesso iterativo para solução das EADs são apresentados nos dois fluxogramas conseguintes.</p><div><center><img src="/PSP/images/stabInit.svg" alt="Inicialização da ferramenta de estabilidade" title="Inicialização da ferramenta de estabilidade"></center></div><div><center><img src="/PSP/images/stabIterative.svg" alt="Processo iterativo para cálculo dos elementos dinâmicos" title="Estrutura geral da ferramenta de estabilidade"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="integração-numérica"></a>Integração numérica<a class="hash-link" href="#integração-numérica" title="Direct link to heading">#</a></h3><p>Para resolver as equações diferenciais da máquina é necessário um método de integração numérica. Tais métodos são classificados em dois grupos: métodos explícitos e métodos implícitos. Os métodos explícitos, devido à sua formulação, calculam diretamente o estado do sistema em um instante de tempo posterior, enquanto métodos implícitos envolvem estados atuais e posteriores em suas equações, exigindo, portanto, um processo iterativo.</p><p>As constantes de tempo presentes no estudo de estabilidade têm uma grande variação em seu valor (podem variar de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">10^{-3}~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">10~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span>). Isso torna o sistema de equações diferenciais da máquina síncrona como um sistema rígido (<em><a href="https://en.wikipedia.org/wiki/Stiff_equation" target="_blank" rel="noopener noreferrer">stiff equation</a></em>). Caso a análise da estabilidade numérica tanto das equações diferenciais rígidas quanto do método de integração obtenham o mesmo comportamento, o método é chamado de absolutamente estável, ou A-estável.</p><p>Métodos de integração numérica explícitos, como por exemplo o <a href="https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods" target="_blank" rel="noopener noreferrer">Runge-Kutta</a> de quarta ordem, não podem ser A-estáveis e, portanto, normalmente possuem comportamento ruim em problemas com equações diferenciais rígidas. Por outro lado, métodos implícitos podem ser A-estáveis. Um método implícito adequado para solução do comportamento dinâmico de sistemas elétricos é o <strong>Trapezoidal Implícito</strong>, por possuir as seguintes vantagens:</p><ul><li>É numericamente estável (A-estável);</li><li>É bastante rápida;</li><li>Possui boa precisão (dependendo somente do passo de integração utilizado).</li></ul><p>Tal método foi implementado no PSP-UFU tanto para solução das equações diferenciais da máquina síncrona quanto nas <a href="/PSP/docs/transferFunction">funções transferência</a> do <a href="/PSP/docs/controlEditor">sistema de controle</a>. Sua formulação é dada pela seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><mi>h</mi><mrow><mo fence="true">(</mo><msubsup><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><msubsup><mi>y</mi><mi>n</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y_{n+1}=y_n + 0{,}5 h \left(y_{n+1}&#x27; + y_{n}&#x27; \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	é o passo de integração</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor do estado no passo anterior (instante <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">t_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>)</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_{n+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor do estado no passo atual (instante <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">t_{n+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8234109999999999em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>)</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>y</mi><mi>n</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">y_{n}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.998892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span>	é a equação diferencial no passo anterior</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">y_{n+1}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.058331em;vertical-align:-0.30643899999999996em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.451892em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.30643899999999996em"><span></span></span></span></span></span></span></span></span></span></span>	é a equação diferencial no passo atual</li></ul><p>Nota-se nessa equação que o método é implícito devido ao termo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> aparecer em ambos os lados da equação, por isso é necessário um processo iterativo para sua solução.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>KUNDUR, P.; et al. Definition and classification of power system stability. IEEE Transactions Power Systems, v. 19, n. 2, mai 2004, p. 1387-1401. doi: <a href="https://doi.org/10.1109/TPWRS.2004.825981" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPWRS.2004.825981</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>PARK, R. H. Two-reactions Theory of Synchronous Machine. Generalized Method of Analysis – Part I. AIEE Transactions, v. 48, n. 3, jul 1929. doi: <a href="https://doi.org/10.1109/T-AIEE.1929.5055275" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/T-AIEE.1929.5055275</a></li><li>ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: <a href="https://doi.org/10.1109/9780470545577" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/9780470545577</a></li><li>SAUER, P. W.; PAI, M. A. Power System Dynamics and Stability. Pretience Hall, Upper Saddle River, 1998.</li><li>KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994.</li><li>DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: <a href="https://doi.org/10.1109/TPAS.1972.293341" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPAS.1972.293341</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/stability.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/harmonics"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Harmônicos</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/simulationConfig"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Configurações da Simulação »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#execução-do-estudo-de-estabilidade-no-psp-ufu" class="table-of-contents__link">Execução do estudo de estabilidade no PSP-UFU</a><ul><li><a href="#inserindo-um-evento-de-estabilidade" class="table-of-contents__link">Inserindo um evento de estabilidade</a></li><li><a href="#erros-comuns-na-execução-do-estudo-de-estabilidade" class="table-of-contents__link">Erros comuns na execução do estudo de estabilidade</a></li></ul></li><li><a href="#estrutura-da-ferramenta-de-estabilidade" class="table-of-contents__link">Estrutura da ferramenta de estabilidade</a><ul><li><a href="#integração-numérica" class="table-of-contents__link">Integração numérica</a></li></ul></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="docMainContainer_3EyW"><div class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Somador</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Como o próprio nome sugere, os somadores são elementos de controle capazes de somar qualquer quantidade de valores reais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-de-dados-do-somador"></a>Formulário de edição de dados do somador<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-do-somador" title="Direct link to heading">#</a></h2><p>A quantidade de entradas, assim como seus sinais são definidas pelo usuário, como mostra a figura abaixo.</p><div><center><img src="/PSP/images/sumForm.png" alt="Formulário de edição de dados do somador no PSP-UFU" title="Formulário de edição de dados do somador no PSP-UFU"></center></div><p>No seu único campo “<strong>Sinais</strong>” é inserida uma lista de sinais <strong>separados por espaços</strong> contendo os símbolos “+” para uma entrada positiva e “-” para negativa.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O usuário pode inserir duas ou mais entradas de sinais no bloco.</p></div></div><p>Uma vez inseridas novas entradas, nós correspondentes serão incluídos no elemento gráfico do somador, o qual poderá ser conectado por meio das linhas de conexão.</p><p>De forma 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Somador</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Como o próprio nome sugere, os somadores são elementos de controle capazes de somar qualquer quantidade de valores reais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-somador"></a>Formulário de edição de dados do somador<a class="hash-link" href="#formulário-de-edição-de-dados-do-somador" title="Direct link to heading">#</a></h2><p>A quantidade de entradas, assim como seus sinais são definidas pelo usuário, como mostra a figura abaixo.</p><div><center><img src="/PSP/images/sumForm.png" alt="Formulário de edição de dados do somador no PSP-UFU" title="Formulário de edição de dados do somador no PSP-UFU"></center></div><p>No seu único campo “<strong>Sinais</strong>” é inserida uma lista de sinais <strong>separados por espaços</strong> contendo os símbolos “+” para uma entrada positiva e “-” para negativa.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O usuário pode inserir duas ou mais entradas de sinais no bloco.</p></div></div><p>Uma vez inseridas novas entradas, nós correspondentes serão incluídos no elemento gráfico do somador, o qual poderá ser conectado por meio das linhas de conexão.</p><p>De forma semelhante, o usuário poderá excluir entradas de sinais pré-existentes, as quais serão removidas no ícone gráfico pela remoção do nó e da linha (caso exista alguma conectada).</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/sum.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Gerador Síncrono</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Uma máquina de corrente alternada na qual a frequência das tensões geradas e a velocidade da máquina estão em uma proporção constante. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=411-31-08" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="gerador-síncrono-no-psp-ufu"></a>Gerador Síncrono no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gerador-síncrono-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Os geradores síncronos são a fonte de potência para o cálculo do fluxo de carga e estabilidade, além de serem um dos principais contribuintes para as correntes de falta.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Somente barramentos que possuem esse elemento conectado podem ser considerados <a href="/PSP/docs/powerFlow">barras de referência</a>. <a href="/PSP/docs/powerFlow">Barras do tipo PV</a> devem conter um gerador síncrono ou um <a href="/PSP/docs/syncMotor">motor síncrono (compensador síncrono)</a>.</p></div></div><p>O formulário de dados é dividido em dois, sendo o primeiro responsável pelos dados gerais, de <a href="/PSP/docs/powerFlow">fluxo de carga</a> e de <a href="/PSP/docs/fault">falta</a> e um segundo pelos dados de <a href="/PSP/docs/stability">estabilidade</a> . Esse último também permite o acesso aos controles das máquinas síncronas manipulados pelo editor de controle.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os dados referentes às <strong>impedâncias de sequência positiva do gerador síncrono</strong> inseridas no <a href="/PSP/docs/syncGenerator#falta">contexto (aba) &quot;Falta&quot;</a> são utilizados <strong>tanto para os estudos de <a href="/PSP/docs/fault">curto-circuito</a> quanto para os estudos de <a href="/PSP/docs/harmonics">harmônicos</a></strong>.</p><p>Esses dados são ignorados nos estudos de <a href="/PSP/docs/powerFlow">fluxo de carga</a> (<em>não são utilizado nesse estudo</em>) e <a href="/PSP/docs/stability">estabilidade</a> (<em>são utilizados dados inseridos em formulário específico</em>).</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="gerador-síncrono-no-fluxo-de-carga"></a>Gerador Síncrono no fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gerador-síncrono-no-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>O gerador síncrono é a fonte de potência do PSP-UFU no estudo de fluxo de carga. Seu comportamento difere de acordo com o tipo de barra conectada:</p><ul><li><a href="/PSP/docs/powerFlow">Barra de referência</a>: Os dados de potência ativa e reativa inseridos são desprezados, uma vez que esse elemento será utilizado para completar o balanço de potência do estudo de fluxo d carga;</li><li><a href="/PSP/docs/powerFlow">Barra PV</a>: O dado de potência ativa é considerado, porém o dados de potência reativa são desprezados. O valor da potência reativa é utilizado para manter o módulo da tensão constante no barramento conectado;</li><li><a href="/PSP/docs/powerFlow">Barra PQ</a>: Para os geradores conectados a essa barra tanto a potência ativa quanto reativa inseridas são consideradas.</li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>Caso o limite de potência reativa seja excedido, o programa automaticamente transforma a <a href="/PSP/docs/powerFlow">Barra PV</a> conectada em uma <a href="/PSP/docs/powerFlow">Barra PQ</a>, utilizando o valor limite de potência reativa que seria ultrapassado.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="gerador-síncrono-no-estudo-de-curto-circuito"></a>Gerador Síncrono no estudo de curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gerador-síncrono-no-estudo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>Enquanto os geradores no estudo de fluxo de carga são modelados somente por correntes injetadas nas barras, para o curto-circuito utiliza-se uma tensão atrás de uma impedância. A figura abaixo mostra o caminho da corrente e o circuito equivalente de cada sequência nos geradores.</p><div><center><img src="/PSP/images/sycGeneratorSeq.svg" alt="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero" title="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero"></center></div><p>As tensões geradas são somente de sequência positiva, uma vez que o gerador fornece sempre tensões trifásicas equilibradas. Portanto a rede de sequência positiva é composta de uma tensão pré-falta atrás de uma impedância de sequência positiva. As redes de sequência negativa e zero não contêm forças eletromotrizes, porém incluem as impedâncias do gerador de sequência negativa e zero.</p><p>A corrente que circula na impedância <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> entre o neutro e a terra é <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">3\dot{I}_{a0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>. Pela figura acima (c), observa-se que a queda de tensão de sequência zero do ponto a para terra (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\dot{V}_{a0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>3</mn><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>g</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\dot{V}_{a0} = -3\dot{I}_{a0}\overline{z}_n - \dot{I}_{a0}\overline{z}_{g0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord">−</span><span class="mord">3</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span></div><p>A rede de sequência zero, que é um circuito monofásico pelo qual se supõe que circule apenas corrente de sequência zero e deve, portanto, ter uma impedância definida pela seguinte equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>0</mn></msub><mo>=</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>g</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{0} = 3\overline{z}_n + \overline{z}_{g0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.916668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a seguinte mensagem de erro pode ser exibida:</p><p><center>&quot;<i>Falha ao inverter a matriz admitância de sequência zero</i>&quot;</center></p><p>Isso ocorre porque a matriz admitância de sequência zero é singular. Para contornar esse problema escolha <strong>uma</strong> das duas soluções abaixo:</p><ol><li>Marque a opção &quot;Neutro aterrado&quot; e insira um alto valor de reatância de aterramento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>9999</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j9999~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo);</li><li><strong>Ou</strong>, na barra do gerador, insira um <a href="/PSP/docs/inductor">reator</a> de baixo valor de potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">1,0~var</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault" style="margin-right:0.03588em">v</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>, por exemplo).</li></ol></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="gerador-síncrono-no-estudo-de-estabilidade"></a>Gerador Síncrono no estudo de estabilidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gerador-síncrono-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A relação de valores observados em testes adequados (definidos na <a href="https://doi.org/10.1109/IEEESTD.2020.9050934" target="_blank" rel="noopener noreferrer">IEEE Std. 115-2019</a>), denominados parâmetros padrões, são utilizados para modelar a máquina síncrona no estudo de estabilidade do PSP-UFU.</p><p>Os parâmetros da máquina síncrona que influenciam rapidamente no decaimento de valores são chamados <strong>subtransitórios</strong> (indicados por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">&#x27;&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>), aqueles que influenciam mais lentamente são chamados <strong>transitórios</strong> (indicados por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>) e, finalmente, aqueles que influenciam continuamente são chamados de parâmetros <strong>síncronos</strong> (sem indicação de sobrescrito).</p><p>Um conjunto de equações algébrico-diferenciais determinam o comportamento da máquina síncrona no estudo de estabilidade:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msub><mi>V</mi><mi>q</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>I</mi><mi>d</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msub><mi>V</mi><mi>d</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>I</mi><mi>q</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msub><mi>V</mi><mi>q</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mi>I</mi><mi>d</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msub><mi>V</mi><mi>d</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mi>I</mi><mi>q</mi><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mrow><mi>f</mi><mi>d</mi></mrow></msub><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>d</mi><mn>0</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msub><mi>s</mi><mi>q</mi></msub><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>q</mi><mn>0</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msubsup><mi>x</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>d</mi><mn>0</mn></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msubsup><mi>x</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>q</mi><mn>0</mn></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mfrac><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>ω</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><msub><mi>ω</mi><mi>r</mi></msub><mrow><mn>2</mn><mi>H</mi></mrow></mfrac><mrow><mo fence="true">[</mo><msub><mi>P</mi><mi>m</mi></msub><mo>−</mo><msub><mi>P</mi><mi>e</mi></msub><mo>−</mo><msub><mi>D</mi><mi>a</mi></msub><mrow><mo fence="true">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow><mo fence="true">]</mo></mrow><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>δ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mrow><mo fence="true">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">E_{q}&#x27; - V_q = r_aI_q - x_{ds}&#x27;Id\\ E_{d}&#x27; - V_d = r_aI_d - x_{qs}&#x27;Iq\\ E_{q}&#x27;&#x27; - V_q = r_aI_q - x_{ds}&#x27;&#x27;Id\\ E_{d}&#x27;&#x27; - V_d = r_aI_d - x_{qs}&#x27;&#x27;Iq\\ ~\\ \frac{dE_{q}&#x27;}{dt} = \frac{V_{fd} + \left( x_d - x_{d}&#x27; \right)I_d - s_d E_{q}&#x27;}{T_{d0}&#x27;}\\ \frac{dE_{d}&#x27;}{dt} = \frac{- \left( x_q - x_{q}&#x27; \right)I_q - s_q E_{d}&#x27;}{T_{q0}&#x27;}\\ \frac{dE_{q}&#x27;&#x27;}{dt} = \frac{s_dE_{q}&#x27; + \left( x_{d}&#x27; - x_{d}&#x27;&#x27; \right)I_d - s_d E_{q}&#x27;&#x27;}{T_{d0}&#x27;&#x27;}\\ \frac{dE_{d}&#x27;&#x27;}{dt} = \frac{s_dE_{d}&#x27; + \left( x_{q}&#x27; - x_{q}&#x27;&#x27; \right)I_q - s_d E_{d}&#x27;&#x27;}{T_{q0}&#x27;&#x27;}\\ ~\\ \frac{d\omega}{dt} = \frac{\omega_r}{2H} \left[ P_m - P_e - D_a \left( \omega - \omega_r \right) \right]\\ \frac{d\delta}{dt} = \Omega_b\left( \omega - \omega_r \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault">d</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault" style="margin-right:0.03588em">q</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault">d</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault" style="margin-right:0.03588em">q</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.211em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.512308em;vertical-align:-0.9873080000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.398692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.30130799999999996em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9873080000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.7115240000000003em;vertical-align:-1.088416em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6231080000000002em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.4336920000000006em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4024159999999999em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.088416em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.211em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.512308em;vertical-align:-0.9873080000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.398692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.30130799999999996em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9873080000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.7115240000000003em;vertical-align:-1.088416em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6231080000000002em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.4336920000000006em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4024159999999999em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.088416em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.7935599999999998em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mclose delimcenter" style="top:0em">]</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03785em">δ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></div><p>As duas últimas equações diferenciais são as equações mecânicas da máquina; e as demais são equações elétricas (consulte essa <a href="https://dx.doi.org/10.14393/ufu.te.2019.2444" target="_blank" rel="noopener noreferrer">tese</a> para maiores detalhes acerca dos parâmetros dessas equações).</p><p>Utilizando as equações transitórias e subtransitórias podem-se definir cinco modelos de distintas complexidades.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O modelo da máquina síncrona é <strong>selecionado automaticamente</strong> de acordo com os dados fornecidos ao programa.</p></div></div><p>Na sequência são apresentados tais modelos, incluídos os efeitos da saturação magnética, em conjunto com seus diagramas de blocos:</p><ul><li><strong>Modelo 1</strong>: Corresponde a uma tensão constante atrás de uma reatância transitória de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">x_{d}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.035em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>), não exigindo equações diferenciais;</li><li><strong>Modelo 2</strong>: São representados os efeitos transitórios de eixo direto, sendo necessária a solução de uma equação diferencial (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model2SyncGenerator.svg" alt="Diagrama de blocos do Modelo 2 das máquinas síncronas" title="Diagrama de blocos do Modelo 2 das máquinas síncronas"></center></div><ul><li><strong>Modelo 3</strong>: São representados os efeitos transitórios de eixo direto e em quadratura, exigindo duas equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model3SyncGenerator.svg" alt="Diagrama de blocos do Modelo 3 das máquinas síncronas" title="Diagrama de blocos do Modelo 3 das máquinas síncronas"></center></div><ul><li><strong>Modelo 4</strong>: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de três equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model4SyncGenerator.svg" alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas"></center></div><ul><li><strong>Modelo 5</strong>: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de quatro equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model5SyncGenerator.svg" alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas"></center></div><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Em todos os modelos as equações diferenciais mecânicas são solucionadas.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="saturação"></a>Saturação<a aria-hidden="true" tabindex="-1" class="hash-link" href="#saturação" title="Direct link to heading">#</a></h4><p>Para representar matematicamente o efeito da saturação nas equações das máquinas síncronas são introduzidos “fatores de saturação” que modificam as impedâncias do circuito equivalente, os quais dependem de uma reatância de dispersão efetiva, chamada de reatância de Potier (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>).</p><p>Tal reatância pode ser obtida por meio de ensaios (utilizando curvas de saturação de circuito aberto e carga de fator de potência zero) ou estimadas de forma aproximada por outros parâmetros da máquina. A reatância de dispersão (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>l</mi></msub></mrow><annotation encoding="application/x-tex">x_l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), aqui substituída de forma aproximada por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, representa a parcela da reatância da máquina originados do fluxo magnético que percorrem o ar na maioria de seu caminho e, portanto, é independente da saturação.</p><p>O método implementado no programa permite reproduzir a saturação em ambos os eixos (direto e em quadratura), diferindo entre si devido à divergência no tamanho do entreferro. É assumido que a soma vetorial das duas componentes do fluxo magnético saturado está em fase com a f.m.m. e proporcional à Tensão de Potier (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">E_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, a qual é a tensão atrás da reatância de Potier).</p><p>Para isso, são utilizados internamente dois fatores de saturação, sendo um no eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">s_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e outro no eixo em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">s_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>). Esses fatores de saturação são <strong>automaticamente calculados</strong> a cada passo de integração e dependem da curva de saturação da máquina definida pelo <a href="/PSP/docs/syncGenerator#fator-de-satura%C3%A7%C3%A3o">fator de saturação inserido no formulário de edição de dados</a>.</p><p>Portanto as reatâncias saturadas, que devem ser inseridas nas equações algébricas da máquina, são definidas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>d</mi></msub><mo>−</mo><msub><mi>x</mi><mi>p</mi></msub></mrow><mrow><msub><mi>s</mi><mi>d</mi></msub><mo>+</mo><msub><mi>x</mi><mi>p</mi></msub></mrow></mfrac><mspace linebreak="newline"></mspace><msub><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>q</mi></msub><mo>−</mo><msub><mi>x</mi><mi>p</mi></msub></mrow><mrow><msub><mi>s</mi><mi>q</mi></msub><mo>+</mo><msub><mi>x</mi><mi>p</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">x_{ds}=\frac{x_d-x_p}{s_d +x_p}\\ x_{qs}=\frac{x_q-x_p}{s_q +x_p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.232438em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.232438em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Essas equações também são utilizadas para as reatâncias transitórias e subtransitórias, visto que o valor da reatância de Potier (ou de dispersão) não é alterada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="barramento-infinito"></a>Barramento infinito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barramento-infinito" title="Direct link to heading">#</a></h4><p>Algumas referências incluem um modelo sem equações diferenciais, em que a máquina é somente representada por uma tensão constante atrás de uma reatância transitória de eixo direto. Tal é utilizado na representação de um <strong>barramento infinito</strong>, o qual é normalmente constituído de um subsistema muito maior àquele simulado.</p><p>No PSP-UFU a <strong>representação de um barramento infinito</strong> pode ser obtido por meio da utilização de uma máquina representada pelo Modelo 1 cujo valor da constante de inércia (H) é infinito ou muito grande (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9999</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">9999~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span>, por exemplo) em relação às demais máquinas do sistema, e o valor de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">x_{d}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.035em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span> deve ser um valor muito pequeno (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">10^{-3}~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="centro-de-inércia"></a>Centro de inércia<a aria-hidden="true" tabindex="-1" class="hash-link" href="#centro-de-inércia" title="Direct link to heading">#</a></h4><p>Normalmente utiliza-se a velocidade de referência como sendo a síncrona e portanto, nesse caso, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>r</mi></msub><mo>=</mo><msub><mi>ω</mi><mi>b</mi></msub><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\omega_r = \omega_b = 1,0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> Essa abordagem, adotada por vários livros de estabilidade, considera como referência uma máquina fictícia girando sempre na velocidade síncrona independente das perturbações aplicadas no sistema. No PSP-UFU foi implementado o conceito de centro de inércia (COI, do inglês, <em>Center of Inertia</em>), que constitui uma soma ponderada das velocidades das máquinas presentes no sistema:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>r</mi></msub><mo>=</mo><mfrac><mrow><mo fence="true">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>H</mi><mi>i</mi></msub><msub><mi>ω</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>H</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\omega_r=\frac{\left( \sum_{i=1}^{n} H_i \omega_i \right)}{\left( \sum_{i=1}^{n} H_i \right)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.488004em;vertical-align:-0.994002em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.494002em"><span style="top:-2.305708em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.994002em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:
-<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">n</span></span></span></span></span>	é o número de máquinas síncronas <strong>conectadas</strong> no sistema.</p><p>A aplicação do COI resulta em dados de saída, como o ângulo do rotor, mais fáceis de serem analisados. Na implementação realizada no programa a utilização ou não desse recurso é opcional e pode ser <a href="/PSP/docs/simulationConfig">definida pelo usuário</a>.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-dos-geradores-síncronos"></a>Formulário de edição dos geradores síncronos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-geradores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos geradores síncronos:</p><div><center><img src="/PSP/images/syncGeneratorForm.png" alt="Formulário dos geradores síncronos no PSP-UFU" title="Formulário dos geradores síncronos no PSP-UFU"></center></div><p>Um segundo pelos dados de estabilidade, como mostra a figura abaixo, acessado ao clicar no botão &quot;Estabilidade&quot; do formulário principal. Nele é possível também acessar aos controles das máquinas síncronas manipulados pelo <a href="/PSP/docs/controlEditor">editor de controle</a>.</p><div><center><img src="/PSP/images/syncGeneratorStabForm.png" alt="Formulário de estabilidade dos geradores síncronos no PSP-UFU" title="Formulário de estabilidade dos geradores síncronos no PSP-UFU"></center></div><p>No formulário de estabilidade pode ser observado o botão &quot;Chaveamento&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do gerador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/syncGeneratorSw.png" alt="Formulário de chaveamento do gerador síncrono" title="Formulário de chaveamento do gerador síncrono"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Falta</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Botão Estabilidade</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Botão Chaveamento</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do gerador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do gerador.</p><p>Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso mais de um gerador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potências-reativas-máxima-e-mínima"></a>Potências reativas máxima e mínima<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-reativas-máxima-e-mínima" title="Direct link to heading">#</a></h4><p>Limites de potência reativa máxima e mínima do gerador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="acesso-aos-controles-da-máquina-síncrona"></a>Acesso aos controles da máquina síncrona<a aria-hidden="true" tabindex="-1" class="hash-link" href="#acesso-aos-controles-da-máquina-síncrona" title="Direct link to heading">#</a></h2><p>Como já <a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">mencionado anteriormente</a>, os reguladores de velocidade e tensão da máquina síncrona podem ser acionados ou inibidos por meio das caixas de seleção &quot;<a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Utilizar AVR e regulador de velocidade</a>&quot;. Ambas as opções irão acessar o <a href="/PSP/docs/controlEditor">editor de controles</a>.</p><p>O acesso aos controles do <strong>AVR</strong> poderão então ser criados e manipulados ao clicar no botão &quot;Editar AVR&quot;, assim como o <strong>Regulador de Velocidade</strong> é acessado no botão &quot;Editar regulador de velocidade&quot;.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU a opção de editar o <strong>AVR</strong> engloba mais que somente o controle de tensão da máquina. Nele <strong>deve</strong> ser inserida a malha de controle da máquina assim como a <strong>excitatriz da máquina síncrona</strong>. Outras estratégias de controle (opcionais), como PSS (<em>Power System Stabilizer</em>) e/ou controles de sobre e sub excitação, são também implementadas em conjunto.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Assim como no AVR, o <strong>Regulador de Velocidade</strong> engloba mais que a regulação primária da máquina. Nessa opção <strong>deve</strong> ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o <strong>modelo da turbina</strong>. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994.</li><li>DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: <a href="https://doi.org/10.1109/TPAS.1972.293341" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPAS.1972.293341</a></li><li>IEEE Std 1110-2002 IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses. IEEE, New York, nov. 2003. doi: <a href="https://doi.org/10.1109/IEEESTD.2003.94408" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/IEEESTD.2003.94408</a></li><li>KIMBARK, E. W. Power System Stability: Volume III – Synchronous Machine. New York: Wiley-IEEE Press, 1995.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/syncGenerator.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/harmSource"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Fonte de Corrente Harmônica</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/syncMotor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Compensador Síncrono »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#gerador-síncrono-no-psp-ufu" class="table-of-contents__link">Gerador Síncrono no PSP-UFU</a><ul><li><a href="#gerador-síncrono-no-fluxo-de-carga" class="table-of-contents__link">Gerador Síncrono no fluxo de carga</a></li><li><a href="#gerador-síncrono-no-estudo-de-curto-circuito" class="table-of-contents__link">Gerador Síncrono no estudo de curto-circuito</a></li><li><a href="#gerador-síncrono-no-estudo-de-estabilidade" class="table-of-contents__link">Gerador Síncrono no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-dos-geradores-síncronos" class="table-of-contents__link">Formulário de edição dos geradores síncronos</a></li><li><a href="#acesso-aos-controles-da-máquina-síncrona" class="table-of-contents__link">Acesso aos controles da máquina síncrona</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Gerador Síncrono</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Uma máquina de corrente alternada na qual a frequência das tensões geradas e a velocidade da máquina estão em uma proporção constante. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=411-31-08" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gerador-síncrono-no-psp-ufu"></a>Gerador Síncrono no PSP-UFU<a class="hash-link" href="#gerador-síncrono-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Os geradores síncronos são a fonte de potência para o cálculo do fluxo de carga e estabilidade, além de serem um dos principais contribuintes para as correntes de falta.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Somente barramentos que possuem esse elemento conectado podem ser considerados <a href="/PSP/docs/powerFlow">barras de referência</a>. <a href="/PSP/docs/powerFlow">Barras do tipo PV</a> devem conter um gerador síncrono ou um <a href="/PSP/docs/syncMotor">motor síncrono (compensador síncrono)</a>.</p></div></div><p>O formulário de dados é dividido em dois, sendo o primeiro responsável pelos dados gerais, de <a href="/PSP/docs/powerFlow">fluxo de carga</a> e de <a href="/PSP/docs/fault">falta</a> e um segundo pelos dados de <a href="/PSP/docs/stability">estabilidade</a> . Esse último também permite o acesso aos controles das máquinas síncronas manipulados pelo editor de controle.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os dados referentes às <strong>impedâncias de sequência positiva do gerador síncrono</strong> inseridas no <a href="/PSP/docs/syncGenerator#falta">contexto (aba) &quot;Falta&quot;</a> são utilizados <strong>tanto para os estudos de <a href="/PSP/docs/fault">curto-circuito</a> quanto para os estudos de <a href="/PSP/docs/harmonics">harmônicos</a></strong>.</p><p>Esses dados são ignorados nos estudos de <a href="/PSP/docs/powerFlow">fluxo de carga</a> (<em>não são utilizado nesse estudo</em>) e <a href="/PSP/docs/stability">estabilidade</a> (<em>são utilizados dados inseridos em formulário específico</em>).</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gerador-síncrono-no-fluxo-de-carga"></a>Gerador Síncrono no fluxo de carga<a class="hash-link" href="#gerador-síncrono-no-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>O gerador síncrono é a fonte de potência do PSP-UFU no estudo de fluxo de carga. Seu comportamento difere de acordo com o tipo de barra conectada:</p><ul><li><a href="/PSP/docs/powerFlow">Barra de referência</a>: Os dados de potência ativa e reativa inseridos são desprezados, uma vez que esse elemento será utilizado para completar o balanço de potência do estudo de fluxo d carga;</li><li><a href="/PSP/docs/powerFlow">Barra PV</a>: O dado de potência ativa é considerado, porém o dados de potência reativa são desprezados. O valor da potência reativa é utilizado para manter o módulo da tensão constante no barramento conectado;</li><li><a href="/PSP/docs/powerFlow">Barra PQ</a>: Para os geradores conectados a essa barra tanto a potência ativa quanto reativa inseridas são consideradas.</li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>Caso o limite de potência reativa seja excedido, o programa automaticamente transforma a <a href="/PSP/docs/powerFlow">Barra PV</a> conectada em uma <a href="/PSP/docs/powerFlow">Barra PQ</a>, utilizando o valor limite de potência reativa que seria ultrapassado.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gerador-síncrono-no-estudo-de-curto-circuito"></a>Gerador Síncrono no estudo de curto-circuito<a class="hash-link" href="#gerador-síncrono-no-estudo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>Enquanto os geradores no estudo de fluxo de carga são modelados somente por correntes injetadas nas barras, para o curto-circuito utiliza-se uma tensão atrás de uma impedância. A figura abaixo mostra o caminho da corrente e o circuito equivalente de cada sequência nos geradores.</p><div><center><img src="/PSP/images/sycGeneratorSeq.svg" alt="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero" title="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero"></center></div><p>As tensões geradas são somente de sequência positiva, uma vez que o gerador fornece sempre tensões trifásicas equilibradas. Portanto a rede de sequência positiva é composta de uma tensão pré-falta atrás de uma impedância de sequência positiva. As redes de sequência negativa e zero não contêm forças eletromotrizes, porém incluem as impedâncias do gerador de sequência negativa e zero.</p><p>A corrente que circula na impedância <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> entre o neutro e a terra é <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">3\dot{I}_{a0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>. Pela figura acima (c), observa-se que a queda de tensão de sequência zero do ponto a para terra (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\dot{V}_{a0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>3</mn><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>g</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\dot{V}_{a0} = -3\dot{I}_{a0}\overline{z}_n - \dot{I}_{a0}\overline{z}_{g0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord">−</span><span class="mord">3</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span></div><p>A rede de sequência zero, que é um circuito monofásico pelo qual se supõe que circule apenas corrente de sequência zero e deve, portanto, ter uma impedância definida pela seguinte equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>0</mn></msub><mo>=</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>g</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{0} = 3\overline{z}_n + \overline{z}_{g0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.916668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a seguinte mensagem de erro pode ser exibida:</p><p><center>&quot;<i>Falha ao inverter a matriz admitância de sequência zero</i>&quot;</center></p><p>Isso ocorre porque a matriz admitância de sequência zero é singular. Para contornar esse problema escolha <strong>uma</strong> das duas soluções abaixo:</p><ol><li>Marque a opção &quot;Neutro aterrado&quot; e insira um alto valor de reatância de aterramento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>9999</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j9999~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo);</li><li><strong>Ou</strong>, na barra do gerador, insira um <a href="/PSP/docs/inductor">reator</a> de baixo valor de potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">1,0~var</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault" style="margin-right:0.03588em">v</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>, por exemplo).</li></ol></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gerador-síncrono-no-estudo-de-estabilidade"></a>Gerador Síncrono no estudo de estabilidade<a class="hash-link" href="#gerador-síncrono-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A relação de valores observados em testes adequados (definidos na <a href="https://doi.org/10.1109/IEEESTD.2020.9050934" target="_blank" rel="noopener noreferrer">IEEE Std. 115-2019</a>), denominados parâmetros padrões, são utilizados para modelar a máquina síncrona no estudo de estabilidade do PSP-UFU.</p><p>Os parâmetros da máquina síncrona que influenciam rapidamente no decaimento de valores são chamados <strong>subtransitórios</strong> (indicados por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">&#x27;&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>), aqueles que influenciam mais lentamente são chamados <strong>transitórios</strong> (indicados por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>) e, finalmente, aqueles que influenciam continuamente são chamados de parâmetros <strong>síncronos</strong> (sem indicação de sobrescrito).</p><p>Um conjunto de equações algébrico-diferenciais determinam o comportamento da máquina síncrona no estudo de estabilidade:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msub><mi>V</mi><mi>q</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>I</mi><mi>d</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msub><mi>V</mi><mi>d</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>I</mi><mi>q</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msub><mi>V</mi><mi>q</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mi>I</mi><mi>d</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msub><mi>V</mi><mi>d</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mi>I</mi><mi>q</mi><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mrow><mi>f</mi><mi>d</mi></mrow></msub><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>d</mi><mn>0</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msub><mi>s</mi><mi>q</mi></msub><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>q</mi><mn>0</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msubsup><mi>x</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>d</mi><mn>0</mn></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msubsup><mi>x</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>q</mi><mn>0</mn></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mfrac><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>ω</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><msub><mi>ω</mi><mi>r</mi></msub><mrow><mn>2</mn><mi>H</mi></mrow></mfrac><mrow><mo fence="true">[</mo><msub><mi>P</mi><mi>m</mi></msub><mo>−</mo><msub><mi>P</mi><mi>e</mi></msub><mo>−</mo><msub><mi>D</mi><mi>a</mi></msub><mrow><mo fence="true">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow><mo fence="true">]</mo></mrow><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>δ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mrow><mo fence="true">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">E_{q}&#x27; - V_q = r_aI_q - x_{ds}&#x27;Id\\ E_{d}&#x27; - V_d = r_aI_d - x_{qs}&#x27;Iq\\ E_{q}&#x27;&#x27; - V_q = r_aI_q - x_{ds}&#x27;&#x27;Id\\ E_{d}&#x27;&#x27; - V_d = r_aI_d - x_{qs}&#x27;&#x27;Iq\\ ~\\ \frac{dE_{q}&#x27;}{dt} = \frac{V_{fd} + \left( x_d - x_{d}&#x27; \right)I_d - s_d E_{q}&#x27;}{T_{d0}&#x27;}\\ \frac{dE_{d}&#x27;}{dt} = \frac{- \left( x_q - x_{q}&#x27; \right)I_q - s_q E_{d}&#x27;}{T_{q0}&#x27;}\\ \frac{dE_{q}&#x27;&#x27;}{dt} = \frac{s_dE_{q}&#x27; + \left( x_{d}&#x27; - x_{d}&#x27;&#x27; \right)I_d - s_d E_{q}&#x27;&#x27;}{T_{d0}&#x27;&#x27;}\\ \frac{dE_{d}&#x27;&#x27;}{dt} = \frac{s_dE_{d}&#x27; + \left( x_{q}&#x27; - x_{q}&#x27;&#x27; \right)I_q - s_d E_{d}&#x27;&#x27;}{T_{q0}&#x27;&#x27;}\\ ~\\ \frac{d\omega}{dt} = \frac{\omega_r}{2H} \left[ P_m - P_e - D_a \left( \omega - \omega_r \right) \right]\\ \frac{d\delta}{dt} = \Omega_b\left( \omega - \omega_r \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault">d</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault" style="margin-right:0.03588em">q</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault">d</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault" style="margin-right:0.03588em">q</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.211em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.512308em;vertical-align:-0.9873080000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.398692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.30130799999999996em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9873080000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.7115240000000003em;vertical-align:-1.088416em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6231080000000002em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.4336920000000006em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4024159999999999em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.088416em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.211em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.512308em;vertical-align:-0.9873080000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.398692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.30130799999999996em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9873080000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.7115240000000003em;vertical-align:-1.088416em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6231080000000002em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.4336920000000006em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4024159999999999em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.088416em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.7935599999999998em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mclose delimcenter" style="top:0em">]</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03785em">δ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></div><p>As duas últimas equações diferenciais são as equações mecânicas da máquina; e as demais são equações elétricas (consulte essa <a href="https://dx.doi.org/10.14393/ufu.te.2019.2444" target="_blank" rel="noopener noreferrer">tese</a> para maiores detalhes acerca dos parâmetros dessas equações).</p><p>Utilizando as equações transitórias e subtransitórias podem-se definir cinco modelos de distintas complexidades.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O modelo da máquina síncrona é <strong>selecionado automaticamente</strong> de acordo com os dados fornecidos ao programa.</p></div></div><p>Na sequência são apresentados tais modelos, incluídos os efeitos da saturação magnética, em conjunto com seus diagramas de blocos:</p><ul><li><strong>Modelo 1</strong>: Corresponde a uma tensão constante atrás de uma reatância transitória de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">x_{d}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.035em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>), não exigindo equações diferenciais;</li><li><strong>Modelo 2</strong>: São representados os efeitos transitórios de eixo direto, sendo necessária a solução de uma equação diferencial (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model2SyncGenerator.svg" alt="Diagrama de blocos do Modelo 2 das máquinas síncronas" title="Diagrama de blocos do Modelo 2 das máquinas síncronas"></center></div><ul><li><strong>Modelo 3</strong>: São representados os efeitos transitórios de eixo direto e em quadratura, exigindo duas equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model3SyncGenerator.svg" alt="Diagrama de blocos do Modelo 3 das máquinas síncronas" title="Diagrama de blocos do Modelo 3 das máquinas síncronas"></center></div><ul><li><strong>Modelo 4</strong>: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de três equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model4SyncGenerator.svg" alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas"></center></div><ul><li><strong>Modelo 5</strong>: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de quatro equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model5SyncGenerator.svg" alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas"></center></div><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Em todos os modelos as equações diferenciais mecânicas são solucionadas.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="saturação"></a>Saturação<a class="hash-link" href="#saturação" title="Direct link to heading">#</a></h4><p>Para representar matematicamente o efeito da saturação nas equações das máquinas síncronas são introduzidos “fatores de saturação” que modificam as impedâncias do circuito equivalente, os quais dependem de uma reatância de dispersão efetiva, chamada de reatância de Potier (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>).</p><p>Tal reatância pode ser obtida por meio de ensaios (utilizando curvas de saturação de circuito aberto e carga de fator de potência zero) ou estimadas de forma aproximada por outros parâmetros da máquina. A reatância de dispersão (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>l</mi></msub></mrow><annotation encoding="application/x-tex">x_l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), aqui substituída de forma aproximada por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, representa a parcela da reatância da máquina originados do fluxo magnético que percorrem o ar na maioria de seu caminho e, portanto, é independente da saturação.</p><p>O método implementado no programa permite reproduzir a saturação em ambos os eixos (direto e em quadratura), diferindo entre si devido à divergência no tamanho do entreferro. É assumido que a soma vetorial das duas componentes do fluxo magnético saturado está em fase com a f.m.m. e proporcional à Tensão de Potier (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">E_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, a qual é a tensão atrás da reatância de Potier).</p><p>Para isso, são utilizados internamente dois fatores de saturação, sendo um no eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">s_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e outro no eixo em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">s_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>). Esses fatores de saturação são <strong>automaticamente calculados</strong> a cada passo de integração e dependem da curva de saturação da máquina definida pelo <a href="/PSP/docs/syncGenerator#fator-de-satura%C3%A7%C3%A3o">fator de saturação inserido no formulário de edição de dados</a>.</p><p>Portanto as reatâncias saturadas, que devem ser inseridas nas equações algébricas da máquina, são definidas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>d</mi></msub><mo>−</mo><msub><mi>x</mi><mi>p</mi></msub></mrow><mrow><msub><mi>s</mi><mi>d</mi></msub><mo>+</mo><msub><mi>x</mi><mi>p</mi></msub></mrow></mfrac><mspace linebreak="newline"></mspace><msub><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>q</mi></msub><mo>−</mo><msub><mi>x</mi><mi>p</mi></msub></mrow><mrow><msub><mi>s</mi><mi>q</mi></msub><mo>+</mo><msub><mi>x</mi><mi>p</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">x_{ds}=\frac{x_d-x_p}{s_d +x_p}\\ x_{qs}=\frac{x_q-x_p}{s_q +x_p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.232438em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.232438em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Essas equações também são utilizadas para as reatâncias transitórias e subtransitórias, visto que o valor da reatância de Potier (ou de dispersão) não é alterada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barramento-infinito"></a>Barramento infinito<a class="hash-link" href="#barramento-infinito" title="Direct link to heading">#</a></h4><p>Algumas referências incluem um modelo sem equações diferenciais, em que a máquina é somente representada por uma tensão constante atrás de uma reatância transitória de eixo direto. Tal é utilizado na representação de um <strong>barramento infinito</strong>, o qual é normalmente constituído de um subsistema muito maior àquele simulado.</p><p>No PSP-UFU a <strong>representação de um barramento infinito</strong> pode ser obtido por meio da utilização de uma máquina representada pelo Modelo 1 cujo valor da constante de inércia (H) é infinito ou muito grande (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9999</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">9999~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span>, por exemplo) em relação às demais máquinas do sistema, e o valor de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">x_{d}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.035em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span> deve ser um valor muito pequeno (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">10^{-3}~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="centro-de-inércia"></a>Centro de inércia<a class="hash-link" href="#centro-de-inércia" title="Direct link to heading">#</a></h4><p>Normalmente utiliza-se a velocidade de referência como sendo a síncrona e portanto, nesse caso, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>r</mi></msub><mo>=</mo><msub><mi>ω</mi><mi>b</mi></msub><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\omega_r = \omega_b = 1,0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> Essa abordagem, adotada por vários livros de estabilidade, considera como referência uma máquina fictícia girando sempre na velocidade síncrona independente das perturbações aplicadas no sistema. No PSP-UFU foi implementado o conceito de centro de inércia (COI, do inglês, <em>Center of Inertia</em>), que constitui uma soma ponderada das velocidades das máquinas presentes no sistema:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>r</mi></msub><mo>=</mo><mfrac><mrow><mo fence="true">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>H</mi><mi>i</mi></msub><msub><mi>ω</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>H</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\omega_r=\frac{\left( \sum_{i=1}^{n} H_i \omega_i \right)}{\left( \sum_{i=1}^{n} H_i \right)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.488004em;vertical-align:-0.994002em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.494002em"><span style="top:-2.305708em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.994002em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:
+<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">n</span></span></span></span></span>	é o número de máquinas síncronas <strong>conectadas</strong> no sistema.</p><p>A aplicação do COI resulta em dados de saída, como o ângulo do rotor, mais fáceis de serem analisados. Na implementação realizada no programa a utilização ou não desse recurso é opcional e pode ser <a href="/PSP/docs/simulationConfig">definida pelo usuário</a>.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-geradores-síncronos"></a>Formulário de edição dos geradores síncronos<a class="hash-link" href="#formulário-de-edição-dos-geradores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos geradores síncronos:</p><div><center><img src="/PSP/images/syncGeneratorForm.png" alt="Formulário dos geradores síncronos no PSP-UFU" title="Formulário dos geradores síncronos no PSP-UFU"></center></div><p>Um segundo pelos dados de estabilidade, como mostra a figura abaixo, acessado ao clicar no botão &quot;Estabilidade&quot; do formulário principal. Nele é possível também acessar aos controles das máquinas síncronas manipulados pelo <a href="/PSP/docs/controlEditor">editor de controle</a>.</p><div><center><img src="/PSP/images/syncGeneratorStabForm.png" alt="Formulário de estabilidade dos geradores síncronos no PSP-UFU" title="Formulário de estabilidade dos geradores síncronos no PSP-UFU"></center></div><p>No formulário de estabilidade pode ser observado o botão &quot;Chaveamento&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do gerador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/syncGeneratorSw.png" alt="Formulário de chaveamento do gerador síncrono" title="Formulário de chaveamento do gerador síncrono"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Chaveamento</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do gerador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do gerador.</p><p>Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso mais de um gerador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-reativas-máxima-e-mínima"></a>Potências reativas máxima e mínima<a class="hash-link" href="#potências-reativas-máxima-e-mínima" title="Direct link to heading">#</a></h4><p>Limites de potência reativa máxima e mínima do gerador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedâncias-de-sequência"></a>Impedâncias de sequência<a class="hash-link" href="#impedâncias-de-sequência" title="Direct link to heading">#</a></h4><p>Valores de resistência e reatância para cálculo das correntes de falta. São inseridos dados de sequência positiva, negativa e zero.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedância-de-aterramento"></a>Impedância de aterramento<a class="hash-link" href="#impedância-de-aterramento" title="Direct link to heading">#</a></h4><p>Valores utilizados para o cálculo das correntes de falta do tipo fase-terra e fase-fase-terra. Caso o neutro do gerador não seja aterrado, o valor inserido nesse campo é ignorado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="neutro-aterrado"></a>Neutro aterrado<a class="hash-link" href="#neutro-aterrado" title="Direct link to heading">#</a></h4><p>Indica se o neutro do gerador é aterrado.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-dados-da-máquina-síncrona"></a>Imprimir dados da máquina síncrona<a class="hash-link" href="#imprimir-dados-da-máquina-síncrona" title="Direct link to heading">#</a></h4><p>Exibe os dados do gerador síncrono nos gráficos no tempo. Os seguintes dados são exibidos:</p><ul><li>tensão terminal</li><li>potências ativa e reativas</li><li>potência mecânica</li><li>frequência</li><li>tensão de campo</li><li>ângulo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03785em">δ</span></span></span></span></span>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-avr-e-regulador-de-velocidade"></a>Utilizar AVR e regulador de velocidade<a class="hash-link" href="#utilizar-avr-e-regulador-de-velocidade" title="Direct link to heading">#</a></h4><p>Esses campos acionam ou inibem os controles da máquina síncrona, assim como modelos de turbina e excitatrizes.</p><p>Caso selecionadas os controles podem ser inseridos e modificados pelos botões “Editar AVR” e “Editar regulador de velocidade”, os quais acessam o <a href="/PSP/docs/controlEditor">editor de controle do PSP-UFU</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-de-armadura"></a>Resistência de armadura<a class="hash-link" href="#resistência-de-armadura" title="Direct link to heading">#</a></h4><p>Resistência de armadura da máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatância-de-potier"></a>Reatância de Potier<a class="hash-link" href="#reatância-de-potier" title="Direct link to heading">#</a></h4><p>Reatância de Potier para cálculo da saturação da máquina, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="fator-de-saturação"></a>Fator de saturação<a class="hash-link" href="#fator-de-saturação" title="Direct link to heading">#</a></h4><p>Valor utilizado no cálculo da curva de saturação.</p><p>Representa o valor (em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) de corrente de campo necessária para atingir 1,2 p.u. de tensão terminal.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Esse valor deve ser <strong>maior que 1,2</strong>, ou irá gerar erros na simulação. Caso não seja informado, a saturação da máquina não é considerada nos cálculos.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="frequência-de-circuito-aberto"></a>Frequência de circuito aberto<a class="hash-link" href="#frequência-de-circuito-aberto" title="Direct link to heading">#</a></h4><p>Indica a velocidade da máquina no caso de início da simulação desconectada da rede.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Essa informação é particularmente útil na análise de conexão de geradores dessincronizados na rede.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatâncias-síncronas"></a>Reatâncias síncronas<a class="hash-link" href="#reatâncias-síncronas" title="Direct link to heading">#</a></h4><p>Valores de reatância síncrona (regime permanente) da máquina. Os valores de eixo direto e em quadratura devem ser iguais ou muito próximos para representação de uma máquina de polos lisos, enquanto para polos salientes esses valores são distintos.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatâncias-e-constantes-de-tempo-transitórias"></a>Reatâncias e constantes de tempo transitórias<a class="hash-link" href="#reatâncias-e-constantes-de-tempo-transitórias" title="Direct link to heading">#</a></h4><p>Parâmetros transitórios da máquina síncrona em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> ou segundos.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O valor da reatância transitória de eixo direto deve ser diferente de zero ou levará o programa a erro.</p></div></div><p>De acordo com a quantidade de parâmetros inseridos é definido internamente pelo programa qual o modelo a ser utilizado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatâncias-e-constantes-de-tempo-subtransitórias"></a>Reatâncias e constantes de tempo subtransitórias<a class="hash-link" href="#reatâncias-e-constantes-de-tempo-subtransitórias" title="Direct link to heading">#</a></h4><p>Parâmetros subtransitórios da máquina síncrona em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> ou segundos, representando em detalhes a presença de enrolamentos amortecedores. Assim como os dados transitórios, esses parâmetros definem o modelo da máquina.</p></div><div role="tabpanel" hidden=""><p>O botão &quot;Chaveamento&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do gerador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="acesso-aos-controles-da-máquina-síncrona"></a>Acesso aos controles da máquina síncrona<a class="hash-link" href="#acesso-aos-controles-da-máquina-síncrona" title="Direct link to heading">#</a></h2><p>Como já <a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">mencionado anteriormente</a>, os reguladores de velocidade e tensão da máquina síncrona podem ser acionados ou inibidos por meio das caixas de seleção &quot;<a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Utilizar AVR e regulador de velocidade</a>&quot;. Ambas as opções irão acessar o <a href="/PSP/docs/controlEditor">editor de controles</a>.</p><p>O acesso aos controles do <strong>AVR</strong> poderão então ser criados e manipulados ao clicar no botão &quot;Editar AVR&quot;, assim como o <strong>Regulador de Velocidade</strong> é acessado no botão &quot;Editar regulador de velocidade&quot;.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU a opção de editar o <strong>AVR</strong> engloba mais que somente o controle de tensão da máquina. Nele <strong>deve</strong> ser inserida a malha de controle da máquina assim como a <strong>excitatriz da máquina síncrona</strong>. Outras estratégias de controle (opcionais), como PSS (<em>Power System Stabilizer</em>) e/ou controles de sobre e sub excitação, são também implementadas em conjunto.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Assim como no AVR, o <strong>Regulador de Velocidade</strong> engloba mais que a regulação primária da máquina. Nessa opção <strong>deve</strong> ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o <strong>modelo da turbina</strong>. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994.</li><li>DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: <a href="https://doi.org/10.1109/TPAS.1972.293341" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPAS.1972.293341</a></li><li>IEEE Std 1110-2002 IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses. IEEE, New York, nov. 2003. doi: <a href="https://doi.org/10.1109/IEEESTD.2003.94408" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/IEEESTD.2003.94408</a></li><li>KIMBARK, E. W. Power System Stability: Volume III – Synchronous Machine. New York: Wiley-IEEE Press, 1995.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/syncGenerator.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/harmSource"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Fonte de Corrente Harmônica</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/syncMotor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Compensador Síncrono »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#gerador-síncrono-no-psp-ufu" class="table-of-contents__link">Gerador Síncrono no PSP-UFU</a><ul><li><a href="#gerador-síncrono-no-fluxo-de-carga" class="table-of-contents__link">Gerador Síncrono no fluxo de carga</a></li><li><a href="#gerador-síncrono-no-estudo-de-curto-circuito" class="table-of-contents__link">Gerador Síncrono no estudo de curto-circuito</a></li><li><a href="#gerador-síncrono-no-estudo-de-estabilidade" class="table-of-contents__link">Gerador Síncrono no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-dos-geradores-síncronos" class="table-of-contents__link">Formulário de edição dos geradores síncronos</a></li><li><a href="#acesso-aos-controles-da-máquina-síncrona" class="table-of-contents__link">Acesso aos controles da máquina síncrona</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Compensador Síncrono</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Uma máquina síncrona operando sem carga mecânica e fornecendo ou absorvendo energia reativa. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=411-34-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="compensador-síncrono-no-psp-ufu"></a>Compensador Síncrono no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#compensador-síncrono-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O compensador síncrono nada mais é que um motor síncrono operando sem carga em seu eixo com o intuito de fornecer ou absorver potência reativa na rede. Em relação ao <a href="/PSP/docs/powerFlow">fluxo de carga</a>, no PSP-UFU esse elemento tem comportamento idêntico ao <a href="/PSP/docs/syncGenerator">gerador síncrono</a>, porém a sua potência ativa é inserida nos algoritmos de solução com sinal negativo.</p><p>Sua parametrização para o estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>, <a href="/PSP/docs/fault">falta</a> e <a href="/PSP/docs/harmonics">harmônicos</a> é idêntica ao <a href="/PSP/docs/syncGenerator">gerador síncrono</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O compensador síncrono <strong>não foi implementado no estudo de <a href="/PSP/docs/stability">estabilidade</a></strong> e sua presença no circuito pode causar erros durante os cálculos.</p><p>Não inclua esse elemento para os estudos de <a href="/PSP/docs/stability">estabilidade</a>. Versões futuras do PSP-UFU irão contemplar esse elemento para esse estudo.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>É possível utilizar um <a href="/PSP/docs/syncGenerator">gerador síncrono</a> com potência ativa <em>nula</em> ou <em>negativa</em> para representar o compensador síncrono nos estudos de <a href="/PSP/docs/stability">estabilidade</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-dos-compensadores-síncronos"></a>Formulário de edição dos compensadores síncronos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-compensadores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos compensadores síncronos:</p><div><center><img src="/PSP/images/syncMotorForm.png" alt="Formulário dos compensadores síncronos no PSP-UFU" title="Formulário dos compensadores síncronos no PSP-UFU"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Falta</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do compensador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do compensador.</p><p>Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso mais de um compensador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potências-reativas-máxima-e-mínima"></a>Potências reativas máxima e mínima<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-reativas-máxima-e-mínima" title="Direct link to heading">#</a></h4><p>Limites de potência reativa máxima e mínima do compensador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do compensador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/syncMotor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/syncGenerator"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Gerador Síncrono</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/indMotor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Motor de Indução »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#compensador-síncrono-no-psp-ufu" class="table-of-contents__link">Compensador Síncrono no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-compensadores-síncronos" class="table-of-contents__link">Formulário de edição dos compensadores síncronos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Compensador Síncrono</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Uma máquina síncrona operando sem carga mecânica e fornecendo ou absorvendo energia reativa. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=411-34-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="compensador-síncrono-no-psp-ufu"></a>Compensador Síncrono no PSP-UFU<a class="hash-link" href="#compensador-síncrono-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O compensador síncrono nada mais é que um motor síncrono operando sem carga em seu eixo com o intuito de fornecer ou absorver potência reativa na rede. Em relação ao <a href="/PSP/docs/powerFlow">fluxo de carga</a>, no PSP-UFU esse elemento tem comportamento idêntico ao <a href="/PSP/docs/syncGenerator">gerador síncrono</a>, porém a sua potência ativa é inserida nos algoritmos de solução com sinal negativo.</p><p>Sua parametrização para o estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>, <a href="/PSP/docs/fault">falta</a> e <a href="/PSP/docs/harmonics">harmônicos</a> é idêntica ao <a href="/PSP/docs/syncGenerator">gerador síncrono</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O compensador síncrono <strong>não foi implementado no estudo de <a href="/PSP/docs/stability">estabilidade</a></strong> e sua presença no circuito pode causar erros durante os cálculos.</p><p>Não inclua esse elemento para os estudos de <a href="/PSP/docs/stability">estabilidade</a>. Versões futuras do PSP-UFU irão contemplar esse elemento para esse estudo.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>É possível utilizar um <a href="/PSP/docs/syncGenerator">gerador síncrono</a> com potência ativa <em>nula</em> ou <em>negativa</em> para representar o compensador síncrono nos estudos de <a href="/PSP/docs/stability">estabilidade</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-compensadores-síncronos"></a>Formulário de edição dos compensadores síncronos<a class="hash-link" href="#formulário-de-edição-dos-compensadores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos compensadores síncronos:</p><div><center><img src="/PSP/images/syncMotorForm.png" alt="Formulário dos compensadores síncronos no PSP-UFU" title="Formulário dos compensadores síncronos no PSP-UFU"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do compensador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do compensador.</p><p>Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso mais de um compensador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-reativas-máxima-e-mínima"></a>Potências reativas máxima e mínima<a class="hash-link" href="#potências-reativas-máxima-e-mínima" title="Direct link to heading">#</a></h4><p>Limites de potência reativa máxima e mínima do compensador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do compensador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedâncias-de-sequência"></a>Impedâncias de sequência<a class="hash-link" href="#impedâncias-de-sequência" title="Direct link to heading">#</a></h4><p>Valores de resistência e reatância para cálculo das correntes de falta. São inseridos dados de sequência positiva, negativa e zero.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedância-de-aterramento"></a>Impedância de aterramento<a class="hash-link" href="#impedância-de-aterramento" title="Direct link to heading">#</a></h4><p>Valores utilizados para o cálculo das correntes de falta do tipo fase-terra e fase-fase-terra. Caso o neutro do gerador não seja aterrado, o valor inserido nesse campo é ignorado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="neutro-aterrado"></a>Neutro aterrado<a class="hash-link" href="#neutro-aterrado" title="Direct link to heading">#</a></h4><p>Indica se o neutro do gerador é aterrado.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/syncMotor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/syncGenerator"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Gerador Síncrono</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/indMotor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Motor de Indução »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#compensador-síncrono-no-psp-ufu" class="table-of-contents__link">Compensador Síncrono no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-compensadores-síncronos" class="table-of-contents__link">Formulário de edição dos compensadores síncronos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Relatórios Tabulares</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Para visualização de todos os resultados dos cálculos de <a href="/PSP/docs/powerFlow">fluxo de carga</a> e <a href="/PSP/docs/fault">curto-circuito</a>, desenvolveu-se um visualizador de relatórios tabulares:</p><div><center><img src="/PSP/images/tabReport.png" alt="Visualizador de dados tabulados no PSP-UFU" title="Visualizador de dados tabulados no PSP-UFU"></center></div><p>Tal visualizador pode ser acessado no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ribbon</a> <strong>Ferramentas</strong> e então no botão <strong>Relatório de dados</strong>, sendo exibido o formulário organizado em abas.</p><p>As grandezas de alguns dados desse relatório podem ser modificadas, sendo selecionadas no cabeçalho da tabela, como mostra a figura acima.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Os valores podem ser exportados para editores de texto e tabelas externos, selecionando e copiando (Ctrl+C) para a área de transferência.</p></div></div></div></article><div class="margin-vert--xl"><div 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class="menu__link" tabindex="-1" href="/PSP/docs/installation">Installation</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mainScreen">Tela Principal</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/cadTools">Ferramentas CAD</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Power Element Editor</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerEditor">Editor de Potência</a></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!" tabindex="-1">Power Elements</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/bus">Barramento</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/line">Linha</a></li><li class="menu__list-item"><a 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href="#!">Control Element Editor</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/controlEditor">Editor de Controle</a></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!" tabindex="-1">Control Elements</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/io">Entrada / Saída</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/transferFunction">Função Transferência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/sum">Somador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/multiplier">Multiplicador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulations</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Relatórios Tabulares</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Para visualização de todos os resultados dos cálculos de <a href="/PSP/docs/powerFlow">fluxo de carga</a> e <a href="/PSP/docs/fault">curto-circuito</a>, desenvolveu-se um visualizador de relatórios tabulares:</p><div><center><img src="/PSP/images/tabReport.png" alt="Visualizador de dados tabulados no PSP-UFU" title="Visualizador de dados tabulados no PSP-UFU"></center></div><p>Tal visualizador pode ser acessado no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ribbon</a> <strong>Ferramentas</strong> e então no botão <strong>Relatório de dados</strong>, sendo exibido o formulário organizado em abas.</p><p>As grandezas de alguns dados desse relatório podem ser modificadas, sendo selecionadas no cabeçalho da tabela, como mostra a figura acima.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Os valores podem ser exportados para editores de texto e tabelas externos, selecionando e copiando (Ctrl+C) para a área de transferência.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/tabularReport.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/text"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Texto Vinculado</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/graphViewer"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Visualizador de Gráficos »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li 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docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Texto Vinculado</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O elemento de texto vinculado (ao elemento de potência) pode ser inserido no editor de <a href="/PSP/docs/powerEditor">elementos de potência</a>. Esse elemento de texto pode ser vinculado a inúmeros resultados e parâmetros do sistema de potência.</p><p>Esse elemento é útil para identificação e análise dos resultados de elementos de potência.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Outra aplicação importante é a atualização de seu valor ao aplicar a “solução contínua” no projeto, auxiliando na visualização das mudanças de estado da rede elétrica.</p><p>Também pode-se utilizar os resultados apresentados por esses elementos juntamente com o diagrama unifilar na criação de relatórios do projeto.</p></div></div><div><center><img src="/PSP/images/textExp.png" alt="Exemplo do elemento de texto no PSP-UFU" title="Exemplo de elemento de texto no PSP-UFU"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="inserindo-elementos-de-texto-no-psp-ufu"></a>Inserindo elementos de texto no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#inserindo-elementos-de-texto-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Por meio do submenu <strong>Ferramentas</strong> pode-se inserir os elementos de texto, como é apresentado na imagem abaixo.</p><div><center><img src="/PSP/images/addTextElement.svg" alt="Adicionar elementos de texto" title="Adicionar elementos de texto"></center></div><p>Ao selecionar o botão &quot;Texto&quot; ou utilizar a <strong>tecla de atalho &quot;A&quot;</strong>, o usuário deve clicar na posição desejada da <a href="/PSP/docs/powerEditor">área de trabalho</a>. Com isso, será adicionado um elemento temporário com o valor &quot;Texto&quot;. Para parametrizar esse elemento clique duas vezes sobre ele.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O elemento de texto, assim como os <a href="/PSP/docs/controlEditor">elementos de controle</a> não exibem automaticamente o formulário de edição de dados ao inseri-lo pela primeira vez.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-de-dados-do-elemento-de-texto"></a>Formulário de edição de dados do elemento de texto<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-do-elemento-de-texto" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados desse componente gráfico.</p><div><center><img src="/PSP/images/textForm.png" alt="Formulário do elemento de texto no PSP-UFU" title="Formulário do elemento de texto no PSP-UFU"></center></div><p>Sua parametrização é realizada por campos que associam o componente aos elementos elétricos de potência, além do formato dos resultados numéricos. Os campos são liberados a medida que os dados anteriores são inseridos e uma mensagem de erro é apresentada ao usuário caso algum parâmetro necessário não seja preenchido.</p><p>O processo de associação se inicia na escolha do tipo de elemento de potência será vinculado ao texto. Em seguida é exibida uma lista com o nome de todos os componentes previamente inseridos no editor de elementos de potência do tipo definido no campo anterior, associando, assim, o texto ao elemento.</p><p>Em “Tipo de texto” define-se qual será a saída apresentada. Tal campo depende do tipo de elemento selecionado, como mostra a tabela abaixo. Alguns outros dados são necessários para os elementos de ramo, como barra de origem e destino, com o intuito de representar o fluxo correto de potência e corrente. Finalmente o usuário deve inserir a quantidade de casas decimais para dados numéricos. A pré-visualização pode ser observada no formulário.</p><table><thead><tr><th align="center">Elemento de potência</th><th>Tipo de texto</th><th>Unidades disponíveis</th></tr></thead><tbody><tr><td align="center"><a href="/PSP/docs/bus">Barra</a></td><td>Nome<br>Tensão<br>Ângulo<br>Corrente de falta<br>Tensão de falta<br>Nível de curto-circuito<br>THD de tensão</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, V, kV<br>Graus, Radianos<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, V, kV<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, VA, kVA, MVA<br>%</td></tr><tr><td align="center"><a href="/PSP/docs/syncGenerator">Gerador</a></td><td>Nome<br>Potência ativa<br>Potência reativa<br>Corrente de falta</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA</td></tr><tr><td align="center"><a href="/PSP/docs/line">Linha</a> e <a href="/PSP/docs/transformer">transformador</a></td><td>Nome<br>Fluxo de potência ativo<br>Fluxo de potência reativo<br>Perdas<br>Corrente<br>Corrente de falta</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA</td></tr><tr><td align="center"><a href="/PSP/docs/load">Carga</a>, <a href="/PSP/docs/indMotor">Motor de indução</a><br>e <a href="/PSP/docs/syncMotor">Compensador síncrono</a></td><td>Nome<br>Potência ativa<br>Potência reativa</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar</td></tr><tr><td align="center"><a href="/PSP/docs/capacitor">Capacitor</a> e <a href="/PSP/docs/inductor">Indutor</a></td><td>Nome<br>Potência reativa</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar</td></tr></tbody></table><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Uma vez inseridos, esses elementos de texto podem ser manipulados da <a href="/PSP/docs/cadTools">mesma forma que os elementos de potência</a>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/text.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 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href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Texto Vinculado</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O elemento de texto vinculado (ao elemento de potência) pode ser inserido no editor de <a href="/PSP/docs/powerEditor">elementos de potência</a>. Esse elemento de texto pode ser vinculado a inúmeros resultados e parâmetros do sistema de potência.</p><p>Esse elemento é útil para identificação e análise dos resultados de elementos de potência.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Outra aplicação importante é a atualização de seu valor ao aplicar a “solução contínua” no projeto, auxiliando na visualização das mudanças de estado da rede elétrica.</p><p>Também pode-se utilizar os resultados apresentados por esses elementos juntamente com o diagrama unifilar na criação de relatórios do projeto.</p></div></div><div><center><img src="/PSP/images/textExp.png" alt="Exemplo do elemento de texto no PSP-UFU" title="Exemplo de elemento de texto no PSP-UFU"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserindo-elementos-de-texto-no-psp-ufu"></a>Inserindo elementos de texto no PSP-UFU<a class="hash-link" href="#inserindo-elementos-de-texto-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Por meio do submenu <strong>Ferramentas</strong> pode-se inserir os elementos de texto, como é apresentado na imagem abaixo.</p><div><center><img src="/PSP/images/addTextElement.svg" alt="Adicionar elementos de texto" title="Adicionar elementos de texto"></center></div><p>Ao selecionar o botão &quot;Texto&quot; ou utilizar a <strong>tecla de atalho &quot;A&quot;</strong>, o usuário deve clicar na posição desejada da <a href="/PSP/docs/powerEditor">área de trabalho</a>. Com isso, será adicionado um elemento temporário com o valor &quot;Texto&quot;. Para parametrizar esse elemento clique duas vezes sobre ele.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O elemento de texto, assim como os <a href="/PSP/docs/controlEditor">elementos de controle</a> não exibem automaticamente o formulário de edição de dados ao inseri-lo pela primeira vez.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-elemento-de-texto"></a>Formulário de edição de dados do elemento de texto<a class="hash-link" href="#formulário-de-edição-de-dados-do-elemento-de-texto" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados desse componente gráfico.</p><div><center><img src="/PSP/images/textForm.png" alt="Formulário do elemento de texto no PSP-UFU" title="Formulário do elemento de texto no PSP-UFU"></center></div><p>Sua parametrização é realizada por campos que associam o componente aos elementos elétricos de potência, além do formato dos resultados numéricos. Os campos são liberados a medida que os dados anteriores são inseridos e uma mensagem de erro é apresentada ao usuário caso algum parâmetro necessário não seja preenchido.</p><p>O processo de associação se inicia na escolha do tipo de elemento de potência será vinculado ao texto. Em seguida é exibida uma lista com o nome de todos os componentes previamente inseridos no editor de elementos de potência do tipo definido no campo anterior, associando, assim, o texto ao elemento.</p><p>Em “Tipo de texto” define-se qual será a saída apresentada. Tal campo depende do tipo de elemento selecionado, como mostra a tabela abaixo. Alguns outros dados são necessários para os elementos de ramo, como barra de origem e destino, com o intuito de representar o fluxo correto de potência e corrente. Finalmente o usuário deve inserir a quantidade de casas decimais para dados numéricos. A pré-visualização pode ser observada no formulário.</p><table><thead><tr><th align="center">Elemento de potência</th><th>Tipo de texto</th><th>Unidades disponíveis</th></tr></thead><tbody><tr><td align="center"><a href="/PSP/docs/bus">Barra</a></td><td>Nome<br>Tensão<br>Ângulo<br>Corrente de falta<br>Tensão de falta<br>Nível de curto-circuito<br>THD de tensão</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, V, kV<br>Graus, Radianos<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, V, kV<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, VA, kVA, MVA<br>%</td></tr><tr><td align="center"><a href="/PSP/docs/syncGenerator">Gerador</a></td><td>Nome<br>Potência ativa<br>Potência reativa<br>Corrente de falta</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA</td></tr><tr><td align="center"><a href="/PSP/docs/line">Linha</a> e <a href="/PSP/docs/transformer">transformador</a></td><td>Nome<br>Fluxo de potência ativo<br>Fluxo de potência reativo<br>Perdas<br>Corrente<br>Corrente de falta</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA</td></tr><tr><td align="center"><a href="/PSP/docs/load">Carga</a>, <a href="/PSP/docs/indMotor">Motor de indução</a><br>e <a href="/PSP/docs/syncMotor">Compensador síncrono</a></td><td>Nome<br>Potência ativa<br>Potência reativa</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar</td></tr><tr><td align="center"><a href="/PSP/docs/capacitor">Capacitor</a> e <a href="/PSP/docs/inductor">Indutor</a></td><td>Nome<br>Potência reativa</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar</td></tr></tbody></table><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Uma vez inseridos, esses elementos de texto podem ser manipulados da <a href="/PSP/docs/cadTools">mesma forma que os elementos de potência</a>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/text.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 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class="docMainContainer_3EyW"><div class="container padding-vert--lg docItemWrapper_1EkI"><div class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Função Transferência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>A Função Transferência é um importante bloco elementar e essencial em um sistema de controle, visto que esse componente possibilita a representação de sistemas dinâmicos por meio de equações algébricas no domínio da frequência.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Nesses blocos é utilizado a letra “<strong>s</strong>” para representação no domínio da frequência, pois essa é a terminologia mais utilizada em sistemas de controle.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="função-transferência-no-psp-ufu"></a>Função Transferência no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#função-transferência-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a inserção da função transferência, o programa a transforma na representação em espaço de estado, visto que sua solução é sistemática tornando-a, assim, mais eficiente computacionalmente.</p><p>Dada uma função transferência genérica abaixo:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>b</mi><mn>0</mn></msub><msup><mi mathvariant="bold">s</mi><mi>k</mi></msup><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><msup><mi mathvariant="bold">s</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>b</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="bold">s</mi><mo>+</mo><msub><mi>b</mi><mi>k</mi></msub></mrow><mrow><msup><mi mathvariant="bold">s</mi><mi>k</mi></msup><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msup><mi mathvariant="bold">s</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="bold">s</mi><mo>+</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{Y(\bold{s})}{U(\bold{s})} = \frac{b_0 \bold{s}^k + b_1 \bold{s}^{k-1} + \cdots + b_{k-1} \bold{s} + b_k}{\bold{s}^k + a_1 \bold{s}^{k-1} + \cdots + a_{k-1} \bold{s} + a_k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em">U</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span></span><span class="mclose">)</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.420439em;vertical-align:-0.894331em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.526108em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751079999999999em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751079999999999em"><span style="top:-2.989em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">s</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">s</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.894331em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A transformação em espaço de estado na <em>forma canônica controlável</em> ficará da seguinte forma:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><msub><mi>u</mi><mi>n</mi></msub><mspace linebreak="newline"></mspace><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>b</mi><mn>0</mn></msub><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \bold{s}x_1\\ \bold{s}x_2\\ \vdots\\ \bold{s}x_{k-1}\\ \bold{s}x_k \end{bmatrix} = \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; \cdots &amp; 0\\ 0 &amp; 0 &amp; 1 &amp; \cdots &amp; 0\\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 1\\ -a_k &amp; -a_{k-1} &amp; -a_{k-2} &amp; \cdots &amp; -a_1 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_{k-1}\\ x_k \end{bmatrix} + \begin{bmatrix} 0\\ 0\\ \vdots\\ 0\\ 1 \end{bmatrix} u_n\\ y_n = \begin{bmatrix} \left( b_n - a_n b_0 \right) &amp; \left( b_{n-1} - a_{n-1} b_0 \right) &amp; \cdots &amp; \left( b_1 - a_1 b_0 \right) \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_k \end{bmatrix} + b_0 u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.240000000000001em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-5.04em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.18em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-1.9800000000000009em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-0.7800000000000009em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953995em"><span style="top:-1.3499850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.4999850000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0959850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6919850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.712975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em"><span style="top:-5.8275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-1.5675000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953995em"><span style="top:-1.3499850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.4999850000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0959850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6919850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.712975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>A equação diferencial matricial é resolvida pelo método <strong>Trapezoidal Implícito</strong>. Um processo iterativo é realizado até que o erro entre os cálculos se torne menor que uma tolerância pré-estipulada e caso as iterações se tornem excessivas, ultrapassando um número máximo definido pelo usuário, o processo é interrompido com erro. Com os valores do vetor de estado (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>) calculados obtém-se o valor da saída do bloco (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>).</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-de-dados-da-função-transferência"></a>Formulário de edição de dados da Função Transferência<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-da-função-transferência" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados da função transferência.</p><div><center><img src="/PSP/images/transferFunctionForm.png" alt="Formulário de edição de dados da função transferência no PSP-UFU" title="Formulário de edição de dados da função transferência no PSP-UFU"></center></div><p>A função transferência é definida pelos coeficientes do numerador e denominador, <strong>separados por espaço</strong>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Os elementos não presentes são representados como coeficientes de valor zero.</p><p>Por exemplo, um parâmetro inserido igual a “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} 1 &amp; 0{,}5 &amp; 0 &amp; 2 \end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2000000000000002em;vertical-align:-0.35000000000000003em"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span></span></span></span></span>” irá gerar: “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>s</mi><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">s^3+0{,}5s^2+2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span>”.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>OGATA, K. Modern Control Engineering. Prentice Hall Inc., New Jersey, 2004.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/transferFunction.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/io"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Entrada / Saída</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/sum"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Somador »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_3klQ"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#função-transferência-no-psp-ufu" class="table-of-contents__link">Função Transferência no PSP-UFU</a></li><li><a href="#formulário-de-edição-de-dados-da-função-transferência" class="table-of-contents__link">Formulário de edição de dados da Função Transferência</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Data Visualization</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Função Transferência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>A Função Transferência é um importante bloco elementar e essencial em um sistema de controle, visto que esse componente possibilita a representação de sistemas dinâmicos por meio de equações algébricas no domínio da frequência.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Nesses blocos é utilizado a letra “<strong>s</strong>” para representação no domínio da frequência, pois essa é a terminologia mais utilizada em sistemas de controle.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="função-transferência-no-psp-ufu"></a>Função Transferência no PSP-UFU<a class="hash-link" href="#função-transferência-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a inserção da função transferência, o programa a transforma na representação em espaço de estado, visto que sua solução é sistemática tornando-a, assim, mais eficiente computacionalmente.</p><p>Dada uma função transferência genérica abaixo:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>b</mi><mn>0</mn></msub><msup><mi mathvariant="bold">s</mi><mi>k</mi></msup><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><msup><mi mathvariant="bold">s</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>b</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="bold">s</mi><mo>+</mo><msub><mi>b</mi><mi>k</mi></msub></mrow><mrow><msup><mi mathvariant="bold">s</mi><mi>k</mi></msup><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msup><mi mathvariant="bold">s</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="bold">s</mi><mo>+</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{Y(\bold{s})}{U(\bold{s})} = \frac{b_0 \bold{s}^k + b_1 \bold{s}^{k-1} + \cdots + b_{k-1} \bold{s} + b_k}{\bold{s}^k + a_1 \bold{s}^{k-1} + \cdots + a_{k-1} \bold{s} + a_k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em">U</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span></span><span class="mclose">)</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.420439em;vertical-align:-0.894331em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.526108em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751079999999999em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751079999999999em"><span style="top:-2.989em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">s</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">s</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.894331em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A transformação em espaço de estado na <em>forma canônica controlável</em> ficará da seguinte forma:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><msub><mi>u</mi><mi>n</mi></msub><mspace linebreak="newline"></mspace><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>b</mi><mn>0</mn></msub><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \bold{s}x_1\\ \bold{s}x_2\\ \vdots\\ \bold{s}x_{k-1}\\ \bold{s}x_k \end{bmatrix} = \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; \cdots &amp; 0\\ 0 &amp; 0 &amp; 1 &amp; \cdots &amp; 0\\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 1\\ -a_k &amp; -a_{k-1} &amp; -a_{k-2} &amp; \cdots &amp; -a_1 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_{k-1}\\ x_k \end{bmatrix} + \begin{bmatrix} 0\\ 0\\ \vdots\\ 0\\ 1 \end{bmatrix} u_n\\ y_n = \begin{bmatrix} \left( b_n - a_n b_0 \right) &amp; \left( b_{n-1} - a_{n-1} b_0 \right) &amp; \cdots &amp; \left( b_1 - a_1 b_0 \right) \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_k \end{bmatrix} + b_0 u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.240000000000001em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-5.04em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.18em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-1.9800000000000009em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-0.7800000000000009em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953995em"><span style="top:-1.3499850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.4999850000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0959850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6919850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.712975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em"><span style="top:-5.8275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-1.5675000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953995em"><span style="top:-1.3499850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.4999850000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0959850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6919850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.712975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>A equação diferencial matricial é resolvida pelo método <strong>Trapezoidal Implícito</strong>. Um processo iterativo é realizado até que o erro entre os cálculos se torne menor que uma tolerância pré-estipulada e caso as iterações se tornem excessivas, ultrapassando um número máximo definido pelo usuário, o processo é interrompido com erro. Com os valores do vetor de estado (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>) calculados obtém-se o valor da saída do bloco (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>).</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-da-função-transferência"></a>Formulário de edição de dados da Função Transferência<a class="hash-link" href="#formulário-de-edição-de-dados-da-função-transferência" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados da função transferência.</p><div><center><img src="/PSP/images/transferFunctionForm.png" alt="Formulário de edição de dados da função transferência no PSP-UFU" title="Formulário de edição de dados da função transferência no PSP-UFU"></center></div><p>A função transferência é definida pelos coeficientes do numerador e denominador, <strong>separados por espaço</strong>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Os elementos não presentes são representados como coeficientes de valor zero.</p><p>Por exemplo, um parâmetro inserido igual a “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} 1 &amp; 0{,}5 &amp; 0 &amp; 2 \end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2000000000000002em;vertical-align:-0.35000000000000003em"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span></span></span></span></span>” irá gerar: “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>s</mi><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">s^3+0{,}5s^2+2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span>”.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>OGATA, K. Modern Control Engineering. Prentice Hall Inc., New Jersey, 2004.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/transferFunction.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/io"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Entrada / Saída</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/sum"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Somador »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#função-transferência-no-psp-ufu" class="table-of-contents__link">Função Transferência no PSP-UFU</a></li><li><a href="#formulário-de-edição-de-dados-da-função-transferência" class="table-of-contents__link">Formulário de edição de dados da Função Transferência</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="row"><div class="col docItemCol_2ASc"><div class="docItemContainer_3QWW"><article><header><h1 class="docTitle_1Lrw">Transformador</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Aparelho estático com dois ou mais enrolamentos que, por indução eletromagnética, transforma um sistema de tensão e corrente alternada em outro sistema de tensão e corrente geralmente de valores diferentes e na mesma frequência com o objetivo de transmitir energia elétrica. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=421-01-01" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="transformador-no-psp-ufu"></a>Transformador no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#transformador-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Transformadores com quaisquer conexões e defasagens entre as tensões primárias e secundárias podem ser inseridos no PSP-UFU. Para isso, os transformadores devem ser modelados representando seu tape (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em"></span><span class="mord mathdefault">t</span></span></span></span></span>) e defasagem (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">ϕ</span></span></span></span></span>), utilizando um transformador ideal de relação de transformação <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>:</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\overline{t}:1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.81508em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> em série com sua impedância (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mo>=</mo><msub><mi>r</mi><mi>T</mi></msub><mo>+</mo><mi>j</mi><msub><mi>x</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">\overline{y}_T = r_T + jx_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8747em;vertical-align:-0.24414em"></span><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.81508em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span> é um número complexo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi mathvariant="normal">∠</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">t\angle\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">t</span><span class="mord">∠</span><span class="mord mathdefault">ϕ</span></span></span></span></span>)</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O modelo utilizado no PSP-UFU <strong>não considera</strong> o ramo magnetizante ou impedâncias mútuas entre as fases.</p></div></div><p>A figura abaixo mostra o modelo do transformador implementado no PSP-UFU:</p><div><center><img src="/PSP/images/transformerModel.svg" alt="Modelo PI da linha" title="Modelo PI da linha"></center></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mi mathvariant="bold">T</mi></msub></mrow><annotation encoding="application/x-tex">\bold{r_T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a resistência total do transformador;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">T</mi></msub></mrow><annotation encoding="application/x-tex">\bold{x_T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a reatância indutiva total do transformador;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">t</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\bold{\overline{t}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8349200000000001em;vertical-align:0em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8349200000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathbf">t</span></span></span><span style="top:-3.7549200000000003em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span></span>	é a o tape complexo do transformador.</li></ul><p>A relação entre as tensões e correntes das barras genéricas <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span></span></span></span></span> é obtida pela equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>j</mi><mi>i</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mi>t</mi><mn>2</mn></msup></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>∗</mo></msup></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mfrac></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>×</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>j</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \dot{I}_{ij}\\ \dot{I}_{ji} \end{bmatrix} = \begin{bmatrix} \displaystyle \frac{\overline{y}_T}{t^2} &amp; -\displaystyle \frac{\overline{y}_T}{\overline{t}^*}\\ -\displaystyle \frac{\overline{y}_T}{\overline{t}} &amp; \overline{y}_T \end{bmatrix} \times \begin{bmatrix} \dot{V}_{i}\\ \dot{V}_{j} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.5603800000000003em;vertical-align:-1.0301900000000004em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.53019em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3298099999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0301900000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.202030000000001em;vertical-align:-1.850025em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000007em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2107020000000004em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.29492em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.70508em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019379999999998em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000007em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.216324em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.893676em"><span style="top:-3.26798em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.783676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2107020000000004em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019379999999998em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.5603800000000003em;vertical-align:-1.0301900000000004em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.53019em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3298099999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0301900000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span></span></span></span></span></div><p>Assim como nas <a href="/PSP/docs/line">linhas</a>, o transformador de dois enrolamentos deve ser inserido entre barras, porém não há a necessidade das tensões nominais desses barramentos serem idênticas.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A primeira barra selecionada será o lado <strong>primário</strong>, o qual é indicado por um círculo. Para alterar o lado primário após a inserção basta desconectar os nós dos transformadores e reconectá-los alternando as barras, utilizando as <a href="/PSP/docs/cadTools#conex%C3%A3o-e-desconex%C3%A3o-de-elemetos-na-barra">ferramentas CAD</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="transformadores-no-estudo-de-curto-circuito"></a>Transformadores no estudo de curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#transformadores-no-estudo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas <a href="/PSP/docs/fault">faltas desbalanceadas</a> é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.</p><p>A matriz admitância de sequência positiva é construída da mesma forma que os estudos de fluxo de carga. Na sequência negativa o ângulo de defasagem entre tensões primária e secundária (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">ϕ</span></span></span></span></span>) deve ser invertido, uma vez que os fasores da sequência negativa <strong>são deslocados na direção oposta</strong>. Com isso, a matriz que representa as admitâncias da equação anterior deve ser substituída pela seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>j</mi><mi>i</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>j</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mi>t</mi><mn>2</mn></msup></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>∗</mo></msup></mfrac></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \overline{Y}_{ii} &amp; \overline{Y}_{ij}\\ \overline{Y}_{ji} &amp; \overline{Y}_{jj} \end{bmatrix} = \begin{bmatrix} \displaystyle \frac{\overline{y}_T}{t^2} &amp; -\displaystyle \frac{\overline{y}_T}{\overline{t}}\\ -\displaystyle \frac{\overline{y}_T}{\overline{t}^*} &amp; \overline{y}_T \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4866600000000005em;vertical-align:-0.9933300000000005em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.49333em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3666699999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9933300000000005em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.49333em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3666699999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9933300000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.202030000000001em;vertical-align:-1.850025em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000003em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2892979999999996em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.216324em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.893676em"><span style="top:-3.26798em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.783676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019380000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000003em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.29492em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.70508em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2892979999999996em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019380000000003em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>A impedância de sequência negativa nos transformadores deve ser tratada de maneira específica devido aos diferentes tipos de conexão. Na figura abaixo são mostrados os circuitos equivalentes para cada tipo de conexão de transformadores de dois enrolamentos. As setas indicam os caminhos possíveis para circulação da corrente de sequência zero.</p><div><center><img src="/PSP/images/seqZeroTrafo.svg" alt="Circuitos equivalentes de sequência zero dos transformadores" title="Circuitos equivalentes de sequência zero dos transformadores"></center></div><p>Para os transformadores com conexão estrela aterrado, se a ligação do neutro para a terra apresentar uma impedância <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, o circuito equivalente de sequência zero deve ter impedância de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">3\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> em série com a resistência e reatância equivalentes do transformador.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="formulário-de-edição-dos-transformadores"></a>Formulário de edição dos transformadores<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-transformadores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos transformadores:</p><div><center><img src="/PSP/images/trafoForm.png" alt="Formulário dos transformadores no PSP-UFU" title="Formulário dos transformadores no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais dos transformadores, informações do fluxo de carga, sua conexão e defasagem;</li><li><strong>Falta</strong>: local onde as impedâncias de sequência zero e impedâncias de aterramento são inseridas.</li></ul><p>Além desses dois contextos, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/trafoSw.png" alt="Formulário de chaveamento do transformador" title="Formulário de chaveamento do transformador"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1w39 tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Falta</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_1w39">Botão Estabilidade</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="tensão-nominal"></a>Tensão nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Campo de informação <em>não editável</em> que apresenta a tensão primária e secundária do transformador. Para alterar esse campo é necessário editar o campo correspondente dos <a href="/PSP/docs/bus#geral">barramento</a> conectados.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="tensão-base"></a>Tensão base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-base" title="Direct link to heading">#</a></h4><p>Tensão utilizada para converter os parâmetros de impedância inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> As tensões selecionadas por uma caixa de escolha são editadas nos formulários dos <a href="/PSP/docs/bus#geral">barramentos</a> conectados.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A tensão base é ignorada caso os dados inseridos estejam em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do transformador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="impedância-do-transformador"></a>Impedância do transformador<a aria-hidden="true" tabindex="-1" class="hash-link" href="#impedância-do-transformador" title="Direct link to heading">#</a></h4><p>Impedância série do transformador (resistência e reatância indutiva), inserida em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span>. Caso as impedâncias sejam inseridas em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> será utilizada a tensão base selecionada para conversão para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="conexão"></a>Conexão<a aria-hidden="true" tabindex="-1" class="hash-link" href="#conexão" title="Direct link to heading">#</a></h4><p>Conexão do transformador utilizada para cálculo das correntes de sequência zero nas faltas desbalanceadas com presença de terra (fase-terra e fase-fase-terra).</p><p>As seguintes conexões estão disponíveis:</p><ol><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y_{aterrado}-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y_{aterrado}-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\Delta-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">Y_{aterrado}-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">Y-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li></ol><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="tap"></a>TAP<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tap" title="Direct link to heading">#</a></h4><p>Tape do transformador <strong>em relação ao primário</strong>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O tape no PSP-UFU deve ser inserido pela relação entre a tensão primária nominal e a tensão de tape <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtext>TAP</mtext><mo>=</mo><mfrac><msub><mi>V</mi><mrow><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>l</mi></mrow></msub><msub><mi>V</mi><mrow><mi>t</mi><mi>a</mi><mi>p</mi><mi>e</mi></mrow></msub></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\text{TAP} = \frac{V_{nominal}}{V_{tape}}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord text"><span class="mord">TAP</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.894191em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em"><span style="top:-2.357em;margin-left:-0.22222em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.41586em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3487714285714287em;margin-left:-0.22222em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5423199999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span></span>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="defasagem"></a>Defasagem<a aria-hidden="true" tabindex="-1" class="hash-link" href="#defasagem" title="Direct link to heading">#</a></h4><p>Defasagem do transformador em relação ao primário, em graus. Esse valor independe do tipo de conexão, portanto para conexões diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span>(aterrado ou não) e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span>, deve-se inserir o ângulo de defasagem correto.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2cZh" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do transformador como base para a conversão das unidades, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/transformer.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Transformador</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Aparelho estático com dois ou mais enrolamentos que, por indução eletromagnética, transforma um sistema de tensão e corrente alternada em outro sistema de tensão e corrente geralmente de valores diferentes e na mesma frequência com o objetivo de transmitir energia elétrica. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=421-01-01" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="transformador-no-psp-ufu"></a>Transformador no PSP-UFU<a class="hash-link" href="#transformador-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Transformadores com quaisquer conexões e defasagens entre as tensões primárias e secundárias podem ser inseridos no PSP-UFU. Para isso, os transformadores devem ser modelados representando seu tape (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em"></span><span class="mord mathdefault">t</span></span></span></span></span>) e defasagem (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">ϕ</span></span></span></span></span>), utilizando um transformador ideal de relação de transformação <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>:</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\overline{t}:1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.81508em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> em série com sua impedância (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mo>=</mo><msub><mi>r</mi><mi>T</mi></msub><mo>+</mo><mi>j</mi><msub><mi>x</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">\overline{y}_T = r_T + jx_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8747em;vertical-align:-0.24414em"></span><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.81508em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span> é um número complexo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi mathvariant="normal">∠</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">t\angle\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">t</span><span class="mord">∠</span><span class="mord mathdefault">ϕ</span></span></span></span></span>)</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O modelo utilizado no PSP-UFU <strong>não considera</strong> o ramo magnetizante ou impedâncias mútuas entre as fases.</p></div></div><p>A figura abaixo mostra o modelo do transformador implementado no PSP-UFU:</p><div><center><img src="/PSP/images/transformerModel.svg" alt="Modelo PI da linha" title="Modelo PI da linha"></center></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mi mathvariant="bold">T</mi></msub></mrow><annotation encoding="application/x-tex">\bold{r_T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a resistência total do transformador;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">T</mi></msub></mrow><annotation encoding="application/x-tex">\bold{x_T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a reatância indutiva total do transformador;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">t</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\bold{\overline{t}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8349200000000001em;vertical-align:0em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8349200000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathbf">t</span></span></span><span style="top:-3.7549200000000003em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span></span>	é a o tape complexo do transformador.</li></ul><p>A relação entre as tensões e correntes das barras genéricas <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span></span></span></span></span> é obtida pela equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>j</mi><mi>i</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mi>t</mi><mn>2</mn></msup></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>∗</mo></msup></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mfrac></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>×</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>j</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \dot{I}_{ij}\\ \dot{I}_{ji} \end{bmatrix} = \begin{bmatrix} \displaystyle \frac{\overline{y}_T}{t^2} &amp; -\displaystyle \frac{\overline{y}_T}{\overline{t}^*}\\ -\displaystyle \frac{\overline{y}_T}{\overline{t}} &amp; \overline{y}_T \end{bmatrix} \times \begin{bmatrix} \dot{V}_{i}\\ \dot{V}_{j} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.5603800000000003em;vertical-align:-1.0301900000000004em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.53019em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3298099999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0301900000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.202030000000001em;vertical-align:-1.850025em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000007em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2107020000000004em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.29492em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.70508em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019379999999998em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000007em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.216324em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.893676em"><span style="top:-3.26798em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.783676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2107020000000004em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019379999999998em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.5603800000000003em;vertical-align:-1.0301900000000004em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.53019em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3298099999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0301900000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span></span></span></span></span></div><p>Assim como nas <a href="/PSP/docs/line">linhas</a>, o transformador de dois enrolamentos deve ser inserido entre barras, porém não há a necessidade das tensões nominais desses barramentos serem idênticas.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A primeira barra selecionada será o lado <strong>primário</strong>, o qual é indicado por um círculo. Para alterar o lado primário após a inserção basta desconectar os nós dos transformadores e reconectá-los alternando as barras, utilizando as <a href="/PSP/docs/cadTools#conex%C3%A3o-e-desconex%C3%A3o-de-elemetos-na-barra">ferramentas CAD</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="transformadores-no-estudo-de-curto-circuito"></a>Transformadores no estudo de curto-circuito<a class="hash-link" href="#transformadores-no-estudo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas <a href="/PSP/docs/fault">faltas desbalanceadas</a> é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.</p><p>A matriz admitância de sequência positiva é construída da mesma forma que os estudos de fluxo de carga. Na sequência negativa o ângulo de defasagem entre tensões primária e secundária (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">ϕ</span></span></span></span></span>) deve ser invertido, uma vez que os fasores da sequência negativa <strong>são deslocados na direção oposta</strong>. Com isso, a matriz que representa as admitâncias da equação anterior deve ser substituída pela seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>j</mi><mi>i</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>j</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mi>t</mi><mn>2</mn></msup></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>∗</mo></msup></mfrac></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \overline{Y}_{ii} &amp; \overline{Y}_{ij}\\ \overline{Y}_{ji} &amp; \overline{Y}_{jj} \end{bmatrix} = \begin{bmatrix} \displaystyle \frac{\overline{y}_T}{t^2} &amp; -\displaystyle \frac{\overline{y}_T}{\overline{t}}\\ -\displaystyle \frac{\overline{y}_T}{\overline{t}^*} &amp; \overline{y}_T \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4866600000000005em;vertical-align:-0.9933300000000005em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.49333em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3666699999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9933300000000005em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.49333em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3666699999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9933300000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.202030000000001em;vertical-align:-1.850025em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000003em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2892979999999996em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.216324em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.893676em"><span style="top:-3.26798em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.783676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019380000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000003em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.29492em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.70508em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2892979999999996em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019380000000003em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>A impedância de sequência negativa nos transformadores deve ser tratada de maneira específica devido aos diferentes tipos de conexão. Na figura abaixo são mostrados os circuitos equivalentes para cada tipo de conexão de transformadores de dois enrolamentos. As setas indicam os caminhos possíveis para circulação da corrente de sequência zero.</p><div><center><img src="/PSP/images/seqZeroTrafo.svg" alt="Circuitos equivalentes de sequência zero dos transformadores" title="Circuitos equivalentes de sequência zero dos transformadores"></center></div><p>Para os transformadores com conexão estrela aterrado, se a ligação do neutro para a terra apresentar uma impedância <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, o circuito equivalente de sequência zero deve ter impedância de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">3\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> em série com a resistência e reatância equivalentes do transformador.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-transformadores"></a>Formulário de edição dos transformadores<a class="hash-link" href="#formulário-de-edição-dos-transformadores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos transformadores:</p><div><center><img src="/PSP/images/trafoForm.png" alt="Formulário dos transformadores no PSP-UFU" title="Formulário dos transformadores no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais dos transformadores, informações do fluxo de carga, sua conexão e defasagem;</li><li><strong>Falta</strong>: local onde as impedâncias de sequência zero e impedâncias de aterramento são inseridas.</li></ul><p>Além desses dois contextos, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/trafoSw.png" alt="Formulário de chaveamento do transformador" title="Formulário de chaveamento do transformador"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-nominal"></a>Tensão nominal<a class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Campo de informação <em>não editável</em> que apresenta a tensão primária e secundária do transformador. Para alterar esse campo é necessário editar o campo correspondente dos <a href="/PSP/docs/bus#geral">barramento</a> conectados.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-base"></a>Tensão base<a class="hash-link" href="#tensão-base" title="Direct link to heading">#</a></h4><p>Tensão utilizada para converter os parâmetros de impedância inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> As tensões selecionadas por uma caixa de escolha são editadas nos formulários dos <a href="/PSP/docs/bus#geral">barramentos</a> conectados.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A tensão base é ignorada caso os dados inseridos estejam em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do transformador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedância-do-transformador"></a>Impedância do transformador<a class="hash-link" href="#impedância-do-transformador" title="Direct link to heading">#</a></h4><p>Impedância série do transformador (resistência e reatância indutiva), inserida em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span>. Caso as impedâncias sejam inseridas em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> será utilizada a tensão base selecionada para conversão para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="conexão"></a>Conexão<a class="hash-link" href="#conexão" title="Direct link to heading">#</a></h4><p>Conexão do transformador utilizada para cálculo das correntes de sequência zero nas faltas desbalanceadas com presença de terra (fase-terra e fase-fase-terra).</p><p>As seguintes conexões estão disponíveis:</p><ol><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y_{aterrado}-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y_{aterrado}-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\Delta-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">Y_{aterrado}-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">Y-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li></ol><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tap"></a>TAP<a class="hash-link" href="#tap" title="Direct link to heading">#</a></h4><p>Tape do transformador <strong>em relação ao primário</strong>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O tape no PSP-UFU deve ser inserido pela relação entre a tensão primária nominal e a tensão de tape <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtext>TAP</mtext><mo>=</mo><mfrac><msub><mi>V</mi><mrow><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>l</mi></mrow></msub><msub><mi>V</mi><mrow><mi>t</mi><mi>a</mi><mi>p</mi><mi>e</mi></mrow></msub></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\text{TAP} = \frac{V_{nominal}}{V_{tape}}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord text"><span class="mord">TAP</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.894191em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em"><span style="top:-2.357em;margin-left:-0.22222em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.41586em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3487714285714287em;margin-left:-0.22222em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5423199999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span></span>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="defasagem"></a>Defasagem<a class="hash-link" href="#defasagem" title="Direct link to heading">#</a></h4><p>Defasagem do transformador em relação ao primário, em graus. Esse valor independe do tipo de conexão, portanto para conexões diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span>(aterrado ou não) e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span>, deve-se inserir o ângulo de defasagem correto.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do transformador como base para a conversão das unidades, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedância-de-sequência-zero-do-transformador"></a>Impedância de sequência zero do transformador<a class="hash-link" href="#impedância-de-sequência-zero-do-transformador" title="Direct link to heading">#</a></h4><p>Esses parâmetros são necessários para o correto cálculo das correntes de <a href="/PSP/docs/fault">falta desbalanceadas</a> (fase-fase, fase-fase-terra e fase-terra) e devem ser inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A não inserção desses dados acarretarão em resultados incorretos para <a href="/PSP/docs/fault">faltas desbalanceadas</a>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedâncias-de-aterramento"></a>Impedâncias de aterramento<a class="hash-link" href="#impedâncias-de-aterramento" title="Direct link to heading">#</a></h4><p>As impedâncias de aterramento são somente utilizadas para conexão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e devem ser inseridas em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p></div><div role="tabpanel" hidden=""><p>O botão &quot;Estabilidade&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/transformer.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/line"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Linha</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/load"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Carga »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#transformador-no-psp-ufu" class="table-of-contents__link">Transformador no PSP-UFU</a><ul><li><a href="#transformadores-no-estudo-de-curto-circuito" class="table-of-contents__link">Transformadores no estudo de curto-circuito</a></li></ul></li><li><a href="#formulário-de-edição-dos-transformadores" class="table-of-contents__link">Formulário de edição dos transformadores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"https://pt.wikipedia.org/wiki/Unicode"}),"Unicode"),"."),Object(n.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(n.b)("h4",{id:"pot\xeancia-nominal"},"Pot\xeancia nominal"),Object(n.b)("p",null,"Pot\xeancia nominal do gerador, inserida em MVA, kVA ou VA."),Object(n.b)("p",null,'Esse campo \xe9 especialmente importante caso a op\xe7\xe3o "Utilizar a pot\xeancia nominal como base" esteja marcada.'),Object(n.b)("h4",{id:"pot\xeancias-ativa-e-reativa"},"Pot\xeancias ativa e reativa"),Object(n.b)("p",null,"Pot\xeancias ativa (inserida em W, kW, MW ou ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math 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Tal fator eleva o valor da resist\xeancia do rotor em condi\xe7\xf5es de grande escorregamento, como ocorre em motores de gaiola dupla ou gaiola profunda. 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Uberlândia</p><p><center><iframe src="https://ghbtns.com/github-btn.html?user=thales1330&amp;repo=psp&amp;type=star&amp;count=true&amp;size=large" frameborder="0" scrolling="0" width="130" height="30" title="GitHub"></iframe></center></p><div class="buttons_1Wc3"><a href="https://github.com/Thales1330/PSP/releases/latest" target="_blank" rel="noopener noreferrer" class="button button--outline button--secondary button--lg">Download</a></div><div class="buttons_1Wc3"><a class="button button--outline button--secondary button--lg" href="/PSP/docs/">User Guide</a></div><div class="buttons_1Wc3"><a href="https://minhaskamal.github.io/DownGit/#/home?url=https://github.com/Thales1330/PSP/tree/master/Project/data/samples" target="_blank" rel="noopener noreferrer" class="button button--outline button--secondary button--lg">Samples</a></div></div></header><main><section class="features_P2SU"><div class="container"><div class="row"><div class="col col--4"><div class="text--center"><img class="featureImage_3Xqx" src="/PSP/img/complete-platform.svg" alt="[object Object]"></div><h3>Complete Platform</h3><p>PSP-UFU is a <b>cross-platform</b>, <b>multilingual</b>, <b>Free and Open-Source Software (FOSS)</b> with <b>advanced GUI (Graphical User Interface)</b> features and <b>CAD (Computer-Aided Design) tools</b> for electrical power system studies.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_3Xqx" src="/PSP/img/advanced-cad.svg" alt="[object Object]"></div><h3>Advanced CAD Features</h3><p>The software allows the construction of any electric <b>transmission network</b> and <b>control systems</b> through the deployment of visual elements.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_3Xqx" src="/PSP/img/visualization.svg" alt="[object Object]"></div><h3>Easy Visualization</h3><p>For the visualization of results, the program offers linked text elements in the main screen, voltage heatmap display, and also table and graph editors.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_3Xqx" src="/PSP/img/application.svg" alt="[object Object]"></div><h3>Application</h3><p>The PSP-UFU aims to provide efficient computer simulation tools for <b>research</b> and <b>education</b> purposes, in addition to <b>industrial applications</b> in electrical power systems.</p></div></div></div></section><div class="container"><h2>Studies Performed</h2></div><section class="features_P2SU"><div class="container"><div class="row"><div class="col col--4"><h3>Power Flow</h3><ul><li>Newton-Raphson</li><li>Gauss-Seidel</li><li>Hybrid Newton-Gauss</li><li>Three-phase induction motors included</li></ul></div><div class="col col--4"><h3>Short-Circuit calculation</h3><ul><li>Balanced</li><li>Unbalanced</li><li>Short-Circuit power in all system buses</li></ul></div><div class="col col--4"><h3>Harmonics</h3><ul><li>Harmonic voltages and THD (Total Harmonic Distortion) calculation</li><li>Frequency scan</li></ul></div><div class="col col--4"><h3>Transient and Dynamic Stability</h3><ul><li>Several synchronous machine models automatically selected</li><li>Three-phase induction motors</li><li>User-defined machine controls, exciters and prime moves created using block diagrams (Exciters, AVR, PSS, Hydro and Thermal turbines, Speed Governor, etc.)</li></ul></div></div></div></section><div class="container"><h2>Published Papers</h2>Further details can be found in the published papers:<blockquote cite="https://doi.org/10.1002/2050-7038.12185">Oliveira, T. L., Guimarães, G. C., &amp; Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. <i>International Transactions on Electrical Energy Systems</i>, e12185. doi: <a href="https://doi.org/10.1002/2050-7038.12185">10.1002/2050-7038.12185</a></blockquote><blockquote cite="https://doi.org/10.1177/0020720919879058">Oliveira, T. L., Guimarães, G. C., Silva, L. R., &amp; Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. <i>The International Journal of Electrical Engineering &amp; Education</i>, 0020720919879058. doi: <a href="https://doi.org/10.1177/0020720919879058">10.1177/0020720919879058</a></blockquote></div><div class="container"><h2>Code Documentation</h2><p>All detailed descriptions of the source-code can be found at <b><a href="https://thales1330.github.io/PSP/doxygen/html/index.html">Online Documentation</a></b>, generated by <a href="http://www.doxygen.org/">Doxygen</a>.</p></div><div class="container"><h2>Screenshots</h2><center><iframe src="gallery.html" style="position:relative;width:100%;height:56.25vw;display:block;overflow:hidden;border:none">Your browser doesn’t support the this feature.</iframe></center></div></main></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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hero--primary heroBanner_3P7f"><div class="container"><h1 class="hero__title">PSP-UFU</h1><p class="hero__subtitle">Power System Platform of Federal University of Uberlândia</p><p><center><iframe src="https://ghbtns.com/github-btn.html?user=thales1330&amp;repo=psp&amp;type=star&amp;count=true&amp;size=large" frameborder="0" scrolling="0" width="130" height="30" title="GitHub"></iframe></center></p><div class="buttons_1r9m"><a href="https://github.com/Thales1330/PSP/releases/latest" target="_blank" rel="noopener noreferrer" class="button button--outline button--secondary button--lg">Download</a></div><div class="buttons_1r9m"><a class="button button--outline button--secondary button--lg" href="/PSP/docs/">User Guide</a></div><div class="buttons_1r9m"><a href="https://minhaskamal.github.io/DownGit/#/home?url=https://github.com/Thales1330/PSP/tree/master/Project/data/samples" target="_blank" rel="noopener noreferrer" class="button button--outline button--secondary button--lg">Samples</a></div></div></header><main><section class="features_3azU"><div class="container"><div class="row"><div class="col col--4"><div class="text--center"><img class="featureImage_ZtzX" src="/PSP/img/complete-platform.svg" alt="[object Object]"></div><h3>Complete Platform</h3><p>PSP-UFU is a <b>cross-platform, multilingual, Free and Open-Source Software (FOSS)</b> with <b>advanced GUI (Graphical User Interface) features</b> and <b>CAD (Computer-Aided Design) tools</b> for electrical power system studies.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_ZtzX" src="/PSP/img/advanced-cad.svg" alt="[object Object]"></div><h3>Advanced CAD Features</h3><p>The software allows the construction of any <b>electric transmission network</b> and <b>control systems</b> through the deployment of visual elements.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_ZtzX" src="/PSP/img/visualization.svg" alt="[object Object]"></div><h3>Easy Visualization</h3><p>For the visualization of results, the program offers linked text elements in the main screen, voltage heatmap display, and also table and graph editors.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_ZtzX" src="/PSP/img/application.svg" alt="[object Object]"></div><h3>Application</h3><p>The PSP-UFU aims to provide efficient computer simulation tools for <b>research and education purposes</b>, in addition to <b>industrial applications</b> in electrical power systems.</p></div></div></div></section><div class="container"><h2>Studies Performed</h2></div><section class="features_3azU"><div class="container"><div class="row"><div class="col col--4"><h3>Power Flow</h3><ul><li>Newton-Raphson</li><li>Gauss-Seidel</li><li>Hybrid Newton-Gauss</li><li>Three-phase induction motors included</li></ul></div><div class="col col--4"><h3>Short-Circuit calculation</h3><ul><li>Balanced</li><li>Unbalanced</li><li>Short-Circuit power in all system buses</li></ul></div><div class="col col--4"><h3>Harmonics</h3><ul><li>Harmonic voltages and THD (Total Harmonic Distortion) calculation</li><li>Frequency scan</li></ul></div><div class="col col--4"><h3>Transient and Dynamic Stability</h3><ul><li>Several synchronous machine models automatically selected</li><li>Three-phase induction motors</li><li>User-defined machine controls, exciters and prime moves created using block diagrams (Exciters, AVR, PSS, Hydro and Thermal turbines, Speed Governor, etc.)</li></ul></div></div></div></section><div class="container"><h2>Published Papers</h2>Further details can be found in the published papers:<blockquote cite="https://doi.org/10.1002/2050-7038.12185">Oliveira, T. L., Guimarães, G. C., &amp; Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. <i>International Transactions on Electrical Energy Systems</i>, e12185. doi: <a href="https://doi.org/10.1002/2050-7038.12185">10.1002/2050-7038.12185</a></blockquote><blockquote cite="https://doi.org/10.1177/0020720919879058">Oliveira, T. L., Guimarães, G. C., Silva, L. R., &amp; Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. <i>The International Journal of Electrical Engineering &amp; Education</i>, 0020720919879058. doi: <a href="https://doi.org/10.1177/0020720919879058">10.1177/0020720919879058</a></blockquote></div><div class="container"><h2>Code Documentation</h2><p>All detailed descriptions of the source-code can be found at <b><a href="https://thales1330.github.io/PSP/doxygen/html/index.html">Online Documentation</a></b>, generated by <a href="http://www.doxygen.org/">Doxygen</a>.</p></div><div class="container"><h2>Screenshots</h2><center><iframe src="gallery.html" style="position:relative;width:100%;height:56.25vw;display:block;overflow:hidden;border:none">Your browser doesn’t support the this feature.</iframe></center></div></main></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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t={pattern:/{\$(?:{(?:{[^{}]+}|[^{}]+)}|[^{}])+}|(^|[^\\{])\$+(?:\w+(?:\[.+?]|->\w+)*)/,lookbehind:!0,inside:{rest:e.languages.php}};e.languages.insertBefore("php","string",{"nowdoc-string":{pattern:/<<<'([^']+)'(?:\r\n?|\n)(?:.*(?:\r\n?|\n))*?\1;/,greedy:!0,alias:"string",inside:{delimiter:{pattern:/^<<<'[^']+'|[a-z_]\w*;$/i,alias:"symbol",inside:{punctuation:/^<<<'?|[';]$/}}}},"heredoc-string":{pattern:/<<<(?:"([^"]+)"(?:\r\n?|\n)(?:.*(?:\r\n?|\n))*?\1;|([a-z_]\w*)(?:\r\n?|\n)(?:.*(?:\r\n?|\n))*?\2;)/i,greedy:!0,alias:"string",inside:{delimiter:{pattern:/^<<<(?:"[^"]+"|[a-z_]\w*)|[a-z_]\w*;$/i,alias:"symbol",inside:{punctuation:/^<<<"?|[";]$/}},interpolation:t}},"single-quoted-string":{pattern:/'(?:\\[\s\S]|[^\\'])*'/,greedy:!0,alias:"string"},"double-quoted-string":{pattern:/"(?:\\[\s\S]|[^\\"])*"/,greedy:!0,alias:"string",inside:{interpolation:t}}}),delete 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n=El;El|=16;for(var r=ou();;)try{lu();break}catch(o){ru(e,o)}if(ea(),El=n,vl.current=r,1===Cl)throw n=_l,nu(e,t),Iu(e,t),Yl(e),n;if(null!==xl)throw Error(i(261));e.finishedWork=e.current.alternate,e.finishedExpirationTime=t,Pl=null,fu(e),Yl(e)}return null}function eu(e,t){var n=El;El|=1;try{return e(t)}finally{0===(El=n)&&Vo()}}function tu(e,t){var n=El;El&=-2,El|=8;try{return e(t)}finally{0===(El=n)&&Vo()}}function nu(e,t){e.finishedWork=null,e.finishedExpirationTime=0;var n=e.timeoutHandle;if(-1!==n&&(e.timeoutHandle=-1,wn(n)),null!==xl)for(n=xl.return;null!==n;){var r=n;switch(r.tag){case 1:null!=(r=r.type.childContextTypes)&&bo();break;case 3:Ia(),uo(po),uo(fo);break;case 5:Fa(r);break;case 4:Ia();break;case 13:case 19:uo(ja);break;case 10:ta(r)}n=n.return}Pl=e,xl=Tu(e.current,null),Tl=t,Cl=wl,_l=null,Rl=Ol=1073741823,Al=null,Nl=0,Il=!1}function ru(e,t){for(;;){try{if(ea(),za.current=gi,Ha)for(var n=Ba.memoizedState;null!==n;){var 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r=e.pingCache;null!==r&&r.delete(t),Pl===e&&Tl===n?Cl===Sl||Cl===kl&&1073741823===Ol&&Uo()-Ll<500?nu(e,Tl):Il=!0:Nu(e,n)&&(0!==(t=e.lastPingedTime)&&t<n||(e.lastPingedTime=n,Yl(e)))}function wu(e,t){var n=e.stateNode;null!==n&&n.delete(t),0===(t=0)&&(t=Wl(t=Vl(),e,null)),null!==(e=Kl(e,t))&&Yl(e)}gl=function(e,t,n){var r=t.expirationTime;if(null!==e){var o=t.pendingProps;if(e.memoizedProps!==o||po.current)Ri=!0;else{if(r<n){switch(Ri=!1,t.tag){case 3:zi(t),_i();break;case 5:if(La(t),4&t.mode&&1!==n&&o.hidden)return t.expirationTime=t.childExpirationTime=1,null;break;case 1:go(t.type)&&wo(t);break;case 4:Na(t,t.stateNode.containerInfo);break;case 10:r=t.memoizedProps.value,o=t.type._context,so(Xo,o._currentValue),o._currentValue=r;break;case 13:if(null!==t.memoizedState)return 0!==(r=t.child.childExpirationTime)&&r>=n?qi(e,t,n):(so(ja,1&ja.current),null!==(t=Qi(e,t,n))?t.sibling:null);so(ja,1&ja.current);break;case 19:if(r=t.childExpirationTime>=n,0!=(64&e.effectTag)){if(r)return 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t=e._ctor;t=t(),e._result=t,t.then((function(t){0===e._status&&(t=t.default,e._status=1,e._result=t)}),(function(t){0===e._status&&(e._status=2,e._result=t)}))}}(o),1!==o._status)throw o._result;switch(o=o._result,t.type=o,a=t.tag=function(e){if("function"==typeof e)return xu(e)?1:0;if(null!=e){if((e=e.$$typeof)===ue)return 11;if(e===de)return 14}return 2}(o),e=Ko(o,e),a){case 0:t=ji(null,t,o,e,n);break e;case 1:t=Di(null,t,o,e,n);break e;case 11:t=Ni(null,t,o,e,n);break e;case 14:t=Ii(null,t,o,Ko(o.type,e),r,n);break e}throw Error(i(306,o,""))}return t;case 0:return r=t.type,o=t.pendingProps,ji(e,t,r,o=t.elementType===r?o:Ko(r,o),n);case 1:return r=t.type,o=t.pendingProps,Di(e,t,r,o=t.elementType===r?o:Ko(r,o),n);case 3:if(zi(t),r=t.updateQueue,null===e||null===r)throw 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Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(m.b)("a",{parentName:"p",href:"https://pt.wikipedia.org/wiki/Unicode"},"Unicode"),"."),Object(m.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(m.b)("h4",{id:"pot\xeancia-ativa"},"Pot\xeancia ativa"),Object(m.b)("p",null,"Parcela de pot\xeancia ativa da carga. 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pot\xeancia constante")," no estudo de ",Object(n.b)("a",{parentName:"p",href:"powerFlow"},"fluxo de carga"),"."),Object(n.b)("p",{parentName:"div"},"A n\xe3o utiliza\xe7\xe3o dessa op\xe7\xe3o poder\xe1 gerar erros de regime permamente no estudo de ",Object(n.b)("a",{parentName:"p",href:"stability"},"estabilidade"),"."))),Object(n.b)("h4",{id:"utilizar-pot\xeancia-nominal-como-base"},"Utilizar pot\xeancia nominal como base"),Object(n.b)("p",null,"Caso essa op\xe7\xe3o seja marcada, o programa ir\xe1 utilizar a pot\xeancia nominal do gerador como base para a convers\xe3o das unidades, inclusive aqueles no formul\xe1rio de estabilidade, caso contr\xe1rio ser\xe1 usada a pot\xeancia base do sistema.")),Object(n.b)(c.a,{value:"stability",mdxType:"TabItem"},Object(n.b)("h4",{id:"imprimir-dados-da-m\xe1quina-de-indu\xe7\xe3o"},"Imprimir dados da m\xe1quina de indu\xe7\xe3o"),Object(n.b)("p",null,"Exibe os dados do gerador s\xedncrono nos gr\xe1ficos no tempo. 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no motor de indu\xe7\xe3o. 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Nessa op\xe7\xe3o ",Object(m.b)("strong",{parentName:"p"},"deve")," ser inserida ao menos a malha de controle da regula\xe7\xe3o prim\xe1ria de velocidade, assim como o ",Object(m.b)("strong",{parentName:"p"},"modelo da turbina"),". Estrat\xe9gias opcionais de controle da velocidade tamb\xe9m s\xe3o inseridas nessa op\xe7\xe3o."))),Object(m.b)("h2",{id:"elementos-de-controle"},"Elementos de Controle"),Object(m.b)("p",null,"O acesso aos elementos de controle \xe9 realizado em uma janela (remov\xedvel e encaix\xe1vel) por meio de \xedcones relacionados."),Object(m.b)("p",null,"Uma vez que o usu\xe1rio clicar no \xedcone desejado, o elemento de controle acompanhar\xe1 o ponteiro do mouse at\xe9 ser efetivamente inserido na posi\xe7\xe3o desejada ao clicar novamente na \xe1rea de trabalho. 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Tal procedimento \xe9 realizado por meio da solu\xe7\xe3o de toda rede de controle at\xe9 que se obtenha sua converg\xeancia, ou seja, a diferen\xe7a absoluta entre as mesmas sa\xeddas de uma solu\xe7\xe3o anterior e uma atual deve ser nula ou muito pr\xf3xima de zero."),Object(m.b)("div",{className:"admonition admonition-info alert alert--info"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"}))),"info")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"O processo de ",Object(m.b)("strong",{parentName:"p"},"inicializa\xe7\xe3o \xe9 realizada automaticamente")," pelo PSP-UFU."),Object(m.b)("p",{parentName:"div"},"Uma vez que a inicializa\xe7\xe3o \xe9 imposta pelas ",Object(m.b)("a",{parentName:"p",href:"io"},"entradas")," do controle, erros de converg\xeancia podem ocorrer devido \xe0 m\xe1 parametriza\xe7\xe3o dos elementos dos sistemas de ",Object(m.b)("strong",{parentName:"p"},"pot\xeancia e controle"),"."))),Object(m.b)("p",null,"Para otimizar e melhorar a estabilidade do processo de inicializa\xe7\xe3o utilizou-se um passo de integra\xe7\xe3o vari\xe1vel dentro de limites, de forma que o passo aumenta em condi\xe7\xf5es de diferen\xe7as menores entre as solu\xe7\xf5es do sistema de controle e diminui caso essa diferen\xe7a se torne elevada. 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As entradas iniciais s\xe3o sempre definidas como zero e assumem valores diferentes ap\xf3s o tempo de in\xedcio;"),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Inclina\xe7\xe3o (",Object(m.b)("span",{parentName:"strong",className:"math 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Um processo iterativo \xe9 realizado at\xe9 que o erro entre os c\xe1lculos se torne menor que uma toler\xe2ncia pr\xe9-estipulada e caso as itera\xe7\xf5es se tornem excessivas, ultrapassando um n\xfamero m\xe1ximo definido pelo usu\xe1rio, o processo \xe9 interrompido com erro. 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programa, as quais permitem ao usu\xe1rio criar e salvar novos projetos, al\xe9m daquelas que possibilitam carregar os projetos j\xe1 existentes e importar arquivos de outros programas. 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Nesse caso, ao criar um novo projeto a tela de trabalho n\xe3o \xe9 exibida ou \xe9 exibida uma tela preta. A solu\xe7\xe3o \xe9 alterar o renderizador para "Device Context". 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Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(r.b)("a",{parentName:"p",href:"https://pt.wikipedia.org/wiki/Unicode"},"Unicode"),"."),Object(r.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(r.b)("h4",{id:"pot\xeancia-nominal"},"Pot\xeancia nominal"),Object(r.b)("p",null,"Pot\xeancia nominal do compensador, inserida em MVA, kVA ou VA."),Object(r.b)("p",null,'Esse campo \xe9 especialmente importante caso a op\xe7\xe3o "Utilizar a pot\xeancia nominal como base" esteja marcada.'),Object(r.b)("h4",{id:"pot\xeancias-ativa-e-reativa"},"Pot\xeancias ativa e reativa"),Object(r.b)("p",null,"Pot\xeancias ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do compensador."),Object(r.b)("p",null,"Caso a barra conectada seja PV o valor de pot\xeancia reativa ser\xe1 ignorado e caso seja de refer\xeancia ambos os valores inseridos ser\xe3o desprezados."),Object(r.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(r.b)("div",{parentName:"div",className:"admonition-heading"},Object(r.b)("h5",{parentName:"div"},Object(r.b)("span",{parentName:"h5",className:"admonition-icon"},Object(r.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(r.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(r.b)("div",{parentName:"div",className:"admonition-content"},Object(r.b)("p",{parentName:"div"},"Caso mais de um compensador esteja conectado na mesma barra, os valores de pot\xeancia reativa (nas barras de refer\xeancia e PV) e ativa (nas barras de refer\xeancia) s\xe3o igualmente distribu\xeddas, respeitando os limites individuais de pot\xeancia reativa."))),Object(r.b)("h4",{id:"pot\xeancias-reativas-m\xe1xima-e-m\xednima"},"Pot\xeancias reativas m\xe1xima e m\xednima"),Object(r.b)("p",null,"Limites de pot\xeancia reativa m\xe1xima e m\xednima do compensador para controle de tens\xe3o em barras PV. 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math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"0")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"0")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"0"))))),", retorna o valor de ",Object(m.b)("inlineCode",{parentName:"td"},"B"),", sen\xe3o retorna ",Object(m.b)("inlineCode",{parentName:"td"},"C"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"imag(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Retorna a parte imagin\xe1ria do n\xfamero complexo ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"int(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Arredonda ",Object(m.b)("inlineCode",{parentName:"td"},"A")," para o inteiro mais pr\xf3ximo.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"log(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Logaritmo natural (base ",Object(m.b)("span",{parentName:"td",className:"math 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",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"log2(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Logaritmo na base ",Object(m.b)("span",{parentName:"td",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"2")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"2")))))," de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"log10(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Logaritmo na base ",Object(m.b)("span",{parentName:"td",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"10")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"10")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"1"),Object(m.b)("span",{parentName:"span",className:"mord"},"0")))))," de A.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"max(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Se ",Object(m.b)("inlineCode",{parentName:"td"},"A>B"),", o resultado \xe9 ",Object(m.b)("inlineCode",{parentName:"td"},"A"),", sen\xe3o \xe9 ",Object(m.b)("inlineCode",{parentName:"td"},"B"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"min(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Se ",Object(m.b)("inlineCode",{parentName:"td"},"A<B"),", o resultado \xe9 ",Object(m.b)("inlineCode",{parentName:"td"},"A"),", sen\xe3o \xe9 ",Object(m.b)("inlineCode",{parentName:"td"},"B"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"polar(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Retorna o n\xfamero complexo de magnitude ",Object(m.b)("inlineCode",{parentName:"td"},"A")," e \xe2ngulo de fase ",Object(m.b)("inlineCode",{parentName:"td"},"B")," (em radianos).")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"pow(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Exponencia\xe7\xe3o (",Object(m.b)("inlineCode",{parentName:"td"},"A")," elevado \xe0 pot\xeancia ",Object(m.b)("inlineCode",{parentName:"td"},"B"),")")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"real(A,B)")),Object(m.b)("td",{parentName:"tr",align:null},"Retorna a parte real do n\xfamero complexo ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"sec(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Secante de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"sin(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Seno de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),". Retorna o \xe2ngulo em radianos.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"sinh(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Mesmo que ",Object(m.b)("inlineCode",{parentName:"td"},"sin()"),", mas para seno hiperb\xf3lico.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"sqrt(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Raiz quadrada de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),".")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"tan(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Tangente de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),". Retorna o \xe2ngulo em radianos.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"tanh(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Mesmo que ",Object(m.b)("inlineCode",{parentName:"td"},"tan()"),", mas para tangente hiperb\xf3lica.")),Object(m.b)("tr",{parentName:"tbody"},Object(m.b)("td",{parentName:"tr",align:null},Object(m.b)("inlineCode",{parentName:"td"},"trunc(A)")),Object(m.b)("td",{parentName:"tr",align:null},"Valor truncado de ",Object(m.b)("inlineCode",{parentName:"td"},"A"),". Retorna o n\xfamero inteiro de ",Object(m.b)("inlineCode",{parentName:"td"},"A")," sem a parcela fracionada.")))),Object(m.b)("p",null,"\xc9 poss\xedvel atribuir novas vari\xe1veis utilizando a seguinte sintaxe: "),Object(m.b)("pre",null,Object(m.b)("code",{parentName:"pre",className:"language-cpp"},"<nome da vari\xe1vel> := <express\xe3o>; <fun\xe7\xe3o>\n")),Object(m.b)("p",null,"Por exemplo:"),Object(m.b)("pre",null,Object(m.b)("code",{parentName:"pre",className:"language-cpp"},"comprimento := sqrt(x*x+y*y); 2*comprimento*sin(comprimento)\n")),Object(m.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Dica")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"O caractere de espa\xe7o e de nova linha s\xe3o ignorados na interpreta\xe7\xe3o da express\xe3o, portanto para maior organiza\xe7\xe3o, o c\xf3digo anterior pode ser escrito da seguinte forma:"),Object(m.b)("pre",{parentName:"div"},Object(m.b)("code",{parentName:"pre",className:"language-cpp"},"comprimento := sqrt(x*x+y*y);\n2*comprimento*sin(comprimento)\n")))),Object(m.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Cuidado!")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Note que a express\xe3o que determina o valor de sa\xedda do bloco ",Object(m.b)("strong",{parentName:"p"},"n\xe3o possui")," o caractere ",Object(m.b)("inlineCode",{parentName:"p"},";")," em seu final."))),Object(m.b)("h2",{id:"formul\xe1rio-de-edi\xe7\xe3o-de-dados-do-bloco-de-express\xe3o-matem\xe1tica"},"Formul\xe1rio de edi\xe7\xe3o de dados do bloco de Express\xe3o Matem\xe1tica"),Object(m.b)("p",null,"O formul\xe1rio de inser\xe7\xe3o e edi\xe7\xe3o das entradas do bloco e da express\xe3o matem\xe1tica gen\xe9rica, assim como ferramentas de aux\xedlio de sua constru\xe7\xe3o \xe9 apresentado na figura abaixo."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(p.a)("images/mathExpressionForm.png"),alt:"Formul\xe1rio de edi\xe7\xe3o de dados do bloco de express\xe3o matem\xe1tica no PSP-UFU",title:"Formul\xe1rio de edi\xe7\xe3o de dados do bloco de express\xe3o matem\xe1tica no PSP-UFU"}))),Object(m.b)("p",null,"No campo de \u201cVari\xe1veis de entrada\u201d \xe9 inserida uma lista com os nomes das entradas separados por espa\xe7os. Um n\xfamero qualquer de entrada pode ser definido nessa lista esses nomes s\xe3o apresentados no \xedcone gr\xe1fico presente no editor de controle, al\xe9m de serem destacados na express\xe3o inserida pelo usu\xe1rio. O n\xfamero de entradas e sa\xeddas se comporta de forma semelhante aos ",Object(m.b)("a",{parentName:"p",href:"sum"},"blocos somadores"),"."),Object(m.b)("p",null,"Abaixo do campo das vari\xe1veis de entrada est\xe1 presente o local para inser\xe7\xe3o da express\xe3o matem\xe1tica. A sintaxe da express\xe3o inserida pelo usu\xe1rio possui realce (por meio de diferentes formas e cores da fonte) para n\xfameros, operadores, vari\xe1veis de entrada, fun\xe7\xf5es e constantes, facilitando a cria\xe7\xe3o, manipula\xe7\xe3o e identifica\xe7\xe3o de erros de digita\xe7\xe3o e l\xf3gica."),Object(m.b)("div",{className:"admonition admonition-info alert alert--info"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"}))),"Informa\xe7\xe3o")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Como ferramenta de aux\xedlio ao usu\xe1rio foi desenvolvida uma verifica\xe7\xe3o da express\xe3o inserida."))),Object(m.b)("p",null,"Tal ferramenta ir\xe1 encontrar erros e indicar\xe1 ao usu\xe1rio qual o tipo do erro, al\xe9m da sua localiza\xe7\xe3o, destacando-o. A figura anterior exemplifica a identifica\xe7\xe3o autom\xe1tica do erro pelo PSP-UFU, assim como sua posi\xe7\xe3o na express\xe3o inserida."),Object(m.b)("h2",{id:"exemplo-de-express\xe3o-matem\xe1tica"},"Exemplo de Express\xe3o Matem\xe1tica"),Object(m.b)("p",null,"A corrente de campo em ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"p"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},"."),Object(m.b)("mi",{parentName:"mrow"},"u"),Object(m.b)("mi",{parentName:"mrow",mathvariant:"normal"},".")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"p.u.")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.19444em"}}),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"p"),Object(m.b)("span",{parentName:"span",className:"mord"},"."),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"u"),Object(m.b)("span",{parentName:"span",className:"mord"},".")))))," pode ser estimada de forma aproximada utilizando as pot\xeancias ativa (",Object(m.b)("span",{parentName:"p",className:"math 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Ao solt\xe1-lo todos os objetos que intercederem o ret\xe2ngulo ser\xe3o selecionados."),Object(n.b)("p",null,"Mantendo pressionado a tecla Control (Ctrl) do teclado pode-se manter sele\xe7\xf5es anteriores enquanto novos elementos s\xe3o selecionados."),Object(n.b)("p",null,"Para desmarcar todos os objetos basta clicar em uma \xe1rea sem objetos."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/select.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/select.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"mover"},"Mover"),Object(n.b)("p",null,"Um elemento poder\xe1 ser movido para qualquer local por meio da a\xe7\xe3o ",Object(n.b)("em",{parentName:"p"},"drag-and-drop")," (clicando e arrastando) ou pela ferramenta mover."),Object(n.b)("p",null,"Ao habilitar o \u201cmodo mover\u201d, movendo todos os componentes selecionados \xe0 medida que a posi\xe7\xe3o do mouse \xe9 alterada. 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O elemento tamb\xe9m pode ser rotacionado pelo ",Object(n.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"submenu Ferramentas")," ou ao acessar o menu de contexto clicando com o bot\xe3o direito sobre o elemento."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"O menu de contexto somente ser\xe1 exibido se o elemento estiver selecionado."))),Object(n.b)("p",null,"Ao utilizar essa ferramenta os objetos ir\xe3o girar 45\xba a partir do seu ponto de origem. Para atingir a angula\xe7\xe3o pretendida basta repetir o processo de rota\xe7\xe3o."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/rotate.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/rotate.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"arrastar"},"Arrastar"),Object(n.b)("p",null,"Tr\xeas ferramentas muito \xfateis na navega\xe7\xe3o do circuito, principalmente em grandes redes, s\xe3o: arrastar, zoom e encaixar."),Object(n.b)("p",null,"A ferramenta arrastar possibilita mover todo o circuito ",Object(n.b)("strong",{parentName:"p"},"preservando as posi\xe7\xf5es relativas entre os elementos"),". Esta ferramenta pode ser acessada pelo submenu \u201cFerramentas\u201d ou ",Object(n.b)("strong",{parentName:"p"},"mantendo pressionada o ",Object(n.b)("em",{parentName:"strong"},"scroll")," do mouse"),", ativando assim o \u201cmodo arrastar\u201d. Ao ativ\xe1-lo todo o circuito ser\xe1 movido clicando e arrastando na dire\xe7\xe3o desejada."),Object(n.b)("p",null,"Para sair do \u201cmodo arrastar\u201d basta pressionar a tecla Escape do teclado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/drag.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/drag.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"zoom"},"Zoom"),Object(n.b)("p",null,"O zoom ser\xe1 aplicado ao utilizar o ",Object(n.b)("strong",{parentName:"p"},"scroll do mouse")," para mais ou menos zoom. A ferramenta zoom ir\xe1 aproximar ou distanciar o circuito a partir da posi\xe7\xe3o do ponteiro do mouse na tela para encontrar objetos ou destaca-los."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/zoom.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/zoom.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"encaixar"},"Encaixar"),Object(n.b)("p",null,"A ferramenta encaixar ir\xe1 mover o circuito e aplicar o zoom necess\xe1rio para que todos os componentes da rede sejam exibidos na tela. Esta ferramenta pode ser acessada atrav\xe9s do submenu \u201cFerramentas\u201d ou pela combina\xe7\xe3o de teclas \u201cShift + F\u201d."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/fit.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/fit.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"copiar-e-colar"},"Copiar e Colar"),Object(n.b)("p",null,"Duas ferramentas importantes durante o processo de cria\xe7\xe3o da rede el\xe9trica de pot\xeancia no ",Object(n.b)("em",{parentName:"p"},"software")," s\xe3o copiar e colar. Qualquer circuito criado no PSP-UFU poder\xe1 ser duplicado completamente ou parcialmente no mesmo projeto ou em m\xfaltiplos atrav\xe9s dessas ferramentas."),Object(n.b)("p",null,"Para copiar basta selecionar os elementos que se deseja duplicar e acessar a ferramenta no submenu \u201cFerramentas\u201d ou pressionando a combina\xe7\xe3o das teclas \u201cCtrl + C\u201d, ent\xe3o os dados do circuito ser\xe3o copiados para a \xe1rea de transfer\xeancia. Ao copiar um circuito ele poder\xe1 ser colado no mesmo projeto ou em um projeto distinto, utilizando o mesmo submenu ou pressionando a combina\xe7\xe3o das teclas \u201cCtrl + V\u201d."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/copyPaste.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/copyPaste.webm"),type:"video/webm"})),Object(n.b)("h3",{id:"excluir"},"Excluir"),Object(n.b)("p",null,"Qualquer objeto poder\xe1 ser exclu\xeddo do projeto por meio da ferramenta deletar, presente no menu Ribbon, pela tecla de atalho Delete ou pelo acesso ao menu de contexto clicando com o bot\xe3o direito sobre o elemento selecionado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/delete.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/delete.webm"),type:"video/webm"})),Object(n.b)("h2",{id:"personaliza\xe7\xe3o-gr\xe1fica-os-elementos"},"Personaliza\xe7\xe3o gr\xe1fica os elementos"),Object(n.b)("p",null,"Alguns elementos permitem sua personaliza\xe7\xe3o gr\xe1fica, como as barras e linhas."),Object(n.b)("h3",{id:"barra"},"Barra"),Object(n.b)("p",null,"No caso dos barramentos, pode-se alterar o ",Object(n.b)("strong",{parentName:"p"},"comprimento da barra")," clicando e arrastando no ",Object(n.b)("em",{parentName:"p"},"pickbox")," exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado."),Object(n.b)("h3",{id:"linha"},"Linha"),Object(n.b)("p",null,"As linhas podem ser inseridas com ",Object(n.b)("strong",{parentName:"p"},'"n\xf3s" (pontos de ancoragem)'),", personalizando a disposi\xe7\xe3o das linhas no diagrama unifilar. Esses n\xf3s s\xe3o inseridos durante o processo de cria\xe7\xe3o do elemento, anteriormente \xe0 sele\xe7\xe3o da segunda barra."),Object(n.b)("p",null,"Ap\xf3s a sua inser\xe7\xe3o, novos n\xf3s podem ser anexados pelo menu de contexto acessado ao clicar com o bot\xe3o direito sobre a linha selecionada. Sua remo\xe7\xe3o tamb\xe9m \xe9 alcan\xe7ada por uma op\xe7\xe3o do menu de contexto."),Object(n.b)("p",null,"Assim como nos barramentos, a altera\xe7\xe3o da posi\xe7\xe3o dos n\xf3s \xe9 obtida clicando e arrastando o ",Object(n.b)("em",{parentName:"p"},"pickbox")," exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado."),Object(n.b)("video",{autoPlay:!0,loop:!0,muted:!0,playsInline:!0,controls:!0},Object(n.b)("source",{src:Object(s.a)("videos/graphicPropsChange.mp4"),type:"video/mp4"}),Object(n.b)("source",{src:Object(s.a)("videos/graphicPropsChange.webm"),type:"video/webm"})),Object(n.b)("h2",{id:"conex\xe3o-e-desconex\xe3o-de-elemetos-na-barra"},"Conex\xe3o e desconex\xe3o de elemetos na barra"),Object(n.b)("p",null,"Os elementos podem ser desconectados e reconectados \xe0s barras ap\xf3s sua inser\xe7\xe3o somente pela ferramenta ",Object(n.b)("em",{parentName:"p"},"drag-and-drop"),". Para isso, o elemento deve ser selecionado e o seu n\xf3 de conex\xe3o deve ser arrastado para o local requerido."),Object(n.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Cuidado!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"O elemento a ser desconectado/reconectado ",Object(n.b)("strong",{parentName:"p"},"deve ser previamente selecionado")," para efetuar a opera\xe7\xe3o. Caso contr\xe1rio a barra ser\xe1 movida ou nenhuma opera\xe7\xe3o ser\xe1 realizada."))),Object(n.b)("p",null,"No caso de desconex\xe3o, o elemento ser\xe1 automaticamente removido da simula\xe7\xe3o, sendo indicado pela sua cor (o elemento passa a ter cor ",Object(n.b)(d,{color:"#7f7f7f",mdxType:"Highlight"},"cinza"),")."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Ao reconectar o elemento ao circuito, deve-se ",Object(n.b)("strong",{parentName:"p"},'atentar para estado do seu "disjuntor"'),", indicado por um quadrado pr\xf3ximo ao seu n\xf3 de conex\xe3o. 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new file mode 100644
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Em contrapartida, caso as vari\xe1veis de estado dos elementos do sistema oscilem ap\xf3s uma perturba\xe7\xe3o e se estabele\xe7am em um novo ponto de opera\xe7\xe3o, mesmo que distinto dos valores iniciais, o sistema \xe9 considerado ",Object(m.b)("strong",{parentName:"p"},"est\xe1vel"),"."),Object(m.b)("h2",{id:"execu\xe7\xe3o-do-estudo-de-estabilidade-no-psp-ufu"},"Execu\xe7\xe3o do estudo de estabilidade no PSP-UFU"),Object(m.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(m.b)("a",{parentName:"p",href:"powerEditor"},"editor de pot\xeancia"),", assim como os ",Object(m.b)("a",{parentName:"p",href:"controlEditor"},"controles")," das ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quinas s\xedncronas"),", a execu\xe7\xe3o da estabilidade \xe9 realizada no ",Object(m.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"menu Simula\xe7\xe3o")," clicando no \xedcone do bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"Executar Estabilidade"),'. Ao clicar na parte inferior do bot\xe3o "Executar Estabilidade" ser\xe1 exibido um menu suspenso com a op\xe7\xe3o ',Object(m.b)("strong",{parentName:"p"},"Lista de eventos de estabilidade"),"."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/menuSimulationStability.svg"),alt:"Menu Simula\xe7\xe3o",title:"Menu Simula\xe7\xe3o"}))),Object(m.b)("p",null,"Ao executar o c\xe1lculo de estabilidade, uma caixa com o status do processo de simula\xe7\xe3o ser\xe1 exibida, indicando primeiramente a inicializa\xe7\xe3o do estudo e posteriormente o tempo de simula\xe7\xe3o calculado."),Object(m.b)("p",null,"Ao clicar na lista de eventos de estabilidade, ser\xe1 exibida uma janela com a descri\xe7\xe3o dos eventos de estabilidade inseridos."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabList.png"),alt:"Lista de eventos de estabilidade",title:"Lista de eventos de estabilidade"}))),Object(m.b)("h3",{id:"inserindo-um-evento-de-estabilidade"},"Inserindo um evento de estabilidade"),Object(m.b)("p",null,"Os dist\xfarbios mais comuns aplicados em estudos de estabilidade transit\xf3ria s\xe3o faltas e chaveamentos. Tais opera\xe7\xf5es s\xe3o facilmente realizadas no PSP-UFU:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Faltas"),": Curtos-circuitos trif\xe1sicos podem ser inseridos nos ",Object(m.b)("a",{parentName:"p",href:"bus"},"barramentos")," do sistema por meio da inclus\xe3o da imped\xe2ncia de falta na matriz admit\xe2ncia de barras da mesma maneira realizada em um elemento ",Object(m.b)("em",{parentName:"p"},"shunt"),". Tal valor \xe9 definido pelo usu\xe1rio e caso seja um curto-circuito franco, um valor n\xe3o nulo, mas suficientemente pr\xf3ximo de zero, \xe9 aplicado, de forma que a tens\xe3o no barramento \xe9 levada a zero durante o dist\xfarbio.")),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Chaveamento de ramo"),": Da mesma forma que as faltas, o chaveamento de ramos \xe9 realizado por meio da altera\xe7\xe3o na matriz admit\xe2ncia, removendo ou inserindo os par\xe2metros do elemento a ser chaveado.\nCada ",Object(m.b)("a",{parentName:"p",href:"powerEditor"},"elemento de pot\xeancia"),' possui um bot\xe3o de "Chaveamento" ou "Estabilidade", em que pode ser inserido os tempos de remo\xe7\xe3o e/ou inser\xe7\xe3o do componente.')),Object(m.b)("li",{parentName:"ul"},Object(m.b)("p",{parentName:"li"},Object(m.b)("strong",{parentName:"p"},"Chaveamento de m\xe1quinas"),": A remo\xe7\xe3o de uma m\xe1quina s\xedncrona \xe9 efetivada com a retirada de sua participa\xe7\xe3o no vetor de correntes, al\xe9m da remo\xe7\xe3o de sua admit\xe2ncia fict\xedcia.\nAssim como os ",Object(m.b)("a",{parentName:"p",href:"powerEditor"},"elementos de pot\xeancia"),", as ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quinas s\xedncronas"),' possui um bot\xe3o "Chaveamento", em que pode ser inserido os tempos de remo\xe7\xe3o e/ou inser\xe7\xe3o do componente.'),Object(m.b)("div",{parentName:"li",className:"admonition admonition-caution alert alert--warning"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Mesmo que removida da barra, os par\xe2metros das ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quinas s\xedncronas")," continuam a ser calculados com a corrente do estator nula, podendo fornecer resultados em uma eventual reconex\xe3o."))))),Object(m.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Dica")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Eventos nos ",Object(m.b)("strong",{parentName:"p"},"sistemas de controle")," podem ser facilmente introduzidos com o bloco de ",Object(m.b)("a",{parentName:"p",href:"mathExpression"},"express\xe3o matem\xe1tica"),". Nesse caso, tais eventos ",Object(m.b)("strong",{parentName:"p"},"n\xe3o")," ser\xe3o exibidos na lista de eventos de estabilidade."))),Object(m.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-estudo-de-estabilidade"},"Erros comuns na execu\xe7\xe3o do estudo de estabilidade"),Object(m.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao estudo de estabilidade."),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-n\xe3o-foi-poss\xedvel-construir-a-matriz-admit\xe2ncia"},'A seguinte mensagem de erro \xe9 exibida: "N\xe3o foi poss\xedvel construir a matriz admit\xe2ncia"'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel construir a matriz admit\xe2ncia de barras. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-erro-ao-inicializar-os-valores-de-satura\xe7\xe3o-do"},'A seguinte mensagem de erro \xe9 exibida: "Erro ao inicializar os valores de satura\xe7\xe3o do..."'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel calcular os fatores de satura\xe7\xe3o da ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quina s\xedncrona"),". As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O fator de satura\xe7\xe3o \xe9 menor que 1,2"),". Esse valor deve ser maior que 1,2, ou ir\xe1 gerar erros na simula\xe7\xe3o. Caso n\xe3o seja informado, a satura\xe7\xe3o da m\xe1quina n\xe3o \xe9 considerada nos c\xe1lculos."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros da m\xe1quina s\xedncrona est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o das m\xe1quinas s\xedncronas sejam inseridos, o c\xe1lculo dos fatores de satura\xe7\xe3o pode divergir. Verifique se os dados foram inseridos corretamente.")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-erro-ao-inicializar-o-avr--regulador-de-velocidade"},'A seguinte mensagem de erro \xe9 exibida: "Erro ao inicializar o AVR / regulador de velocidade..."'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel ",Object(m.b)("a",{parentName:"p",href:"controlEditor#inicializa%C3%A7%C3%A3o-do-sistema-de-controle"},"inicializar o sistema de controle")," de uma ",Object(m.b)("a",{parentName:"p",href:"syncGenerator"},"m\xe1quina s\xedncrona"),". As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O sistema de controle est\xe1 vazio"),". Caso esteja habilitado o AVR e/ou o regulador de velocidade e o controle n\xe3o foi inserido, esse erro pode ser acionado. Insira o controle da m\xe1quina ou desmarque a op\xe7\xe3o de utiliza\xe7\xe3o do AVR e/ou regulador de velocidade."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O sistema de controle n\xe3o possui ao menos uma entrada e uma sa\xedda"),". O sistema de controle deve ter ao menos uma ",Object(m.b)("a",{parentName:"li",href:"io"},"entrada e uma sa\xedda"),", caso contr\xe1rio apresentar\xe1 erro de execu\xe7\xe3o."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O passo de integra\xe7\xe3o est\xe1 muito pequeno"),". Caso o passo de integra\xe7\xe3o esteja muito pequeno, os c\xe1lculos gerar\xe3o erros e ir\xe3o divergir. Reduza o passo de integra\xe7\xe3o nas ",Object(m.b)("a",{parentName:"li",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),".")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-imposs\xedvel-resolver-as-m\xe1quinas-do-sistema"},'A seguinte mensagem de erro \xe9 exibida: "Imposs\xedvel resolver as m\xe1quinas do sistema"'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel resolver as EADs das m\xe1quinas s\xedncronas inseridas no sistemas. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros da m\xe1quina s\xedncrona est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o das m\xe1quinas s\xedncronas sejam inseridos, o c\xe1lculo das EADs pode se tornar imposs\xedvel. Verifique se os dados foram inseridos corretamente."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"O passo de integra\xe7\xe3o est\xe1 muito pequeno"),". Caso o passo de integra\xe7\xe3o esteja muito pequeno, os c\xe1lculos gerar\xe3o erros e ir\xe3o divergir. Reduza o passo de integra\xe7\xe3o nas ",Object(m.b)("a",{parentName:"li",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),"."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-erro-ao-inicializar-o-escorregamento-do-motor"},'A seguinte mensagem de erro \xe9 exibida: "Erro ao inicializar o escorregamento do motor..."'),Object(m.b)("p",null,"Essa mensagem de erro \xe9 exibida quando n\xe3o \xe9 poss\xedvel calcular o valor de escorregamento inicial do motor de indu\xe7\xe3o. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Os par\xe2metros de estabilidade do motor est\xe3o incorretos"),". Caso par\xe2metros muito fora dos valores padr\xe3o dos motores de indu\xe7\xe3o sejam inseridos, o c\xe1lculo do escorregamento pode se tornar imposs\xedvel. Verifique se os dados foram inseridos corretamente.")),Object(m.b)("h2",{id:"estrutura-da-ferramenta-de-estabilidade"},"Estrutura da ferramenta de estabilidade"),Object(m.b)("p",null,"A estabilidade de um SEP \xe9 um problema din\xe2mico e necessita de modelos mais elaborados de elementos de pot\xeancia comparados \xe0queles apresentados nos outros estudos. Esses modelos s\xe3o descritos individualmente, com destaque \xe0s ",Object(m.b)("a",{parentName:"p",href:"syncGenerator#gerador-s%C3%ADncrono-no-estudo-de-estabilidade"},"m\xe1quinas s\xedncronas"),", ",Object(m.b)("a",{parentName:"p",href:"load#carga-no-estudo-de-estabilidade"},"cargas ZIP")," e ",Object(m.b)("a",{parentName:"p",href:"indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-estudo-de-estabilidade"},"motores de indu\xe7\xe3o"),"."),Object(m.b)("p",null,"A representa\xe7\xe3o dos demais componentes do sistema el\xe9trico: ",Object(m.b)("a",{parentName:"p",href:"line"},"linhas de transmiss\xe3o"),", ",Object(m.b)("a",{parentName:"p",href:"transformer"},"transformadores")," e elementos ",Object(m.b)("em",{parentName:"p"},"shunt")," (com exce\xe7\xe3o de cargas ZIP), que formam a rede de transmiss\xe3o ou distribui\xe7\xe3o balanceada, \xe9 realizada utilizando os mesmos modelos do ",Object(m.b)("a",{parentName:"p",href:"powerFlow"},"fluxo de carga"),"."),Object(m.b)("p",null,"Os modelos din\xe2micos para a an\xe1lise no dom\xednio do tempo s\xe3o na forma de um sistema de equa\xe7\xf5es alg\xe9brico-diferenciais (EADs), descritas a seguir:"),Object(m.b)("div",{className:"math 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mathdefault",style:{marginRight:"0.03588em"}},"y"),Object(m.b)("span",{parentName:"span",className:"mclose"},")")))))),Object(m.b)("p",null,"Em que:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("span",{parentName:"li",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"x")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"x")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.43056em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord 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alg\xe9bricas"),Object(m.b)("li",{parentName:"ul"},Object(m.b)("span",{parentName:"li",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"u")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"u")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.43056em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"u"))))),"\ts\xe3o as vari\xe1veis de entrada")),Object(m.b)("p",null,"As vari\xe1veis de entrada s\xe3o inseridas pelo usu\xe1rio e permanecem constantes durante todo o processo de c\xe1lculo. Para o problema de estabilidade, as vari\xe1veis alg\xe9bricas iniciais correspondem \xe0s tens\xf5es e \xe2ngulos das barras calculadas no ",Object(m.b)("a",{parentName:"p",href:"powerFlow"},"fluxo de carga"),". As vari\xe1veis de estado iniciais s\xe3o estimadas ap\xf3s a converg\xeancia do fluxo de pot\xeancia (inicializa\xe7\xe3o dos elementos din\xe2micos)."),Object(m.b)("p",null,"Na sequ\xeancia \xe9 apresentado, na forma de fluxogramas, a estrutura do m\xf3dulo de estabilidade implementado no PSP-UFU. O fluxograma abaixo mostra a estrutura geral da ferramenta de estabilidade."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabGeneral.svg"),alt:"Estrutura geral da ferramenta de estabilidade",title:"Estrutura geral da ferramenta de estabilidade"}))),Object(m.b)("p",null,"A inicializa\xe7\xe3o da rede el\xe9trica e dos elementos din\xe2micos, assim como o precesso iterativo para solu\xe7\xe3o das EADs s\xe3o apresentados nos dois fluxogramas conseguintes."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabInit.svg"),alt:"Inicializa\xe7\xe3o da ferramenta de estabilidade",title:"Inicializa\xe7\xe3o da ferramenta de estabilidade"}))),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(r.a)("images/stabIterative.svg"),alt:"Processo iterativo para c\xe1lculo dos elementos din\xe2micos",title:"Estrutura geral da ferramenta de estabilidade"}))),Object(m.b)("h3",{id:"integra\xe7\xe3o-num\xe9rica"},"Integra\xe7\xe3o num\xe9rica"),Object(m.b)("p",null,"Para resolver as equa\xe7\xf5es diferenciais da m\xe1quina \xe9 necess\xe1rio um m\xe9todo de integra\xe7\xe3o num\xe9rica. Tais m\xe9todos s\xe3o classificados em dois grupos: m\xe9todos expl\xedcitos e m\xe9todos impl\xedcitos. Os m\xe9todos expl\xedcitos, devido \xe0 sua formula\xe7\xe3o, calculam diretamente o estado do sistema em um instante de tempo posterior, enquanto m\xe9todos impl\xedcitos envolvem estados atuais e posteriores em suas equa\xe7\xf5es, exigindo, portanto, um processo iterativo."),Object(m.b)("p",null,"As constantes de tempo presentes no estudo de estabilidade t\xeam uma grande varia\xe7\xe3o em seu valor (podem variar de ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"1"),Object(m.b)("msup",{parentName:"mrow"},Object(m.b)("mn",{parentName:"msup"},"0"),Object(m.b)("mrow",{parentName:"msup"},Object(m.b)("mo",{parentName:"mrow"},"\u2212"),Object(m.b)("mn",{parentName:"mrow"},"3"))),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"s")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"10^{-3}~s")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.8141079999999999em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"1"),Object(m.b)("span",{parentName:"span",className:"mord"},Object(m.b)("span",{parentName:"span",className:"mord"},"0"),Object(m.b)("span",{parentName:"span",className:"msupsub"},Object(m.b)("span",{parentName:"span",className:"vlist-t"},Object(m.b)("span",{parentName:"span",className:"vlist-r"},Object(m.b)("span",{parentName:"span",className:"vlist",style:{height:"0.8141079999999999em"}},Object(m.b)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},Object(m.b)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),Object(m.b)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},Object(m.b)("span",{parentName:"span",className:"mord mtight"},Object(m.b)("span",{parentName:"span",className:"mord mtight"},"\u2212"),Object(m.b)("span",{parentName:"span",className:"mord mtight"},"3"))))))))),Object(m.b)("span",{parentName:"span",className:"mspace nobreak"},"\xa0"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"s")))))," a ",Object(m.b)("span",{parentName:"p",className:"math math-inline"},Object(m.b)("span",{parentName:"span",className:"katex"},Object(m.b)("span",{parentName:"span",className:"katex-mathml"},Object(m.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"10"),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"s")),Object(m.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"10~s")))),Object(m.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(m.b)("span",{parentName:"span",className:"base"},Object(m.b)("span",{parentName:"span",className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}}),Object(m.b)("span",{parentName:"span",className:"mord"},"1"),Object(m.b)("span",{parentName:"span",className:"mord"},"0"),Object(m.b)("span",{parentName:"span",className:"mspace nobreak"},"\xa0"),Object(m.b)("span",{parentName:"span",className:"mord mathdefault"},"s"))))),"). Isso torna o sistema de equa\xe7\xf5es diferenciais da m\xe1quina s\xedncrona como um sistema r\xedgido (",Object(m.b)("em",{parentName:"p"},Object(m.b)("a",{parentName:"em",href:"https://en.wikipedia.org/wiki/Stiff_equation"},"stiff equation")),"). Caso a an\xe1lise da estabilidade num\xe9rica tanto das equa\xe7\xf5es diferenciais r\xedgidas quanto do m\xe9todo de integra\xe7\xe3o obtenham o mesmo comportamento, o m\xe9todo \xe9 chamado de absolutamente est\xe1vel, ou A-est\xe1vel."),Object(m.b)("p",null,"M\xe9todos de integra\xe7\xe3o num\xe9rica expl\xedcitos, como por exemplo o ",Object(m.b)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods"},"Runge-Kutta")," de quarta ordem, n\xe3o podem ser A-est\xe1veis e, portanto, normalmente possuem comportamento ruim em problemas com equa\xe7\xf5es diferenciais r\xedgidas. Por outro lado, m\xe9todos impl\xedcitos podem ser A-est\xe1veis. 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AIEE Transactions, v. 48, n. 3, jul 1929. doi: ",Object(m.b)("a",{parentName:"li",href:"https://doi.org/10.1109/T-AIEE.1929.5055275"},"https://doi.org/10.1109/T-AIEE.1929.5055275")),Object(m.b)("li",{parentName:"ol"},"ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: ",Object(m.b)("a",{parentName:"li",href:"https://doi.org/10.1109/9780470545577"},"https://doi.org/10.1109/9780470545577")),Object(m.b)("li",{parentName:"ol"},"SAUER, P. W.; PAI, M. A. Power System Dynamics and Stability. Pretience Hall, Upper Saddle River, 1998."),Object(m.b)("li",{parentName:"ol"},"KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994."),Object(m.b)("li",{parentName:"ol"},"DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. 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PSP-UFU ",Object(m.b)("strong",{parentName:"p"},"n\xe3o considera")," o ramo magnetizante ou imped\xe2ncias m\xfatuas entre as fases."))),Object(m.b)("p",null,"A figura abaixo mostra o modelo do transformador implementado no PSP-UFU:"),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(p.a)("images/transformerModel.svg"),alt:"Modelo PI da linha",title:"Modelo PI da linha"}))),Object(m.b)("p",null,"Em que:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("span",{parentName:"li",className:"math 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Esse formul\xe1rio, comum a v\xe1rios outros elementos, permite a inser\xe7\xe3o e/ou remo\xe7\xe3o do transformador durante o estudo de ',Object(m.b)("a",{parentName:"p",href:"stability"},"estabilidade"),"."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(p.a)("images/trafoSw.png"),alt:"Formul\xe1rio de chaveamento do transformador",title:"Formul\xe1rio de chaveamento do transformador"}))),Object(m.b)(b.a,{groupId:"transformer-tabs",defaultValue:"general",values:[{label:"Geral",value:"general"},{label:"Falta",value:"fault"},{label:"Bot\xe3o Estabilidade",value:"stability"}],mdxType:"Tabs"},Object(m.b)(r.a,{value:"general",mdxType:"TabItem"},Object(m.b)("h4",{id:"nome"},"Nome"),Object(m.b)("p",null,"Identifica\xe7\xe3o do elemento el\xe9trico. Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(m.b)("a",{parentName:"p",href:"https://pt.wikipedia.org/wiki/Unicode"},"Unicode"),"."),Object(m.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(m.b)("h4",{id:"tens\xe3o-nominal"},"Tens\xe3o nominal"),Object(m.b)("p",null,"Campo de informa\xe7\xe3o ",Object(m.b)("em",{parentName:"p"},"n\xe3o edit\xe1vel")," que apresenta a tens\xe3o prim\xe1ria e secund\xe1ria do transformador. 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O execut\xe1vel est\xe1 presente na pasta "',Object(r.b)("em",{parentName:"p"},"<local de instala\xe7\xe3o>/PSP-UFU/bin/PSP-UFU.exe"),'".'),Object(r.b)("h3",{id:"problemas-recorrentes-no-windows"},"Problemas recorrentes no Windows"),Object(r.b)("ul",null,Object(r.b)("li",{parentName:"ul"},"Ao baixar o navegador pode ",Object(r.b)("strong",{parentName:"li"},"falsamente"),' sinalizar que \xe9 um arquivo perigoso, basta clicar em "Manter arquivo";'),Object(r.b)("li",{parentName:"ul"},'Ao abrir o instalador, novamente o Windows sinaliza que o arquivo \xe9 perigoso, basta clicar em "Executar assim mesmo";'),Object(r.b)("li",{parentName:"ul"},"Ao abrir o programa e alterar o seu idioma, uma mensagem de erro \xe9 exibida. Para resolver esse problema basta executar o PSP-UFU como administrador (clicar com bot\xe3o direito no atalho do programa e posteriormente em executar como administrador). 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alert--danger"},Object(r.b)("div",{parentName:"div",className:"admonition-heading"},Object(r.b)("h5",{parentName:"div"},Object(r.b)("span",{parentName:"h5",className:"admonition-icon"},Object(r.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(r.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Cuidado!")),Object(r.b)("div",{parentName:"div",className:"admonition-content"},Object(r.b)("p",{parentName:"div"},"Utilizando esse m\xe9todo os passos abaixo devem ser executados ",Object(r.b)("strong",{parentName:"p"},"todas as vezes")," que voc\xea 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Sua unidade pode ser selecionada, sendo expressa em V ou em kV."),Object(r.b)("p",null,"A modifica\xe7\xe3o desse par\xe2metro ir\xe1 alterar toda a tens\xe3o do trecho conectado por linhas el\xe9tricas, sendo emitido um alerta ao usu\xe1rio."),Object(r.b)("h4",{id:"tens\xe3o-controlada"},"Tens\xe3o controlada"),Object(r.b)("p",null,"Caracteriza o barramento como ",Object(r.b)("strong",{parentName:"p"},"barra de tens\xe3o controlada (",Object(r.b)("a",{parentName:"strong",href:"powerFlow"},"Barra PV"),")"),","),Object(r.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(r.b)("div",{parentName:"div",className:"admonition-heading"},Object(r.b)("h5",{parentName:"div"},Object(r.b)("span",{parentName:"h5",className:"admonition-icon"},Object(r.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(r.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(r.b)("div",{parentName:"div",className:"admonition-content"},Object(r.b)("p",{parentName:"div"},"Essa op\xe7\xe3o \xe9 somente v\xe1lida caso alguma m\xe1quina s\xedncrona esteja conectado, caso contr\xe1rio esse valor ser\xe1 ignorado. 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Como qualquer falta assim\xe9trica provoca fluxo de corrente desequilibrada \xe9 necess\xe1rio empregar o m\xe9todo das componentes sim\xe9tricas. Esse m\xe9todo permite o estudo de sistemas balanceados em conjunto cargas desbalanceadas."),Object(m.b)("h2",{id:"execu\xe7\xe3o-do-c\xe1lculo-de-curto-circuito-no-psp-ufu"},"Execu\xe7\xe3o do c\xe1lculo de curto-circuito no PSP-UFU"),Object(m.b)("p",null,"Existem duas formas de se calcular o curto-circuito no PSP-UFU:"),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Falta"),": Calcula a falta inserida nas ",Object(m.b)("a",{parentName:"li",href:"bus"},"barras"),". Nesse tipo de c\xe1lculo \xe9 poss\xedvel calcular faltas ",Object(m.b)("em",{parentName:"li"},"shunt")," nos ",Object(m.b)("a",{parentName:"li",href:"bus"},"barramentos")," balanceadas e desbalanceadas."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"N\xedvel de curto-circuito"),": Calcula o n\xedvel de curto-circuito (falta trif\xe1sica) em todos ",Object(m.b)("a",{parentName:"li",href:"bus"},"barramentos")," do sistema.")),Object(m.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(m.b)("a",{parentName:"p",href:"powerEditor"},"editor de pot\xeancia"),", a execu\xe7\xe3o do c\xe1lculo de curto-circuito \xe9 realizada no ",Object(m.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"menu Simula\xe7\xe3o")," clicando no bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"Falta"),". 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Com essa op\xe7\xe3o, os c\xe1lculos est\xe1ticos selecionados nas ",Object(m.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o")," s\xe3o automaticamente realizados ao modificar quaisquer par\xe2metros da rede, como dados el\xe9tricos e acionamento dos disjuntores dos elementos (remo\xe7\xe3o ou inser\xe7\xe3o)."),Object(m.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(m.b)("div",{parentName:"div",className:"admonition-heading"},Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",{parentName:"h5",className:"admonition-icon"},Object(m.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(m.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Cuidado!")),Object(m.b)("div",{parentName:"div",className:"admonition-content"},Object(m.b)("p",{parentName:"div"},"Os c\xe1lculos de curtos-circuitos n\xe3o s\xe3o habilitados por padr\xe3o no c\xe1lculo cont\xednuo e devem ser inseridos nas ",Object(m.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),"."))),Object(m.b)("p",null,"Os resultados do c\xe1lculo de curto-circuito s\xe3o exibidos nos ",Object(m.b)("a",{parentName:"p",href:"text"},"elementos de texto vinculado"),", ao posicionar o mouse sobre os barramentos e em ",Object(m.b)("a",{parentName:"p",href:"tabularReport"},"relat\xf3rios tabulares"),"."),Object(m.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-c\xe1lculo-de-curto-circuito"},"Erros comuns na execu\xe7\xe3o do c\xe1lculo de curto-circuito"),Object(m.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao calculo de curto-circuito."),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-falha-ao-inverter-a-matriz-admit\xe2ncia-de-sequ\xeancia-zero"},'A seguinte mensagem de erro \xe9 exibida: "Falha ao inverter a matriz admit\xe2ncia de sequ\xeancia zero"'),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Impossibilidade de circula\xe7\xe3o da corrente de sequ\xeancia zero"),". Caso o gerador n\xe3o seja aterrado, n\xe3o circular\xe1 corrente de sequ\xeancia zero por ele. Nesse caso, dependendo da conex\xe3o do transformador pr\xf3ximo ao gerador sem aterramento, a matriz admit\xe2ncia de sequ\xeancia zero \xe9 singular. 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Como qualquer falta assim\xe9trica provoca fluxo de corrente desequilibrada \xe9 necess\xe1rio empregar o m\xe9todo das componentes sim\xe9tricas. Esse m\xe9todo permite o estudo de sistemas balanceados com cargas desbalanceadas."),Object(m.b)("h3",{id:"m\xe9todo-das-componentes-sim\xe9tricas"},"M\xe9todo das componentes sim\xe9tricas"),Object(m.b)("p",null,"Esse m\xe9todo proposto por C. L. Fortescue, permite definir um sistema de n fasores desbalanceados em n \u2013 1 sistemas de n fases balanceados e um sistema de fase zero. O sistema de fase zero \xe9 definido por todas as fases de mesmo m\xf3dulo e \xe2ngulo.\nPara um sistema trif\xe1sico pode-se definir tr\xeas componentes de sequ\xeancia:"),Object(m.b)("ol",null,Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia positiva, constituindo em tr\xeas fasores iguais em m\xf3dulo, 120\xba defasados entre si, e tendo a mesma sequ\xeancia de fase que os fasores originais;"),Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia negativa, constituindo em tr\xeas fasores iguais em m\xf3dulo, 120\xba defasados entre si, e tendo a sequ\xeancia de fase oposta \xe0 dos fasores originais."),Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia zero, constituindo em tr\xeas fasores iguais em m\xf3dulo e com defasagem nula entre si.")),Object(m.b)("p",null,"Com isso pode-se decompor as tens\xf5es de fase em componentes sim\xe9tricas pelas seguintes equa\xe7\xf5es:"),Object(m.b)("div",{className:"math 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Seu comportamento difere de acordo com o tipo de barra conectada:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("a",{parentName:"li",href:"powerFlow"},"Barra de refer\xeancia"),": Os dados de pot\xeancia ativa e reativa inseridos s\xe3o desprezados, uma vez que esse elemento ser\xe1 utilizado para completar o balan\xe7o de pot\xeancia do estudo de fluxo d carga;"),Object(n.b)("li",{parentName:"ul"},Object(n.b)("a",{parentName:"li",href:"powerFlow"},"Barra PV"),": O dado de pot\xeancia ativa \xe9 considerado, por\xe9m o dados de pot\xeancia reativa s\xe3o desprezados. O valor da pot\xeancia reativa \xe9 utilizado para manter o m\xf3dulo da tens\xe3o constante no barramento conectado;"),Object(n.b)("li",{parentName:"ul"},Object(n.b)("a",{parentName:"li",href:"powerFlow"},"Barra PQ"),": Para os geradores conectados a essa barra tanto a pot\xeancia ativa quanto reativa inseridas s\xe3o consideradas.")),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Caso o limite de pot\xeancia reativa seja excedido, o programa automaticamente transforma a ",Object(n.b)("a",{parentName:"p",href:"powerFlow"},"Barra PV")," conectada em uma ",Object(n.b)("a",{parentName:"p",href:"powerFlow"},"Barra PQ"),", utilizando o valor limite de pot\xeancia reativa que seria ultrapassado."))),Object(n.b)("h3",{id:"gerador-s\xedncrono-no-estudo-de-curto-circuito"},"Gerador S\xedncrono no estudo de curto-circuito"),Object(n.b)("p",null,"Enquanto os geradores no estudo de fluxo de carga s\xe3o modelados somente por correntes injetadas nas barras, para o curto-circuito utiliza-se uma tens\xe3o atr\xe1s de uma imped\xe2ncia. 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Tal \xe9 utilizado na representa\xe7\xe3o de um ",Object(n.b)("strong",{parentName:"p"},"barramento infinito"),", o qual \xe9 normalmente constitu\xeddo de um subsistema muito maior \xe0quele simulado."),Object(n.b)("p",null,"No PSP-UFU a ",Object(n.b)("strong",{parentName:"p"},"representa\xe7\xe3o de um barramento infinito")," pode ser obtido por meio da utiliza\xe7\xe3o de uma m\xe1quina representada pelo Modelo 1 cujo valor da constante de in\xe9rcia (H) \xe9 infinito ou muito grande (",Object(n.b)("span",{parentName:"p",className:"math 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serem analisados. Na implementa\xe7\xe3o realizada no programa a utiliza\xe7\xe3o ou n\xe3o desse recurso \xe9 opcional e pode ser ",Object(n.b)("a",{parentName:"p",href:"simulationConfig"},"definida pelo usu\xe1rio"),"."),Object(n.b)("h2",{id:"formul\xe1rio-de-edi\xe7\xe3o-dos-geradores-s\xedncronos"},"Formul\xe1rio de edi\xe7\xe3o dos geradores s\xedncronos"),Object(n.b)("p",null,"A imagem abaixo apresenta o formul\xe1rio de inser\xe7\xe3o/altera\xe7\xe3o de dados dos geradores s\xedncronos:"),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(p.a)("images/syncGeneratorForm.png"),alt:"Formul\xe1rio dos geradores s\xedncronos no PSP-UFU",title:"Formul\xe1rio dos geradores s\xedncronos no PSP-UFU"}))),Object(n.b)("p",null,'Um segundo pelos dados de estabilidade, como mostra a figura abaixo, acessado ao clicar no bot\xe3o "Estabilidade" do formul\xe1rio principal. Nele \xe9 poss\xedvel tamb\xe9m acessar aos controles das m\xe1quinas s\xedncronas manipulados pelo ',Object(n.b)("a",{parentName:"p",href:"controlEditor"},"editor de controle"),"."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(p.a)("images/syncGeneratorStabForm.png"),alt:"Formul\xe1rio de estabilidade dos geradores s\xedncronos no PSP-UFU",title:"Formul\xe1rio de estabilidade dos geradores s\xedncronos no PSP-UFU"}))),Object(n.b)("p",null,'No formul\xe1rio de estabilidade pode ser observado o bot\xe3o "Chaveamento" na parte inferior esquerda do formul\xe1rio. Esse formul\xe1rio, comum a v\xe1rios outros elementos, permite a inser\xe7\xe3o e/ou remo\xe7\xe3o do gerador durante o estudo de ',Object(n.b)("a",{parentName:"p",href:"stability"},"estabilidade"),"."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(p.a)("images/syncGeneratorSw.png"),alt:"Formul\xe1rio de chaveamento do gerador s\xedncrono",title:"Formul\xe1rio de chaveamento do gerador s\xedncrono"}))),Object(n.b)(b.a,{groupId:"syncGenerator-tabs",defaultValue:"general",values:[{label:"Geral",value:"general"},{label:"Falta",value:"fault"},{label:"Bot\xe3o Estabilidade",value:"stability"},{label:"Bot\xe3o Chaveamento",value:"switching"}],mdxType:"Tabs"},Object(n.b)(r.a,{value:"general",mdxType:"TabItem"},Object(n.b)("h4",{id:"nome"},"Nome"),Object(n.b)("p",null,"Identifica\xe7\xe3o do elemento el\xe9trico. Podem ser inseridos quaisquer n\xfameros de caracteres no padr\xe3o ",Object(n.b)("a",{parentName:"p",href:"https://pt.wikipedia.org/wiki/Unicode"},"Unicode"),"."),Object(n.b)("p",null,"Todos os componentes de pot\xeancia do PSP-UFU possuem esse campo."),Object(n.b)("h4",{id:"pot\xeancia-nominal"},"Pot\xeancia nominal"),Object(n.b)("p",null,"Pot\xeancia nominal do gerador, inserida em MVA, kVA ou VA."),Object(n.b)("p",null,'Esse campo \xe9 especialmente importante caso a op\xe7\xe3o "Utilizar a pot\xeancia nominal como base" esteja marcada.'),Object(n.b)("h4",{id:"pot\xeancias-ativa-e-reativa"},"Pot\xeancias ativa e reativa"),Object(n.b)("p",null,"Pot\xeancias ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do gerador."),Object(n.b)("p",null,"Caso a barra conectada seja PV o valor de pot\xeancia reativa ser\xe1 ignorado e caso seja de refer\xeancia ambos os valores inseridos ser\xe3o desprezados."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Caso mais de um gerador esteja conectado na mesma barra, os valores de pot\xeancia reativa (nas barras de refer\xeancia e PV) e ativa (nas barras de refer\xeancia) s\xe3o igualmente distribu\xeddas, respeitando os limites individuais de pot\xeancia reativa."))),Object(n.b)("h4",{id:"pot\xeancias-reativas-m\xe1xima-e-m\xednima"},"Pot\xeancias reativas m\xe1xima e m\xednima"),Object(n.b)("p",null,"Limites de pot\xeancia reativa m\xe1xima e m\xednima do gerador para controle de tens\xe3o em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade ser\xe1 limitado ao valor inserido e a barra conectada ser\xe1 transformada em PQ, n\xe3o controlando a tens\xe3o estabelecida."),Object(n.b)("h4",{id:"utilizar-pot\xeancia-nominal-como-base"},"Utilizar pot\xeancia nominal como base"),Object(n.b)("p",null,"Caso essa op\xe7\xe3o seja marcada, o programa ir\xe1 utilizar a pot\xeancia nominal do gerador como base para a convers\xe3o das unidades, inclusive aqueles no formul\xe1rio de estabilidade, caso contr\xe1rio ser\xe1 usada a ",Object(n.b)("a",{parentName:"p",href:"simulationConfig"},"pot\xeancia base do sistema"),".")),Object(n.b)(r.a,{value:"fault",mdxType:"TabItem"},Object(n.b)("h4",{id:"imped\xe2ncias-de-sequ\xeancia"},"Imped\xe2ncias de sequ\xeancia"),Object(n.b)("p",null,"Valores de resist\xeancia e reat\xe2ncia para c\xe1lculo das correntes de falta. 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mathdefault"},"p"),Object(n.b)("span",{parentName:"span",className:"mord"},"."),Object(n.b)("span",{parentName:"span",className:"mord mathdefault"},"u"),Object(n.b)("span",{parentName:"span",className:"mord"},"."))))),") de corrente de campo necess\xe1ria para atingir 1,2 p.u. de tens\xe3o terminal."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 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Ambas as op\xe7\xf5es ir\xe3o acessar o ',Object(n.b)("a",{parentName:"p",href:"controlEditor"},"editor de controles"),"."),Object(n.b)("p",null,"O acesso aos controles do ",Object(n.b)("strong",{parentName:"p"},"AVR"),' poder\xe3o ent\xe3o ser criados e manipulados ao clicar no bot\xe3o "Editar AVR", assim como o ',Object(n.b)("strong",{parentName:"p"},"Regulador de Velocidade"),' \xe9 acessado no bot\xe3o "Editar regulador de velocidade".'),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"No PSP-UFU a op\xe7\xe3o de editar o ",Object(n.b)("strong",{parentName:"p"},"AVR")," engloba mais que somente o controle de tens\xe3o da m\xe1quina. 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Power System Stability and Control. McGraw-Hill, New York, 1994."),Object(n.b)("li",{parentName:"ol"},"DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1109/TPAS.1972.293341"},"https://doi.org/10.1109/TPAS.1972.293341")),Object(n.b)("li",{parentName:"ol"},"IEEE Std 1110-2002 IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses. IEEE, New York, nov. 2003. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1109/IEEESTD.2003.94408"},"https://doi.org/10.1109/IEEESTD.2003.94408")),Object(n.b)("li",{parentName:"ol"},"KIMBARK, E. W. Power System Stability: Volume III \u2013 Synchronous Machine. 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Essa barra \xe9 necessariamente geradora, uma vez que ela \xe9 respons\xe1vel pelo equil\xedbrio do balan\xe7o de pot\xeancia do sistema."))),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"O tipo de barra deve ser definido no ",Object(n.b)("a",{parentName:"p",href:"bus"},"elemento barramento"),"."),Object(n.b)("p",{parentName:"div"},"Note que ",Object(n.b)("strong",{parentName:"p"},"o sistema deve possuir somente uma barra de refer\xeancia"),"."))),Object(n.b)("p",null,"Como mencionado anteriormente, as equa\xe7\xf5es s\xe3o n\xe3o-lineares e a solu\xe7\xe3o anal\xedtica n\xe3o \xe9 pr\xe1tica. As solu\xe7\xf5es dessas equa\xe7\xf5es seguem ",Object(n.b)("strong",{parentName:"p"},Object(n.b)("a",{parentName:"strong",href:"https://en.wikipedia.org/wiki/Iterative_method"},"processos iterativos")),", em que s\xe3o atribu\xeddos valores estimados (ou iniciais) para as barras com tens\xf5es desconhecidas e, baseado na pot\xeancia ativa e reativa e m\xf3dulo da tens\xe3o especificados, calcula-se por meio das equa\xe7\xf5es previamente apresentadas as novas tens\xf5es complexas em cada n\xf3 do sistema."),Object(n.b)("p",null,"Na sequ\xeancia, esse conjunto de valores para as tens\xf5es em cada barra \xe9 utilizado para novamente calcular outro grupo de tens\xf5es. Cada c\xe1lculo de um novo conjunto de tens\xf5es \xe9 chamado itera\xe7\xe3o. O processo iterativo \xe9 repetido at\xe9 que as mudan\xe7as em todas as barras sejam menores do que um valor pr\xe9-estipulado, obtendo assim a converg\xeancia."),Object(n.b)("h2",{id:"execu\xe7\xe3o-do-fluxo-de-carga-no-psp-ufu"},"Execu\xe7\xe3o do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(n.b)("a",{parentName:"p",href:"powerEditor"},"editor de pot\xeancia"),", a execu\xe7\xe3o do fluxo de carga \xe9 realizada no ",Object(n.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"menu Simula\xe7\xe3o")," clicando no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Fluxo de carga"),"."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(p.a)("images/menuSimulationPF.svg"),alt:"Menu Simula\xe7\xe3o",title:"Menu Simula\xe7\xe3o"}))),Object(n.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Dica")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Caso o fluxo de carga tenha sido executado com sucesso, as setas de pot\xeancia ser\xe3o exibidas, a barra de status indicar\xe1 sucesso na opera\xe7\xe3o e os ",Object(n.b)("a",{parentName:"p",href:"text"},"elementos de texto")," ser\xe3o atualizados."))),Object(n.b)("p",null,Object(n.b)("strong",{parentName:"p"},"Outra possibilidade")," \xe9 a execu\xe7\xe3o por meio do c\xe1lculo cont\xednuo, tamb\xe9m presente no ",Object(n.b)("a",{parentName:"p",href:"mainScreen#menu-ribbon"},"menu Simula\xe7\xe3o")," e seu acionamento \xe9 realizado co clicar no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Habilitar solu\xe7\xe3o"),". Com essa op\xe7\xe3o, os c\xe1lculos est\xe1ticos selecionados nas ",Object(n.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o")," s\xe3o automaticamente realizados ao modificar quaisquer par\xe2metros da rede, como dados el\xe9tricos e acionamento dos disjuntores dos elementos (remo\xe7\xe3o ou inser\xe7\xe3o)."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",{parentName:"div",className:"admonition-heading"},Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",{parentName:"h5",className:"admonition-icon"},Object(n.b)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"},Object(n.b)("path",{parentName:"svg",fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"}))),"Aten\xe7\xe3o!")),Object(n.b)("div",{parentName:"div",className:"admonition-content"},Object(n.b)("p",{parentName:"div"},"Evite construir o circuito com o c\xe1lculo cont\xednuo habilitado, uma vez que configura\xe7\xf5es tempor\xe1rias podem levar a erros de execu\xe7\xe3o da simula\xe7\xe3o."),Object(n.b)("p",{parentName:"div"},"Para desabilitar o c\xe1lculo cont\xednuo clique no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Desabilitar solu\xe7\xe3o"),"."))),Object(n.b)("p",null,"Os resultados do fluxo de carga s\xe3o exibidos nos ",Object(n.b)("a",{parentName:"p",href:"text"},"elementos de texto vinculado"),", ao posicionar o mouse sobre os elementos e em ",Object(n.b)("a",{parentName:"p",href:"tabularReport"},"relat\xf3rios tabulares"),"."),Object(n.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-fluxo-de-carga"},"Erros comuns na execu\xe7\xe3o do fluxo de carga"),Object(n.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao fluxo de carga."),Object(n.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-o-n\xfamero-m\xe1ximo-de-itera\xe7\xf5es-foi-alcan\xe7ado"},'A seguinte mensagem de erro \xe9 exibida: "O n\xfamero m\xe1ximo de itera\xe7\xf5es foi alcan\xe7ado"'),Object(n.b)("p",null,"Essa mensagem de erro \xe9 exibida quando o m\xe9todo de solu\xe7\xe3o num\xe9rica selecionado nas ",Object(n.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o")," atinge o n\xfamero m\xe1ximo de itera\xe7\xf5es inserido. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"O n\xfamero m\xe1ximo de itera\xe7\xf5es est\xe1 muito baixo"),". Alguns circuitos exigem um n\xfamero maior de itera\xe7\xf5es, portanto altere o valor do m\xe1ximo de itera\xe7\xf5es nas ",Object(n.b)("a",{parentName:"li",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),". Tamb\xe9m tente alternar entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros de simula\xe7\xe3o est\xe3o inadequados"),". Caso um par\xe2metro do m\xe9todo de solu\xe7\xe3o esteja inadequado, como fator de acelera\xe7\xe3o ou toler\xe2ncia, o c\xe1lculo pode n\xe3o alcan\xe7ar a converg\xeancia. Altere esses par\xe2metros nas ",Object(n.b)("a",{parentName:"li",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),".")),Object(n.b)("h4",{id:"os-dados-de-sa\xedda-s\xe3o-exibidos-como-nan-ou-nan"},'Os dados de sa\xedda s\xe3o exibidos como "NaN" ou "nan"'),Object(n.b)("p",null,'Isso ocorre devido a erros de opera\xe7\xf5es matem\xe1ticas nos c\xe1lculos de fluxo de carga. "NaN" significa ',Object(n.b)("em",{parentName:"p"},"Not a Number"),"."),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(n.b)("h2",{id:"m\xe9todos-de-solu\xe7\xe3o-num\xe9rica-do-fluxo-de-carga-no-psp-ufu"},"M\xe9todos de solu\xe7\xe3o num\xe9rica do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Os m\xe9todos implementados no programa para solu\xe7\xe3o do problema de fluxo de carga no PSP-UFU s\xe3o ",Object(n.b)("strong",{parentName:"p"},"Gauss-Seidel (GS)")," e ",Object(n.b)("strong",{parentName:"p"},"Newton-Raphson (NR)"),". Al\xe9m desses m\xe9todos cl\xe1ssicos, um m\xe9todo h\xedbrido pode ser utilizado (definido nas ",Object(n.b)("a",{parentName:"p",href:"simulationConfig"},"configura\xe7\xf5es de simula\xe7\xe3o"),"), em que \xe9 utilizado inicialmente o GS e na sequ\xeancia o NR, aumentando a chance de converg\xeancia do NR."),Object(n.b)("h3",{id:"gauss-seidel"},"Gauss-Seidel"),Object(n.b)("p",null,"O m\xe9todo de Gauss-Seidel tem sido bastante utilizado nas \xfaltimas d\xe9cadas para solu\xe7\xe3o do problema de fluxo de carga, uma vez que n\xe3o h\xe1 a necessidade de fatorar a matrizes, reduzindo o esfor\xe7o computacional. Atualmente, restri\xe7\xf5es computacionais s\xe3o menos problem\xe1ticas e outros m\xe9todos s\xe3o normalmente escolhidos, por\xe9m o Gauss-Seidel ainda possui valor did\xe1tico e, visto que o PSP-UFU tamb\xe9m possui fins educacionais, optou-se pela implementa\xe7\xe3o desse m\xe9todo."),Object(n.b)("p",null,"Para iniciar as itera\xe7\xf5es do m\xe9todo s\xe3o necess\xe1rios valores iniciais para as tens\xf5es (",Object(n.b)("span",{parentName:"p",className:"math math-inline"},Object(n.b)("span",{parentName:"span",className:"katex"},Object(n.b)("span",{parentName:"span",className:"katex-mathml"},Object(n.b)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mover",{parentName:"msubsup",accent:"true"},Object(n.b)("mi",{parentName:"mover"},"V"),Object(n.b)("mo",{parentName:"mover"},"\u02d9")),Object(n.b)("mi",{parentName:"msubsup"},"i"),Object(n.b)("mn",{parentName:"msubsup"},"0"))),Object(n.b)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\dot{V}_i^0")))),Object(n.b)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},Object(n.b)("span",{parentName:"span",className:"base"},Object(n.b)("span",{parentName:"span",className:"strut",style:{height:"1.178854em",verticalAlign:"-0.258664em"}}),Object(n.b)("span",{parentName:"span",className:"mord"},Object(n.b)("span",{parentName:"span",className:"mord accent"},Object(n.b)("span",{parentName:"span",className:"vlist-t"},Object(n.b)("span",{parentName:"span",className:"vlist-r"},Object(n.b)("span",{parentName:"span",className:"vlist",style:{height:"0.9201900000000001em"}},Object(n.b)("span",{parentName:"span",style:{top:"-3em"}},Object(n.b)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),Object(n.b)("span",{parentName:"span",className:"mord"},Object(n.b)("span",{parentName:"span",className:"mord mathdefault",style:{marginRight:"0.22222em"}},"V"))),Object(n.b)("span",{parentName:"span",style:{top:"-3.25233em"}},Object(n.b)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),Object(n.b)("span",{parentName:"span",className:"accent-body",style:{left:"-0.13889em"}},Object(n.b)("span",{parentName:"span",className:"mord"},"\u02d9"))))))),Object(n.b)("span",{parentName:"span",className:"msupsub"},Object(n.b)("span",{parentName:"span",className:"vlist-t vlist-t2"},Object(n.b)("span",{parentName:"span",className:"vlist-r"},Object(n.b)("span",{parentName:"span",className:"vlist",style:{height:"0.8141079999999999em"}},Object(n.b)("span",{parentName:"span",style:{top:"-2.441336em",marginRight:"0.05em"}},Object(n.b)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),Object(n.b)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},Object(n.b)("span",{parentName:"span",className:"mord mathdefault mtight"},"i"))),Object(n.b)("span",{parentName:"span",style:{top:"-3.063em",marginRight:"0.05em"}},Object(n.b)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),Object(n.b)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},Object(n.b)("span",{parentName:"span",className:"mord mtight"},"0")))),Object(n.b)("span",{parentName:"span",className:"vlist-s"},"\u200b")),Object(n.b)("span",{parentName:"span",className:"vlist-r"},Object(n.b)("span",{parentName:"span",className:"vlist",style:{height:"0.258664em"}},Object(n.b)("span",{parentName:"span"})))))))))),") que devem ser calculadas. 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No PSP-UFU foi implementada a \xfaltima estrat\xe9gia, uma vez que separa os conceitos de c\xe1lculo e de verifica\xe7\xe3o de limites, tornando mais f\xe1cil o desenvolvimento de novos m\xe9todos num\xe9ricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situa\xe7\xe3o n\xe3o violadora o c\xe1lculo iterativo deve ser retomado at\xe9 que obtenha novamente a converg\xeancia."),Object(n.b)("h2",{id:"refer\xeancias"},"Refer\xeancias"),Object(n.b)("ol",null,Object(n.b)("li",{parentName:"ol"},"MONTICELLI, A. J. Fluxo de Carga em Redes de Energia El\xe9trica. S\xe3o Paulo: Edgar Bl\xfccher, 1983."),Object(n.b)("li",{parentName:"ol"},"STEVENSON JR.; WILLIAN, D. Elementos de An\xe1lise de Sistemas de Pot\xeancia. 2\xaa ed. S\xe3o Paulo: McGraw-Hill, 1986."),Object(n.b)("li",{parentName:"ol"},"MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1007/978-3-642-13669-6"},"https://doi.org/10.1007/978-3-642-13669-6")),Object(n.b)("li",{parentName:"ol"},"ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: ",Object(n.b)("a",{parentName:"li",href:"https://doi.org/10.1002/9781118878286"},"https://doi.org/10.1002/9781118878286")),Object(n.b)("li",{parentName:"ol"},"TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton\u2019s Method. IEEE Transaction on Power Apparatus and Systems, v. 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Ba(e),null;case 13:return fo(Ua),4096&(t=e.effectTag)?(e.effectTag=-4097&t|64,e):null;case 19:return fo(Ua),null;case 4:return ja(),null;case 10:return aa(e),null;default:return null}}function nl(e,t){return{value:e,source:t,stack:be(t)}}qi=function(e,t){for(var n=t.child;null!==n;){if(5===n.tag||6===n.tag)e.appendChild(n.stateNode);else if(4!==n.tag&&null!==n.child){n.child.return=n,n=n.child;continue}if(n===t)break;for(;null===n.sibling;){if(null===n.return||n.return===t)return;n=n.return}n.sibling.return=n.return,n=n.sibling}},Hi=function(e,t,n,r,a){var i=e.memoizedProps;if(i!==r){var l,u,s=t.stateNode;switch(Da(Na.current),e=null,n){case"input":i=ke(s,i),r=ke(s,r),e=[];break;case"option":i=Oe(s,i),r=Oe(s,r),e=[];break;case"select":i=o({},i,{value:void 0}),r=o({},r,{value:void 0}),e=[];break;case"textarea":i=_e(s,i),r=_e(s,r),e=[];break;default:"function"!=typeof i.onClick&&"function"==typeof r.onClick&&(s.onclick=cn)}for(l in 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diff --git a/docs/pt/assets/js/main.168dd2d0.js.LICENSE.txt b/docs/pt/assets/js/main.168dd2d0.js.LICENSE.txt
new file mode 100644
index 0000000..4a630d7
--- /dev/null
+++ b/docs/pt/assets/js/main.168dd2d0.js.LICENSE.txt
@@ -0,0 +1,44 @@
+/*
+object-assign
+(c) Sindre Sorhus
+@license MIT
+*/
+
+/* NProgress, (c) 2013, 2014 Rico Sta. Cruz - http://ricostacruz.com/nprogress
+ * @license MIT */
+
+/** @license React v0.19.1
+ * scheduler.production.min.js
+ *
+ * Copyright (c) Facebook, Inc. and its affiliates.
+ *
+ * This source code is licensed under the MIT license found in the
+ * LICENSE file in the root directory of this source tree.
+ */
+
+/** @license React v16.13.1
+ * react-dom.production.min.js
+ *
+ * Copyright (c) Facebook, Inc. and its affiliates.
+ *
+ * This source code is licensed under the MIT license found in the
+ * LICENSE file in the root directory of this source tree.
+ */
+
+/** @license React v16.13.1
+ * react-is.production.min.js
+ *
+ * Copyright (c) Facebook, Inc. and its affiliates.
+ *
+ * This source code is licensed under the MIT license found in the
+ * LICENSE file in the root directory of this source tree.
+ */
+
+/** @license React v16.13.1
+ * react.production.min.js
+ *
+ * Copyright (c) Facebook, Inc. and its affiliates.
+ *
+ * This source code is licensed under the MIT license found in the
+ * LICENSE file in the root directory of this source tree.
+ */
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properly);</li><li>✔️ Added Visual Studio solution to the GitHub project;</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2020-06-13-2020w24a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="margin-vert--xl"><nav class="pagination-nav" aria-label="Blog post page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" 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bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2020-07-09-2020w28a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="margin-vert--xl"><nav class="pagination-nav" aria-label="Blog post page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/blog/2020/08/01/2020w31a-beta"><div class="pagination-nav__sublabel">Newer Post</div><div class="pagination-nav__label">« 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class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><article><header><h1 class="margin-bottom--sm blogPostTitle_GeHD">2020w31a-beta</h1><div class="margin-vert--md"><time datetime="2020-08-01T00:00:00.000Z" class="blogPostDate_fNvV">31 de julho de 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>);</li><li>✔️ New fancy <a href="https://thales1330.github.io/PSP/" target="_blank" rel="noopener noreferrer">PSP-UFU website</a> and <a href="https://thales1330.github.io/PSP/blog" target="_blank" rel="noopener noreferrer">changelog</a> using <a href="https://v2.docusaurus.io/" target="_blank" rel="noopener noreferrer">docusaurus</a>;</li><li>✔️ Text element performance improved for OpenGL render;</li><li>🐛 Several major and minor bug fixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2020-08-01-2020w31a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="margin-vert--xl"><nav class="pagination-nav" aria-label="Blog post page navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/blog/2020/07/09/2020w28a-beta"><div class="pagination-nav__sublabel">Older Post</div><div class="pagination-nav__label">2020w28a-beta »</div></a></div></nav></div></main><div class="col col--2"><div class="tableOfContents_35-E thin-scrollbar"></div></div></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 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class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. Construído com Docusaurus.</div></div></div></footer></div>
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+<?xml version="1.0" encoding="utf-8"?>
+<feed xmlns="http://www.w3.org/2005/Atom">
+    <id>https://thales1330.github.io/blog</id>
+    <title>PSP-UFU Blog</title>
+    <updated>2020-08-01T00:00:00.000Z</updated>
+    <generator>https://github.com/jpmonette/feed</generator>
+    <link rel="alternate" href="https://thales1330.github.io/blog"/>
+    <subtitle>PSP-UFU Blog</subtitle>
+    <icon>https://thales1330.github.io/img/favicon.ico</icon>
+    <entry>
+        <title type="html"><![CDATA[2020w31a-beta]]></title>
+        <id>2020w31a-beta</id>
+        <link href="https://thales1330.github.io/PSP/pt/blog/2020/08/01/2020w31a-beta"/>
+        <updated>2020-08-01T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- 🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on GitHub or Twitter);]]></summary>
+    </entry>
+    <entry>
+        <title type="html"><![CDATA[2020w28a-beta]]></title>
+        <id>2020w28a-beta</id>
+        <link href="https://thales1330.github.io/PSP/pt/blog/2020/07/09/2020w28a-beta"/>
+        <updated>2020-07-09T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- 🔥 Voltage heatmap implemented (OpenGL enabled);]]></summary>
+    </entry>
+    <entry>
+        <title type="html"><![CDATA[2020w24a-beta]]></title>
+        <id>2020w24a-beta</id>
+        <link href="https://thales1330.github.io/PSP/pt/blog/2020/06/13/2020w24a-beta"/>
+        <updated>2020-06-13T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- ✔️ Device context render implemented (to avoid use OpenGL - some machines can't use it properly);]]></summary>
+    </entry>
+    <entry>
+        <title type="html"><![CDATA[22019w44a-beta]]></title>
+        <id>22019w44a-beta</id>
+        <link href="https://thales1330.github.io/PSP/pt/blog/2019/11/01/2019w44a-beta"/>
+        <updated>2019-11-01T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- ✔️ Minor changes to 32bit MSW version for compatibility;]]></summary>
+    </entry>
+    <entry>
+        <title type="html"><![CDATA[2019w42a-beta]]></title>
+        <id>2019w42a-beta</id>
+        <link href="https://thales1330.github.io/PSP/pt/blog/2019/10/13/2019w42a-beta"/>
+        <updated>2019-10-13T00:00:00.000Z</updated>
+        <summary type="html"><![CDATA[- 🔥 Harmonic study implemented;]]></summary>
+    </entry>
+</feed>
\ No newline at end of file
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new file mode 100644
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+<meta name="generator" content="Docusaurus v2.0.0-alpha.71">
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+<link rel="alternate" type="application/atom+xml" href="/PSP/pt/blog/atom.xml" title="PSP-UFU Blog Atom Feed">
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0-.773.327-1.58.876-2.156a3.21 3.21 0 011.229-.799l.082 4.277a2.773 2.773 0 01-1.243.131zm2.427-.553l.046-4.109c.084-.004.166-.01.252-.01.773 0 1.494.145 1.885.361.391.217-1.023 2.713-2.183 3.758zm-8.95-7.668a.196.196 0 00-.196-.145h-1.95a.194.194 0 00-.194.144L.008 16.916c-.017.051-.011.076.062.076h1.733c.075 0 .099-.023.114-.072l1.008-3.318h3.496l1.008 3.318c.016.049.039.072.113.072h1.734c.072 0 .078-.025.062-.076-.014-.05-3.083-9.741-3.494-11.04zm-2.618 6.318l1.447-5.25 1.447 5.25H3.226z"></path></svg><span>Languages</span></span></a><ul class="menu__list"><li class="menu__list-item"><a href="/PSP/blog" target="_self" rel="noopener noreferrer" class="menu__link">English</a></li><li class="menu__list-item"><a href="/PSP/pt/blog" target="_self" rel="noopener noreferrer" class="menu__link dropdown__link--active">Português</a></li></ul></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">GitHub</a></li></ul></div></div></div></nav><div class="main-wrapper blog-wrapper"><div class="container margin-vert--lg"><div class="row"><div class="col col--3"><div class="sidebar_2ahu thin-scrollbar"><h3 class="sidebarItemTitle_2hhb">Recent posts</h3><ul class="sidebarItemList_2xAf"><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2020/08/01/2020w31a-beta">2020w31a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2020/07/09/2020w28a-beta">2020w28a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2020/06/13/2020w24a-beta">2020w24a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2019/11/01/2019w44a-beta">22019w44a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2020/08/01/2020w31a-beta">2020w31a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-08-01T00:00:00.000Z" class="blogPostDate_fNvV">31 de julho de 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>);</li><li>✔️ New fancy <a href="https://thales1330.github.io/PSP/" target="_blank" rel="noopener noreferrer">PSP-UFU website</a> and <a href="https://thales1330.github.io/PSP/blog" target="_blank" rel="noopener noreferrer">changelog</a> using <a href="https://v2.docusaurus.io/" target="_blank" rel="noopener noreferrer">docusaurus</a>;</li><li>✔️ Text element performance improved for OpenGL render;</li><li>🐛 Several major and minor bug fixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2020/07/09/2020w28a-beta">2020w28a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_fNvV">8 de julho de 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Voltage heatmap implemented (OpenGL enabled);</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2020/06/13/2020w24a-beta">2020w24a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_fNvV">12 de junho de 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Device context render implemented (to avoid use OpenGL - some machines can&#x27;t use it properly);</li><li>✔️ Added Visual Studio solution to the GitHub project;</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2019/11/01/2019w44a-beta">22019w44a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_fNvV">31 de outubro de 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Minor changes to 32bit MSW version for compatibility;</li><li>✔️ Travis CI (Continuous Integration) implemented (Linux).</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_fNvV">12 de outubro de 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Harmonic study implemented;</li><li>🔥 Autolayout from importation implemented;</li><li>🔥 Importation of Matpower files;</li><li>🐛 Several minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"></div></nav></main></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. Construído com Docusaurus.</div></div></div></footer></div>
+<script src="/PSP/pt/assets/js/styles.cc4b2cfe.js"></script>
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+</html>
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+        <title>PSP-UFU Blog</title>
+        <link>https://thales1330.github.io/blog</link>
+        <description>PSP-UFU Blog</description>
+        <lastBuildDate>Sat, 01 Aug 2020 00:00:00 GMT</lastBuildDate>
+        <docs>https://validator.w3.org/feed/docs/rss2.html</docs>
+        <generator>https://github.com/jpmonette/feed</generator>
+        <item>
+            <title><![CDATA[2020w31a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/pt/blog/2020/08/01/2020w31a-beta</link>
+            <guid>2020w31a-beta</guid>
+            <pubDate>Sat, 01 Aug 2020 00:00:00 GMT</pubDate>
+            <description><![CDATA[- 🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on GitHub or Twitter);]]></description>
+        </item>
+        <item>
+            <title><![CDATA[2020w28a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/pt/blog/2020/07/09/2020w28a-beta</link>
+            <guid>2020w28a-beta</guid>
+            <pubDate>Thu, 09 Jul 2020 00:00:00 GMT</pubDate>
+            <description><![CDATA[- 🔥 Voltage heatmap implemented (OpenGL enabled);]]></description>
+        </item>
+        <item>
+            <title><![CDATA[2020w24a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/pt/blog/2020/06/13/2020w24a-beta</link>
+            <guid>2020w24a-beta</guid>
+            <pubDate>Sat, 13 Jun 2020 00:00:00 GMT</pubDate>
+            <description><![CDATA[- ✔️ Device context render implemented (to avoid use OpenGL - some machines can't use it properly);]]></description>
+        </item>
+        <item>
+            <title><![CDATA[22019w44a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/pt/blog/2019/11/01/2019w44a-beta</link>
+            <guid>22019w44a-beta</guid>
+            <pubDate>Fri, 01 Nov 2019 00:00:00 GMT</pubDate>
+            <description><![CDATA[- ✔️ Minor changes to 32bit MSW version for compatibility;]]></description>
+        </item>
+        <item>
+            <title><![CDATA[2019w42a-beta]]></title>
+            <link>https://thales1330.github.io/PSP/pt/blog/2019/10/13/2019w42a-beta</link>
+            <guid>2019w42a-beta</guid>
+            <pubDate>Sun, 13 Oct 2019 00:00:00 GMT</pubDate>
+            <description><![CDATA[- 🔥 Harmonic study implemented;]]></description>
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class="menu__link">GitHub</a></li></ul></div></div></div></nav><div class="main-wrapper blog-wrapper"><div class="container margin-vert--lg"><div class="row"><div class="col col--3"><div class="sidebar_2ahu thin-scrollbar"><h3 class="sidebarItemTitle_2hhb">Recent posts</h3><ul class="sidebarItemList_2xAf"><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2020/08/01/2020w31a-beta">2020w31a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2020/07/09/2020w28a-beta">2020w28a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2020/06/13/2020w24a-beta">2020w24a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2019/11/01/2019w44a-beta">22019w44a-beta</a></li><li class="sidebarItem_2UVv"><a class="sidebarItemLink_1RT6" href="/PSP/pt/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></li></ul></div></div><main class="col col--7"><h1>5 posts tagged with &quot;psp-ufu&quot;</h1><a href="/PSP/pt/blog/tags">View All Tags</a><div class="margin-vert--xl"><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2020/08/01/2020w31a-beta">2020w31a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-08-01T00:00:00.000Z" class="blogPostDate_fNvV">31 de julho de 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 PSP-UFU online user guide finally written (only in Brazilian Portuguese, if you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>);</li><li>✔️ New fancy <a href="https://thales1330.github.io/PSP/" target="_blank" rel="noopener noreferrer">PSP-UFU website</a> and <a href="https://thales1330.github.io/PSP/blog" target="_blank" rel="noopener noreferrer">changelog</a> using <a href="https://v2.docusaurus.io/" target="_blank" rel="noopener noreferrer">docusaurus</a>;</li><li>✔️ Text element performance improved for OpenGL render;</li><li>🐛 Several major and minor bug fixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w31a-beta" href="/PSP/pt/blog/2020/08/01/2020w31a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2020/07/09/2020w28a-beta">2020w28a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_fNvV">8 de julho de 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Voltage heatmap implemented (OpenGL enabled);</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w28a-beta" href="/PSP/pt/blog/2020/07/09/2020w28a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2020/06/13/2020w24a-beta">2020w24a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_fNvV">12 de junho de 2020 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Device context render implemented (to avoid use OpenGL - some machines can&#x27;t use it properly);</li><li>✔️ Added Visual Studio solution to the GitHub project;</li><li>🐛 Several major and minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w24a-beta" href="/PSP/pt/blog/2020/06/13/2020w24a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2019/11/01/2019w44a-beta">22019w44a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_fNvV">31 de outubro de 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>✔️ Minor changes to 32bit MSW version for compatibility;</li><li>✔️ Travis CI (Continuous Integration) implemented (Linux).</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 22019w44a-beta" href="/PSP/pt/blog/2019/11/01/2019w44a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_GeHD"><a href="/PSP/pt/blog/2019/10/13/2019w42a-beta">2019w42a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_fNvV">12 de outubro de 2019 · One min read</time></div><div class="avatar margin-vert--md"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer" class="avatar__photo-link avatar__photo"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&amp;v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noopener noreferrer">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><div class="markdown"><ul><li>🔥 Harmonic study implemented;</li><li>🔥 Autolayout from importation implemented;</li><li>🔥 Importation of Matpower files;</li><li>🐛 Several minor bugfixes.</li></ul></div><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/pt/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2019w42a-beta" href="/PSP/pt/blog/2019/10/13/2019w42a-beta"><strong>Read More</strong></a></div></footer></article></div></main></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. Construído com Docusaurus.</div></div></div></footer></div>
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diff --git a/docs/pt/docs/bus/index.html b/docs/pt/docs/bus/index.html
new file mode 100644
index 0000000..f690436
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+++ b/docs/pt/docs/bus/index.html
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+<!doctype html>
+<html lang="pt" dir="ltr">
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href="/PSP/pt/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Barramento</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><p>Condutor de baixa impedância ao qual vários circuitos elétricos podem ser conectados em pontos separados.
+Nota - Em muitos casos, o barramento consiste em uma barra. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=151-12-30" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barramento-no-psp-ufu"></a>Barramento no PSP-UFU<a class="hash-link" href="#barramento-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento <strong>barramento</strong>, ou simplesmente <strong>barra</strong>, é um conector ou nó do diagrama unifilar do PSP-UFU. Essa barra pode representar um PAC (Ponto de Acoplamento Comum), um poste de distribuição, uma subestação, um barramento da subestação, entre inúmeros outros tipos pontos de análise e conexão entre elementos.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-barramentos"></a>Formulário de edição dos barramentos<a class="hash-link" href="#formulário-de-edição-dos-barramentos" title="Direct link to heading">#</a></h2><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O barramento deve ser o <strong>primeiro elemento elétrico a ser inserido no diagrama de potência</strong>, uma vez que os demais componentes de potência são conectados nele.</p></div></div><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das barras:</p><div><center><img src="/PSP/pt/images/busForm.png" alt="Formulário dos barramentos no PSP-UFU" title="Formulário dos barramentos no PSP-UFU"></center></div><p>Esse formulário é subdividido em quatro contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais da barra e informações do fluxo de carga;</li><li><strong>Falta</strong>: local onde o curto-circuito <em>shunt</em> deve ser inserido;</li><li><strong>Estabilidade</strong>: contendo opções de visualização de dados da barra em gráficos no tempo e inserção de faltas trifásicas no cálculo de estabilidade transitória;</li><li><strong>Qualidade de energia</strong>: contém a opção de de visualização da impedância harmônica vista pela barra.</li></ul><hr><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Qualidade de energia</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-nominal"></a>Tensão nominal<a class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Utilizado para cálculo da impedância base de alguns elementos conectados, além do cálculo da relação de transformação dos transformadores conectados entre barras. Sua unidade pode ser selecionada, sendo expressa em V ou em kV.</p><p>A modificação desse parâmetro irá alterar toda a tensão do trecho conectado por linhas elétricas, sendo emitido um alerta ao usuário.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-controlada"></a>Tensão controlada<a class="hash-link" href="#tensão-controlada" title="Direct link to heading">#</a></h4><p>Caracteriza o barramento como <strong>barra de tensão controlada (<a href="/PSP/pt/docs/powerFlow">Barra PV</a>)</strong>,</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Essa opção é somente válida caso alguma máquina síncrona esteja conectado, caso contrário esse valor será ignorado. Caso o limite de potência reativa da máquina síncrona conectada seja ultrapassado esse valor também é ignorado.</p></div></div><p>O valor poderá ser inserido em p.u. ou em volts (ou kV caso a tensão nominal esteja nesta unidade).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barra-de-referência"></a>Barra de referência<a class="hash-link" href="#barra-de-referência" title="Direct link to heading">#</a></h4><p>Caracteriza o barramento como <strong>barra de referência (<a href="/PSP/pt/docs/powerFlow">Barra de oscilação</a>)</strong>. Essa opção é somente válida caso esteja conectado um gerador síncrono, caso contrário uma mensagem de erro será exibida ao usuário ao realizar algum dos cálculos do programa.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserir-falta"></a>Inserir falta<a class="hash-link" href="#inserir-falta" title="Direct link to heading">#</a></h4><p>Indica se existe um curto-circuito na barra nos estudos de <a href="/PSP/pt/docs/fault">curto-circuito</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Nota-se que essa opção irá inserir uma falta no cálculo de <a href="/PSP/pt/docs/fault">curto-circuito</a> e <strong>não será considerado no estudo de estabilidade</strong>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tipo-de-falta"></a>Tipo de falta<a class="hash-link" href="#tipo-de-falta" title="Direct link to heading">#</a></h4><p>Seleciona o tipo de falha shunt da barra:</p><ul><li>trifásico para a terra</li><li>fase-fase</li><li>fase-fase-terra</li><li>fase-terra</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="local-da-falta"></a>Local da falta<a class="hash-link" href="#local-da-falta" title="Direct link to heading">#</a></h4><p>Seleciona a fase em que se situa a falta (ou combinação delas no caso de falha entre duas fases), sendo essa opção desabilitada para o tipo de falta trifásica.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-e-reatância-de-falta"></a>Resistência e reatância de falta<a class="hash-link" href="#resistência-e-reatância-de-falta" title="Direct link to heading">#</a></h4><p>Representam a impedância da falta. Curtos-circuitos francos (sem impedância de falta) são representados inserindo valor zero em ambos os campos.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-dados"></a>Imprimir dados<a class="hash-link" href="#imprimir-dados" title="Direct link to heading">#</a></h4><p>Mostra os dados da barra editada no gráfico gerado pelo estudo de estabilidade transitória. Os seguintes dados são exibidos:</p><ul><li>módulo da tensão do barramento</li><li>ângulo da tensão do barramento.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserir-falta-1"></a>Inserir falta<a class="hash-link" href="#inserir-falta-1" title="Direct link to heading">#</a></h4><p>Insere uma falta trifásica na barra no cálculo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Nota-se que essa opção irá inserir uma falta no cálculo de <a href="/PSP/pt/docs/stability">estabilidade</a> e <strong>não será considerado no estudo de <a href="/PSP/pt/docs/fault">curto-circuito</a></strong>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tempo"></a>Tempo<a class="hash-link" href="#tempo" title="Direct link to heading">#</a></h4><p>Instante no tempo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">t_{falta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) em que ocorre a falta durante os estudos de <a href="/PSP/pt/docs/stability">estabilidade</a>. Esse campo e os próximos são somente acessados caso seja inserida uma falta no estudo de estabilidade.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="duração-da-falta"></a>Duração da falta<a class="hash-link" href="#duração-da-falta" title="Direct link to heading">#</a></h4><p>Duração da falta inserida (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta t_{falta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) no estudo de estabilidade. A falta irá iniciar no instante <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">t_{falta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e será interrompida em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>t</mi><mrow><mi>f</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">t_{falta}+\Delta t_{falta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-e-reatância-de-falta-1"></a>Resistência e reatância de falta<a class="hash-link" href="#resistência-e-reatância-de-falta-1" title="Direct link to heading">#</a></h4><p>Impedância de falta nos estudos de estabilidade. Uma falta franca é representada pela inserção de um valor de impedância shunt de valor muito baixo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j10^{-5} p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>).</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-impedância-harmônica-da-barra"></a>Imprimir impedância harmônica da barra<a class="hash-link" href="#imprimir-impedância-harmônica-da-barra" title="Direct link to heading">#</a></h4><p>Imprime a impedância harmônica vista por essa barra referente ao estudo de <a href="/PSP/pt/docs/harmonics">Resposta na Frequência</a>.</p></div></div></div><hr><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Resumo de como definir o tipo de barra:</p><ul><li>Para definir o barramento como sendo uma <strong><a href="/PSP/pt/docs/powerFlow">Barra PQ</a></strong>, as opções &quot;Barra de referência&quot; e &quot;Tensão controlada&quot; devem estar <strong>desmarcadas</strong>; </li><li>Para definir o barramento como sendo uma <strong><a href="/PSP/pt/docs/powerFlow">Barra PV</a></strong>, deve-se <strong>marcar somente</strong> a opção &quot;Tensão controlada&quot;, mantendo a opção &quot;Barra de referência&quot; desmarcada;</li><li>Para definir o barramento como sendo uma <strong><a href="/PSP/pt/docs/powerFlow">Barra de Referência</a></strong>, deve-se marcar a opção &quot;Barra de referência&quot;. Caso a opção &quot;Tensão controlada&quot; esteja desmarcada será admitido um valor de tensão controlada de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">1{,}0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></li></ul></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p><strong>O sistema deve possuir somente uma barra de referência</strong>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/bus.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/powerEditor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Editor de Potência</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/line"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Linha »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#barramento-no-psp-ufu" class="table-of-contents__link">Barramento no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-barramentos" class="table-of-contents__link">Formulário de edição dos barramentos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Ferramentas CAD</h1></header><div class="markdown"><blockquote><p>Desenho assistido por computador (DAC; em inglês: <em>computer aided design</em> - CAD) é o nome genérico de sistemas computacionais (<em>software</em>) utilizados pela engenharia, geologia, geografia, arquitetura e design para facilitar o projeto e desenho técnicos. No caso do <em>design</em>, este pode estar ligado especificamente a todas as suas vertentes (produtos como vestuário, eletroeletrônicos, automobilísticos, etc.), de modo que os jargões de cada especialidade são incorporados na interface de cada programa. <a href="https://pt.wikipedia.org/wiki/Desenho_assistido_por_computador" target="_blank" rel="noopener noreferrer">Wikipedia</a></p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="ferramentas-do-menu"></a>Ferramentas do menu<a class="hash-link" href="#ferramentas-do-menu" title="Direct link to heading">#</a></h2><p>Todos os objetos criados contêm vários atributos gráficos, como: posição, orientação, cor, tamanho, etc. Para modificá-los foram inseridas algumas ferramentas intuitivas e de fácil utilização, as quais são: <a href="/PSP/pt/docs/cadTools#selecionar"><strong>selecionar</strong></a>, <a href="/PSP/pt/docs/cadTools#mover"><strong>mover</strong></a>, <a href="/PSP/pt/docs/cadTools#girar"><strong>girar</strong></a>, <a href="/PSP/pt/docs/cadTools#arrastar"><strong>arrastar</strong></a>, <a href="/PSP/pt/docs/cadTools#zoom"><strong>zoom</strong></a>, <a href="/PSP/pt/docs/cadTools#encaixar"><strong>encaixar</strong></a>, <a href="/PSP/pt/docs/cadTools#copiar-e-colar"><strong>copiar, colar</strong></a> e <a href="/PSP/pt/docs/cadTools#excluir"><strong>excluir</strong></a>.</p><p>Essas ferramentas podem ser acessadas por suas teclas de atalho assim como pelo <a href="/PSP/pt/docs/mainScreen#menu-ribbon">submenu Ferramentas</a>.</p><p>Tais ferramentas permitem a <strong>personalização</strong>, <strong>navegação</strong> e auxiliam a <strong>criação</strong> de redes elétricas de potência, assim como diagramas de controle na plataforma.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="selecionar"></a>Selecionar<a class="hash-link" href="#selecionar" title="Direct link to heading">#</a></h3><p>A seleção de um único objeto é realizada clicando com o botão esquerdo do mouse sobre o elemento, que será identificado pelo contorno em <span style="background-color:#80bfff;border-radius:2px;color:#000;padding:0.2rem">azul</span> do objeto.</p><p>Para a seleção de vários objetos simultaneamente é necessário clicar com o botão esquerdo do mouse em um local vazio da tela e arrastá-lo, criando assim um <strong>retângulo de seleção</strong> que irá acompanhar o ponteiro do mouse enquanto o botão esquerdo estiver pressionado. Ao soltá-lo todos os objetos que intercederem o retângulo serão selecionados.</p><p>Mantendo pressionado a tecla Control (Ctrl) do teclado pode-se manter seleções anteriores enquanto novos elementos são selecionados.</p><p>Para desmarcar todos os objetos basta clicar em uma área sem objetos.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/select.mp4" type="video/mp4"><source src="/PSP/pt/videos/select.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="mover"></a>Mover<a class="hash-link" href="#mover" title="Direct link to heading">#</a></h3><p>Um elemento poderá ser movido para qualquer local por meio da ação <em>drag-and-drop</em> (clicando e arrastando) ou pela ferramenta mover.</p><p>Ao habilitar o “modo mover”, movendo todos os componentes selecionados à medida que a posição do mouse é alterada. Para desabilitar esse modo basta pressionar a tecla Escape do teclado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/move.mp4" type="video/mp4"><source src="/PSP/pt/videos/move.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="girar"></a>Girar<a class="hash-link" href="#girar" title="Direct link to heading">#</a></h3><p>Alguns objetos poderão ser rotacionados através da ferramenta girar, acessado pela tecla de atalho “R”, girando o objeto no sentido horário, ou “Shift + R”, para o sentido anti-horário (essa opção irá afetar todos os objetos selecionados). O elemento também pode ser rotacionado pelo <a href="/PSP/pt/docs/mainScreen#menu-ribbon">submenu Ferramentas</a> ou ao acessar o menu de contexto clicando com o botão direito sobre o elemento.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O menu de contexto somente será exibido se o elemento estiver selecionado.</p></div></div><p>Ao utilizar essa ferramenta os objetos irão girar 45º a partir do seu ponto de origem. Para atingir a angulação pretendida basta repetir o processo de rotação.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/rotate.mp4" type="video/mp4"><source src="/PSP/pt/videos/rotate.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="arrastar"></a>Arrastar<a class="hash-link" href="#arrastar" title="Direct link to heading">#</a></h3><p>Três ferramentas muito úteis na navegação do circuito, principalmente em grandes redes, são: arrastar, zoom e encaixar.</p><p>A ferramenta arrastar possibilita mover todo o circuito <strong>preservando as posições relativas entre os elementos</strong>. Esta ferramenta pode ser acessada pelo submenu “Ferramentas” ou <strong>mantendo pressionada o <em>scroll</em> do mouse</strong>, ativando assim o “modo arrastar”. Ao ativá-lo todo o circuito será movido clicando e arrastando na direção desejada.</p><p>Para sair do “modo arrastar” basta pressionar a tecla Escape do teclado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/drag.mp4" type="video/mp4"><source src="/PSP/pt/videos/drag.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="zoom"></a>Zoom<a class="hash-link" href="#zoom" title="Direct link to heading">#</a></h3><p>O zoom será aplicado ao utilizar o <strong>scroll do mouse</strong> para mais ou menos zoom. A ferramenta zoom irá aproximar ou distanciar o circuito a partir da posição do ponteiro do mouse na tela para encontrar objetos ou destaca-los.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/zoom.mp4" type="video/mp4"><source src="/PSP/pt/videos/zoom.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="encaixar"></a>Encaixar<a class="hash-link" href="#encaixar" title="Direct link to heading">#</a></h3><p>A ferramenta encaixar irá mover o circuito e aplicar o zoom necessário para que todos os componentes da rede sejam exibidos na tela. Esta ferramenta pode ser acessada através do submenu “Ferramentas” ou pela combinação de teclas “Shift + F”.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/fit.mp4" type="video/mp4"><source src="/PSP/pt/videos/fit.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="copiar-e-colar"></a>Copiar e Colar<a class="hash-link" href="#copiar-e-colar" title="Direct link to heading">#</a></h3><p>Duas ferramentas importantes durante o processo de criação da rede elétrica de potência no <em>software</em> são copiar e colar. Qualquer circuito criado no PSP-UFU poderá ser duplicado completamente ou parcialmente no mesmo projeto ou em múltiplos através dessas ferramentas.</p><p>Para copiar basta selecionar os elementos que se deseja duplicar e acessar a ferramenta no submenu “Ferramentas” ou pressionando a combinação das teclas “Ctrl + C”, então os dados do circuito serão copiados para a área de transferência. Ao copiar um circuito ele poderá ser colado no mesmo projeto ou em um projeto distinto, utilizando o mesmo submenu ou pressionando a combinação das teclas “Ctrl + V”.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/copyPaste.mp4" type="video/mp4"><source src="/PSP/pt/videos/copyPaste.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="excluir"></a>Excluir<a class="hash-link" href="#excluir" title="Direct link to heading">#</a></h3><p>Qualquer objeto poderá ser excluído do projeto por meio da ferramenta deletar, presente no menu Ribbon, pela tecla de atalho Delete ou pelo acesso ao menu de contexto clicando com o botão direito sobre o elemento selecionado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/delete.mp4" type="video/mp4"><source src="/PSP/pt/videos/delete.webm" type="video/webm"></video><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="personalização-gráfica-os-elementos"></a>Personalização gráfica os elementos<a class="hash-link" href="#personalização-gráfica-os-elementos" title="Direct link to heading">#</a></h2><p>Alguns elementos permitem sua personalização gráfica, como as barras e linhas.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barra"></a>Barra<a class="hash-link" href="#barra" title="Direct link to heading">#</a></h3><p>No caso dos barramentos, pode-se alterar o <strong>comprimento da barra</strong> clicando e arrastando no <em>pickbox</em> exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="linha"></a>Linha<a class="hash-link" href="#linha" title="Direct link to heading">#</a></h3><p>As linhas podem ser inseridas com <strong>&quot;nós&quot; (pontos de ancoragem)</strong>, personalizando a disposição das linhas no diagrama unifilar. Esses nós são inseridos durante o processo de criação do elemento, anteriormente à seleção da segunda barra.</p><p>Após a sua inserção, novos nós podem ser anexados pelo menu de contexto acessado ao clicar com o botão direito sobre a linha selecionada. Sua remoção também é alcançada por uma opção do menu de contexto.</p><p>Assim como nos barramentos, a alteração da posição dos nós é obtida clicando e arrastando o <em>pickbox</em> exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/graphicPropsChange.mp4" type="video/mp4"><source src="/PSP/pt/videos/graphicPropsChange.webm" type="video/webm"></video><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="conexão-e-desconexão-de-elemetos-na-barra"></a>Conexão e desconexão de elemetos na barra<a class="hash-link" href="#conexão-e-desconexão-de-elemetos-na-barra" title="Direct link to heading">#</a></h2><p>Os elementos podem ser desconectados e reconectados às barras após sua inserção somente pela ferramenta <em>drag-and-drop</em>. Para isso, o elemento deve ser selecionado e o seu nó de conexão deve ser arrastado para o local requerido.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O elemento a ser desconectado/reconectado <strong>deve ser previamente selecionado</strong> para efetuar a operação. Caso contrário a barra será movida ou nenhuma operação será realizada.</p></div></div><p>No caso de desconexão, o elemento será automaticamente removido da simulação, sendo indicado pela sua cor (o elemento passa a ter cor <span style="background-color:#7f7f7f;border-radius:2px;color:#000;padding:0.2rem">cinza</span>).</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Ao reconectar o elemento ao circuito, deve-se <strong>atentar para estado do seu &quot;disjuntor&quot;</strong>, indicado por um quadrado próximo ao seu nó de conexão. Um elemento reconectado volta ao circuito com seu disjuntor aberto (<span style="background-color:#ff1919;border-radius:2px;color:#000;padding:0.2rem">vermelho</span>), sendo necessário clicar sobre o mesmo a fim de inserir o elemento no circuito.</p></div></div><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/disconnect.mp4" type="video/mp4"><source src="/PSP/pt/videos/disconnect.webm" type="video/webm"></video></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/cadTools.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/mainScreen"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Tela Principal</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/powerEditor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Editor de Potência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#ferramentas-do-menu" class="table-of-contents__link">Ferramentas do menu</a><ul><li><a href="#selecionar" class="table-of-contents__link">Selecionar</a></li><li><a href="#mover" class="table-of-contents__link">Mover</a></li><li><a href="#girar" class="table-of-contents__link">Girar</a></li><li><a href="#arrastar" class="table-of-contents__link">Arrastar</a></li><li><a href="#zoom" class="table-of-contents__link">Zoom</a></li><li><a href="#encaixar" class="table-of-contents__link">Encaixar</a></li><li><a href="#copiar-e-colar" class="table-of-contents__link">Copiar e Colar</a></li><li><a href="#excluir" class="table-of-contents__link">Excluir</a></li></ul></li><li><a href="#personalização-gráfica-os-elementos" class="table-of-contents__link">Personalização gráfica os elementos</a><ul><li><a href="#barra" class="table-of-contents__link">Barra</a></li><li><a href="#linha" class="table-of-contents__link">Linha</a></li></ul></li><li><a href="#conexão-e-desconexão-de-elemetos-na-barra" class="table-of-contents__link">Conexão e desconexão de elemetos na barra</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Capacitor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><p>Um dispositivo constituído essencialmente por dois eletrodos separados por um dielétrico. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=436-01-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="capacitor-no-psp-ufu"></a>Capacitor no PSP-UFU<a class="hash-link" href="#capacitor-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento capacitor representa, geralmente, um banco de capacitores <em>shunt</em> no circuito do PSP-UFU.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/pt/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/pt/docs/bus">barramento</a> e um <a href="/PSP/pt/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/pt/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-capacitores"></a>Formulário de edição dos capacitores<a class="hash-link" href="#formulário-de-edição-dos-capacitores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos capacitores:</p><div><center><img src="/PSP/pt/images/capacitorForm.png" alt="Formulário dos capacitores no PSP-UFU" title="Formulário dos capacitores no PSP-UFU"></center></div><p>Além do único contexto geral, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do capacitor durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/pt/images/swCapacitor.png" alt="Formulário de chaveamento do capacitor" title="Formulário de chaveamento do capacitor"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-reativa"></a>Potência reativa<a class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base de potência do sistema). </p></div><div role="tabpanel" hidden=""><p>O botão &quot;Estabilidade&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/capacitor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/load"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Carga</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/inductor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Indutor »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#capacitor-no-psp-ufu" class="table-of-contents__link">Capacitor no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-capacitores" class="table-of-contents__link">Formulário de edição dos capacitores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Constante</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>As constantes são valores numéricos reais que permanecem inalteradas em todo o período de simulação. Esses elementos possuem somente um nó de saída, o qual poderá ser conectado a qualquer outro nó de entrada dos elementos de controle.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Esse elemento pode, por exemplo, ser utilizado para definir valores alvos ou deslocamentos de sinal.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/const.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/rateLimiter"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Limitador de taxa</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/gain"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Ganho »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Editor de Controle</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Os controles das <a href="/PSP/pt/docs/syncGenerator">máquinas síncronas</a> são criados, modificados, importados e exportados em um editor distinto dos elementos de potência, apresentado na figura abaixo.</p><div><center><img src="/PSP/pt/images/controlEditor.png" alt="Editor de elementos de controle do PSP-UFU" title="Editor de elementos de controle do PSP-UFU"></center></div><p>O acesso ao editor é realizado por meio de dois botões presentes no <a href="/PSP/pt/docs/syncGenerator#formul%C3%A1rio-de-edi%C3%A7%C3%A3o-dos-geradores-s%C3%ADncronos">formulário de edição dos dados de estabilidade dos geradores síncronos</a>:</p><ul><li><a href="/PSP/pt/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Editar AVR</a></li><li><a href="/PSP/pt/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Editar regulador de velocidade</a></li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU a opção de editar o <strong>AVR</strong> engloba mais que somente o controle de tensão da máquina. Nele <strong>deve</strong> ser inserida a malha de controle da máquina assim como a <strong>excitatriz da máquina síncrona</strong>. Outras estratégias de controle (opcionais), como PSS (<em>Power System Stabilizer</em>) e/ou controles de sobre e sub excitação, são também implementadas em conjunto (como apresentado na figura anterior).</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Assim como no AVR, o <strong>Regulador de Velocidade</strong> engloba mais que a regulação primária da máquina. Nessa opção <strong>deve</strong> ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o <strong>modelo da turbina</strong>. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="elementos-de-controle"></a>Elementos de Controle<a class="hash-link" href="#elementos-de-controle" title="Direct link to heading">#</a></h2><p>O acesso aos elementos de controle é realizado em uma janela (removível e encaixável) por meio de ícones relacionados.</p><p>Uma vez que o usuário clicar no ícone desejado, o elemento de controle acompanhará o ponteiro do mouse até ser efetivamente inserido na posição desejada ao clicar novamente na área de trabalho. Os elementos são então conectados por “linhas de conexão” inseridas ao clicar nos nós dos componentes previamente adicionados, permitindo a construção da rede de controle genérica.</p><p>As ferramentas de manipulação e navegação, como arrastar, mover e excluir são <a href="/PSP/pt/docs/cadTools">herdadas</a> do <a href="/PSP/pt/docs/powerEditor">editor de elementos de potência</a>, possuindo comportamento idêntico. A <strong>edição dos dados</strong> dos componentes inseridos também é realizada com <strong>duplo clique</strong> sobre o elemento inserido, exibindo um formulário de edição de dados.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/timelapseControlEditor.mp4" type="video/mp4"><source src="/PSP/videos/timelapseControlEditor.webm" type="video/webm"></video><p>Os seguintes blocos de controle estão presentes no PSP-UFU:</p><ul><li><a href="/PSP/pt/docs/io">Entrada e Saída</a></li><li><a href="/PSP/pt/docs/transferFunction">Função Transferência</a></li><li>Matemática<ul><li><a href="/PSP/pt/docs/sum">Somador</a></li><li><a href="/PSP/pt/docs/multiplier">Multiplicador</a></li><li><a href="/PSP/pt/docs/divider">Divisor</a></li><li><a href="/PSP/pt/docs/exponential">Exponencial</a></li></ul></li><li>Limitadores<ul><li><a href="/PSP/pt/docs/limiter">Limitador absoluto</a></li><li><a href="/PSP/pt/docs/rateLimiter">Limitador de taxa</a></li></ul></li><li>Constantes<ul><li><a href="/PSP/pt/docs/const">Constante</a></li><li><a href="/PSP/pt/docs/gain">Ganho</a></li></ul></li><li><a href="/PSP/pt/docs/mathExpression">Expressão Matemática</a></li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inicialização-do-sistema-de-controle"></a>Inicialização do sistema de controle<a class="hash-link" href="#inicialização-do-sistema-de-controle" title="Direct link to heading">#</a></h2><p>Caso as entradas do sistema de controle apresentem valores diferentes de zero ou a primeira operação entre elas não resulte em uma saída nula, o sistema necessita de inicialização, de forma a adequar os valores de entradas e saída dos blocos elementares e dos vetores de estado das <a href="/PSP/pt/docs/transferFunction">funções transferências</a> presentes. Tal procedimento é realizado por meio da solução de toda rede de controle até que se obtenha sua convergência, ou seja, a diferença absoluta entre as mesmas saídas de uma solução anterior e uma atual deve ser nula ou muito próxima de zero.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>info</h5></div><div class="admonition-content"><p>O processo de <strong>inicialização é realizada automaticamente</strong> pelo PSP-UFU.</p><p>Uma vez que a inicialização é imposta pelas <a href="/PSP/pt/docs/io">entradas</a> do controle, erros de convergência podem ocorrer devido à má parametrização dos elementos dos sistemas de <strong>potência e controle</strong>.</p></div></div><p>Para otimizar e melhorar a estabilidade do processo de inicialização utilizou-se um passo de integração variável dentro de limites, de forma que o passo aumenta em condições de diferenças menores entre as soluções do sistema de controle e diminui caso essa diferença se torne elevada. A implementação dessa abordagem reduziu significativamente o encerramento do processo com erro causado pela instabilidade numérica, além de acelerar a inicialização.</p><p>O fluxograma abaixo evidencia o processo de inicialização implementado no PSP-UFU:</p><div><center><img src="/PSP/pt/images/controlInit.png" alt="Estrutura da inicialização do sistema de controle" title="Estrutura da inicialização do sistema de controle"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="teste-do-diagrama-de-controle"></a>Teste do diagrama de controle<a class="hash-link" href="#teste-do-diagrama-de-controle" title="Direct link to heading">#</a></h2><p>Na parte inferior esquerda do Editor de Controle está presente o botão &quot;<strong>Testar Sistema...</strong>&quot;. O botão acessa o formulário de teste de controles, como é indicado na figura abaixo:</p><div><center><img src="/PSP/pt/images/testControl.png" alt="Formulário de teste de controles" title="Formulário de teste de controles"></center></div><p>Nesse formulário é possível inserir o comportamento de <strong>todas as entradas do diagrama</strong>:</p><ul><li><strong>Tipo de entrada</strong>: Define um dos tipos de entradas possíveis no PSP-UFU: &quot;Passo (<em>Step</em>)&quot;, Rampa (<em>Ramp</em>) ou Quadrática (<em>Quadratic</em>);</li><li><strong>Tempo de início (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">t_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>)</strong>: Define o tempo de início na simulação da entrada definida. As entradas iniciais são sempre definidas como zero e assumem valores diferentes após o tempo de início;</li><li><strong>Inclinação (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span>)</strong>: Valor relacionado ao tipo de entrada:<ul><li>Para entrada do tipo &quot;Passo&quot; - O valor da inclinação define o valor final do passo:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>&lt;</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t &lt; t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li><li>Para entrada do tipo &quot;Rampa&quot; - define o coeficiente de inclinação da reta:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>&lt;</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>×</mo><mi>t</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t &lt; t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li><li>Para entrada do tipo &quot;Quadrática&quot; - define o coeficiente de crescimento da curva:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>&lt;</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>×</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t &lt; t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t^2 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li></ul></li><li><strong>Passo de integração (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>)</strong>: Passo do método de integração Trapezoidal Implícito;</li><li><strong>Tempo de simulação (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">t_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>)</strong>: Tempo total de simulação.</li></ul><p>Após a simulação os resultados são exibidos no <a href="/PSP/pt/docs/graphViewer">visualizador de gráficos</a>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A mesma parametrização de entrada é aplicada em todos os blocos de <a href="/PSP/pt/docs/io">entrada</a> do sistema.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>É possível aplicar <strong>várias entradas distintas</strong> (inclusive com condicionais!) utilizando o bloco de <a href="/PSP/pt/docs/mathExpression">expressão matemática</a>.</p></div></div><p>Utilizando o bloco de <a href="/PSP/pt/docs/mathExpression">expressão matemática</a> é possível inserir entradas complexas, como por exemplo testar o diagrama de blocos (AVR DC1C + PSS1A - <a href="https://doi.org/10.1109/IEEESTD.2016.7553421" target="_blank" rel="noopener noreferrer">IEEE Std. 421.5-2016</a>) da figura abaixo:</p><div><center><img src="/PSP/pt/images/exControl.svg" alt="Sistema de controle sob teste" title="Sistema de controle sob teste"></center></div><div><center><img src="/PSP/pt/images/compControl.svg" alt="Comparação do sistema de controle sob teste" title="Comparação do sistema de controle sob teste"></center></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/controlEditor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/indMotor"><div 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Divisor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Esse elemento é bastante simples e cumpre com a função matemáticas de seu nome.</p><p>Possui duas entradas de valores reais e sua saída é computada na sequência do nó superior pelo inferior, ou seja, a saída será o valor presente no nó 1 dividido pelo nó 2.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Nós não conectados no divisor são considerados como valor nulo. Caso a divisão possua numerador zero, o valor da saída é definida como sendo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">10^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span></span>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/divider.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/multiplier"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Multiplicador</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/exponential"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Exponencial »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Exponencial</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>A função exponencial dos elementos de controle realiza essa operação com uma entrada real, obedecendo a expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mi mathvariant="bold">A</mi><msup><mi>e</mi><mrow><mi mathvariant="bold">B</mi><msub><mi>u</mi><mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">y_n = \bold{A} e^{\bold{B} u_n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8932769999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">A</span></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8932769999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">B</span></span><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a entrada e a saída do bloco exponencial, respectivamente;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\bold{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">A</span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">B</mi></mrow><annotation encoding="application/x-tex">\bold{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">B</span></span></span></span></span></span>	são constantes definidas pelo usuário.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-bloco-exponencial"></a>Formulário de edição de dados do bloco Exponencial<a class="hash-link" href="#formulário-de-edição-de-dados-do-bloco-exponencial" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco exponencial.</p><div><center><img src="/PSP/pt/images/expForm.png" alt="Formulário de edição de dados do bloco exponencial no PSP-UFU" title="Formulário de edição de dados do bloco exponencial no PSP-UFU"></center></div><p>Esse bloco não linear é definido pelas constantes A e B, inseridas pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tais blocos são úteis na representação de não linearidades, como, por exemplo, modelagem da saturação de máquinas de corrente contínuas presentes em alguns reguladores automáticos de tensão.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/exponential.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/divider"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Divisor</div></a></div><div 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Potência</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Curto-Circuito</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>O principal objetivo da análise de curto-circuito é o cálculo das correntes e tensões de falta para especificação de disjuntores, transformadores de corrente e a parametrização de relés de proteção. De 70 a 80% das faltas em linhas de transmissão são entre uma fase e terra, as quais ocorrem devido ao centelhamento de apenas uma fase da linha para a torre e então para a terra.O menor número de faltas, cerca de 5%, envolve todas as três fases, chamadas de faltas trifásicas. Os outros tipos de faltas envolvem duas fases e duas fases e a terra.</p><p>Todas essas falhas, exceto a trifásica, são assimétricas e causam desequilíbrio entre as fases.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="cálculo-de-curto-circuito-no-psp-ufu"></a>Cálculo de Curto-Circuito no PSP-UFU<a class="hash-link" href="#cálculo-de-curto-circuito-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O primeiro estágio do cálculo de curto-circuito é a determinação das tensões pré-falta, das potências de geração e cargas do sistema. Esses dados são obtidos por meio do estudo de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Atualmente, o PSP-UFU fornece resultado para os seguintes tipos de falta:</p><ul><li>Falta Trifásica (3F-T);</li><li>Falta Fase-Terra (F-T);</li><li>Falta Fase-Fase (F-F);</li><li>Falta Fase-Fase-Terra (F-F-T).</li></ul></div></div><p>Os modelos dos elementos elétricos que constituem um sistema de potência para o estudo de curto-circuito são semelhantes aos do fluxo de carga, apresentando algumas divergências para as faltas desbalanceadas (F-T, F-F e F-F-T).</p><p>As faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados em conjunto cargas desbalanceadas.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="execução-do-cálculo-de-curto-circuito-no-psp-ufu"></a>Execução do cálculo de curto-circuito no PSP-UFU<a class="hash-link" href="#execução-do-cálculo-de-curto-circuito-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Existem duas formas de se calcular o curto-circuito no PSP-UFU:</p><ul><li><strong>Falta</strong>: Calcula a falta inserida nas <a href="/PSP/pt/docs/bus">barras</a>. Nesse tipo de cálculo é possível calcular faltas <em>shunt</em> nos <a href="/PSP/pt/docs/bus">barramentos</a> balanceadas e desbalanceadas.</li><li><strong>Nível de curto-circuito</strong>: Calcula o nível de curto-circuito (falta trifásica) em todos <a href="/PSP/pt/docs/bus">barramentos</a> do sistema.</li></ul><p>Após a construção do diagrama unifilar no <a href="/PSP/pt/docs/powerEditor">editor de potência</a>, a execução do cálculo de curto-circuito é realizada no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Falta</strong>. Para calcular o nível de curto-circuito (falta trifásica) em todos <a href="/PSP/pt/docs/bus">barramentos</a> do sistema, basta clicar no botão <strong>Nível de curto-circuito</strong>.</p><div><center><img src="/PSP/pt/images/menuSimulationFaulta.svg" alt="Execução dos cálculos de curto-circuito" title="Execução dos cálculos de curto-circuito"></center></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>É possível calcular as faltas sem a execução do <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>, porém <strong>não é recomendável</strong>, visto que os valores das correntes de falta são significativamente alteradas.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Os cálculos de curtos-circuitos não são habilitados por padrão no cálculo contínuo e devem ser inseridos nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a>.</p></div></div><p>Os resultados do cálculo de curto-circuito são exibidos nos <a href="/PSP/pt/docs/text">elementos de texto vinculado</a>, ao posicionar o mouse sobre os barramentos e em <a href="/PSP/pt/docs/tabularReport">relatórios tabulares</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="erros-comuns-na-execução-do-cálculo-de-curto-circuito"></a>Erros comuns na execução do cálculo de curto-circuito<a class="hash-link" href="#erros-comuns-na-execução-do-cálculo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao calculo de curto-circuito.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-falha-ao-inverter-a-matriz-admitância-de-sequência-zero"></a>A seguinte mensagem de erro é exibida: &quot;Falha ao inverter a matriz admitância de sequência zero&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-falha-ao-inverter-a-matriz-admitância-de-sequência-zero" title="Direct link to heading">#</a></h4><ul><li><strong>Impossibilidade de circulação da corrente de sequência zero</strong>. Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a matriz admitância de sequência zero é singular. Para contornar esse problema escolha uma das duas soluções abaixo:<ul><li>Marque a opção &quot;Neutro aterrado&quot; e insira um alto valor de reatância de aterramento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>9999</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j9999~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo);</li><li>Ou, na barra do gerador, insira um reator de baixo valor de potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">1{,}0~var</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault" style="margin-right:0.03588em">v</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>, por exemplo).</li></ul></li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="o-cálculo-de-curto-circuito"></a>O cálculo de curto-circuito<a class="hash-link" href="#o-cálculo-de-curto-circuito" title="Direct link to heading">#</a></h2><p>Como já foi apresentado anteriormente, as faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados com cargas desbalanceadas.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="método-das-componentes-simétricas"></a>Método das componentes simétricas<a class="hash-link" href="#método-das-componentes-simétricas" title="Direct link to heading">#</a></h3><p>Esse método proposto por C. L. Fortescue, permite definir um sistema de n fasores desbalanceados em n – 1 sistemas de n fases balanceados e um sistema de fase zero. O sistema de fase zero é definido por todas as fases de mesmo módulo e ângulo.
+Para um sistema trifásico pode-se definir três componentes de sequência:</p><ol><li>Componentes de sequência positiva, constituindo em três fasores iguais em módulo, 120º defasados entre si, e tendo a mesma sequência de fase que os fasores originais;</li><li>Componentes de sequência negativa, constituindo em três fasores iguais em módulo, 120º defasados entre si, e tendo a sequência de fase oposta à dos fasores originais.</li><li>Componentes de sequência zero, constituindo em três fasores iguais em módulo e com defasagem nula entre si.</li></ol><p>Com isso pode-se decompor as tensões de fase em componentes simétricas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>a</mi></msub><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>b</mi></msub><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>b</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>b</mi><mn>2</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>b</mi><mn>0</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>c</mi></msub><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>c</mi><mn>2</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>c</mi><mn>0</mn></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \dot{V}_a = \dot{V}_{a1} + \dot{V}_{a2} + \dot{V}_{a0}\\ \dot{V}_b = \dot{V}_{b1} + \dot{V}_{b2} + \dot{V}_{b0}\\ \dot{V}_c = \dot{V}_{c1} + \dot{V}_{c2} + \dot{V}_{c0} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" 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class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A figura abaixo apresenta um exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados.</p><div><center><img src="/PSP/pt/images/fortescue.svg" alt="Exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados" title="Exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados"></center></div><p>Para simplificar os cálculos adota-se um operador “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63056em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span>”, com o intuito de indicar a rotação de um fasor. Tal operador é um número complexo de módulo unitário e ângulo de 120º:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mn>1</mn><mi mathvariant="normal">∠</mi><mn>12</mn><msup><mn>0</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>=</mo><mn>1</mn><msup><mi>e</mi><mrow><mi>j</mi><mn>2</mn><mi>π</mi><mi mathvariant="normal">/</mi><mn>3</mn></mrow></msup><mo>=</mo><mo>−</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>866</mn></mrow><annotation encoding="application/x-tex">\overline{a} = 1 \angle 120^{\circ} = 1 e^{j2\pi/3} = -0{,}5 + j0{,}866</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63056em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.724115em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">∠</span><span class="mord">1</span><span class="mord">2</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.724115em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mtight">2</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">π</span><span class="mord mtight">/</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">−</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">8</span><span class="mord">6</span><span class="mord">6</span></span></span></span></span></div><p>Com isso pode-se utilizar as equações (de tensão apresentadas  em conjunto com o operador “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63056em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span>” para construir a seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>a</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>b</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>c</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mover><mover><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msup></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo stretchy="true">⏞</mo></mover><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow></mover><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \dot{V}_a\\ \dot{V}_b\\ \dot{V}_c \end{bmatrix} = \overbrace{ \begin{bmatrix} 1 &amp; 1 &amp; 1\\ 1 &amp; \overline{a}^2 &amp; \overline{a}\\ 1 &amp; \overline{a} &amp; \overline{a}^2 \end{bmatrix} }^{\left[ \bold{A} \right]} \begin{bmatrix} \dot{V}_{a0}\\ \dot{V}_{a1}\\ \dot{V}_{a2} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.8405700000000005em;vertical-align:-1.6702850000000007em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.170285em"><span style="top:-4.250095em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.9699049999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.6897149999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6702850000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:5.2692950000000005em;vertical-align:-1.6702850000000007em"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.59901em"><span style="top:-4.6990099999999995em"><span class="pstrut" style="height:4.6990099999999995em"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.69901em"><span style="top:-4.05101em"><span class="pstrut" style="height:4.05101em"></span><span class="mord"><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span></span></span><span class="svg-align" style="top:-6.202019999999999em"><span class="pstrut" style="height:4.05101em"></span><span class="stretchy" style="height:0.548em;min-width:1.6em"><span class="brace-left" style="height:0.548em"><svg width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
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+c-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14
+-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7
+ 311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11
+ 12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z"></path></svg></span><span class="brace-right" style="height:0.548em"><svg width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M400000 542l
+-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
+s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
+c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span style="top:-7.773019999999999em"><span class="pstrut" style="height:4.6990099999999995em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="minner mtight"><span class="mopen mtight delimcenter" style="top:0em"><span class="mtight">[</span></span><span class="mord mtight"><span class="mord mathbf mtight">A</span></span><span class="mclose mtight delimcenter" style="top:0em"><span class="mtight">]</span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.170285em"><span style="top:-4.250095em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.9699049999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.6897149999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6702850000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Considerando a matriz quadrada da equação anterior sendo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left[ \bold{A} \right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathbf">A</span></span><span class="mclose delimcenter" style="top:0em">]</span></span></span></span></span></span>, pode-se encontrar as componentes simétricas pré-multiplicando ambos os lados dessa mesma equação por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\left[ \bold{A} \right]^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.204008em;vertical-align:-0.25em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathbf">A</span></span><span class="mclose delimcenter" style="top:0em">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954008em"><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span>.</p><p>Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas faltas assimétricas (F-T, F-F e F-F-T) é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As informações a respeito das particularidades dos modelos para o estudo de curto-circuito são apresentados individualmente nos <a href="/PSP/pt/docs/powerEditor#editando-dados-el%C3%A9tricos">elementos de potência</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="equações-do-curto-circuito"></a>Equações do curto-circuito<a class="hash-link" href="#equações-do-curto-circuito" title="Direct link to heading">#</a></h3><p>Primeiramente será tratado o equacionamento para faltas balanceadas e então os estudos serão estendidos para as faltas desbalanceadas por meio da utilização do método das componentes simétricas.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="faltas-balanceadas"></a>Faltas balanceadas<a class="hash-link" href="#faltas-balanceadas" title="Direct link to heading">#</a></h4><p>Utiliza-se da matriz impedância de barras para o cálculo de curto-circuito, definida pela seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Z</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\dot{V}] = [Z_{bus}][\dot{I}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Z</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[Z_{bus}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span> é a inversa da matriz admitância de barras, chamada de matriz impedância de barras.</li></ul><p>Por meio da expansão da equação anterior é possível calcular a corrente de falta trifásica na barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi></msub><mo>=</mo><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\dot{I}_f = \frac{\dot{E}_i}{\overline{z}_{ii}+\overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.569298em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\dot{I}_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> é a corrente de falta trifásica na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\dot{E}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão pré-falta na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{ii}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a impedância equivalente de Thevenin vista pela barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>, retirada da matriz impedância</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{f}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.916668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> é a impedância de falta</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="faltas-desbalanceadas"></a>Faltas desbalanceadas<a class="hash-link" href="#faltas-desbalanceadas" title="Direct link to heading">#</a></h4><p>O desenvolvimento das equações do cálculo de curto-circuito para faltas desbalanceadas é realizado seguindo o seguinte procedimento:</p><ol><li>Definir os diagramas no ponto da falta, mostrando as conexões de todas fases para a falta. Assume-se que apenas impedâncias balanceadas estão presentes em ambos os lados do ponto da falta e o equivalente Thevenin até esse ponto é conhecido;</li><li>Escrever as condições de contorno relacionando as tensões e corrente conhecidas para o tipo de falta estudada;</li><li>Transformar as correntes e tensões do item 2 de a-b-c para o sistema de coordenadas 0-1-2;</li><li>Encontrar a corrente do curto-circuito em estudo baseado no seguinte sistema de equações (para a fase A):<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>a</mi></msub><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>0</mn></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \dot{V}_{a1} = \dot{E}_a - \dot{I}_{a1} \overline{z}_1\\ \dot{V}_{a2} = - \dot{I}_{a2} \overline{z}_2\\ \dot{V}_{a0} = - \dot{I}_{a0} \overline{z}_0 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li></ol><p>A tabela abaixo apresenta as equações para as faltas desbalanceadas após a execução do procedimento apresenteado:</p><table><thead><tr><th><strong>Falta</strong></th><th align="center"><strong>Seq. Positiva (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th><th align="center"><strong>Seq. Negativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th><th align="center"><strong>Seq. Zero (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th></tr></thead><tbody><tr><td><strong>F-T</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6078659999999996em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td></tr><tr><td><strong>F-F</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6078659999999996em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">- \dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span></td></tr><tr><td><strong>F-F-T</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo fence="true">)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i \left( \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f} \right)}{\overline{z}_{ii}^{1} \overline{z}_{ii}^{2} + 3 \overline{z}_{ii}^{2} \overline{z}_{f} + \overline{z}_{ii}^{2} \overline{z}_{ii}^{0} + 3 \overline{z}_{ii}^{1} \overline{z}_{f} + \overline{z}_{ii}^{1} \overline{z}_{ii}^{0} }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.670876em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6602000000000001em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7400100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span 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mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span 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style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.971568em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mo>−</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">- \dfrac{\dot{E}_i - \overline{z}_{ii}^{1} \dot{I}_{f}^{1}}{\overline{z}_{ii}^{0} + 3 \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.740082em;vertical-align:-1.010676em"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.729406em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.809216em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td></tr></tbody></table><p>Para obter os valores em a-b-c é usada a equação matricial apresentada anteriormente, encerrando o cálculo de curto-circuito.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.</li><li>ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: <a href="https://doi.org/10.1109/9780470545577" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/9780470545577</a></li><li>FORTESCUE, C. L. Method of Symmetrical Coordinates Applied to the Solution of Polyphase Networks. Trans. AIEE, v. 37, p.1027-1140, 1918. doi: <a href="https://doi.org/10.1109/T-AIEE.1918.4765570" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/T-AIEE.1918.4765570</a></li><li>ANDERSON, P. M. Analysis of faulted power systems. New York: IEEE Press, 1995.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/fault.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/powerFlow"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Fluxo de Potência</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/harmonics"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Harmônicos »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#cálculo-de-curto-circuito-no-psp-ufu" class="table-of-contents__link">Cálculo de Curto-Circuito no PSP-UFU</a></li><li><a href="#execução-do-cálculo-de-curto-circuito-no-psp-ufu" class="table-of-contents__link">Execução do cálculo de curto-circuito no PSP-UFU</a><ul><li><a href="#erros-comuns-na-execução-do-cálculo-de-curto-circuito" class="table-of-contents__link">Erros comuns na execução do cálculo de curto-circuito</a></li></ul></li><li><a href="#o-cálculo-de-curto-circuito" class="table-of-contents__link">O cálculo de curto-circuito</a><ul><li><a href="#método-das-componentes-simétricas" class="table-of-contents__link">Método das componentes simétricas</a></li><li><a href="#equações-do-curto-circuito" class="table-of-contents__link">Equações do curto-circuito</a></li></ul></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Visualizador de Gráficos</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Para exibição dos resultados originados dos cálculos de <a href="/PSP/pt/docs/stability">estabilidade</a> e <a href="/PSP/pt/docs/harmonics">varredura de frequências</a>, desenvolveu-se um visualizador de gráficos, como mostra a figura abaixo:</p><div><center><img src="/PSP/pt/images/graphViewer.png" alt="Visualizador de gráficos do PSP-UFU" title="Visualizador de gráficos do PSP-UFU"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="criação-manipulação-e-navegação-de-curvas"></a>Criação, manipulação e navegação de curvas<a class="hash-link" href="#criação-manipulação-e-navegação-de-curvas" title="Direct link to heading">#</a></h2><p>Tal visualizador utiliza a biblioteca <a href="https://wxmathplot.sourceforge.io/" target="_blank" rel="noopener noreferrer">wxMathPlot</a>, e permite ao usuário imprimir todas as curvas no tempo originadas dos resultados de <a href="/PSP/pt/docs/stability">estabilidade</a> e <a href="/PSP/pt/docs/harmonics">varredura de frequências</a>, acessados em dados hierárquicos em forma de árvore, identificados pelo tipo e nome do elemento de potência, além do tipo de curva.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As curvas podem ser plotadas com <strong>clique duplo</strong> sobre ela ou ao selecionar a caixa &quot;Plotar&quot; nas propriedades do traço.</p></div></div><p>Os gráficos impressos podem ser caracterizados pela edição de suas propriedades do traço, ou seja, pode-se alterar a cor, espessura e tipo de traço (seis tipos disponíveis). Por padrão o eixo vertical é o tempo (em estabilidade) ou frequência (varredura de frequências), porém o usuário pode alterar para qualquer curva disponível.</p><p>Várias propriedades gerais do gráfico podem ser editadas: título do gráfico, rótulos dos eixos x e y, margens e limites inferiores e superiores de ambos os eixos.</p><p>O menu <strong>Visualização</strong> permite a inserção de exibição de legenda, linhas de grade e coordenadas do ponteiro do mouse (utilizada para destacar pontos na curva), além da possibilidade de alternar o tema de impressão entre escuro e claro, facilitando a visualização ou exportação do gráfico.</p><p>No menu <strong>Arquivo</strong>, pode-se salvar o gráfico na forma de imagem (<em>.png</em>, <em>.jpg</em> ou <em>.bmp</em>) ou enviá-lo diretamente para a área de transferência. No mesmo menu também é permitido a exportação no formato <em><a href="https://en.wikipedia.org/wiki/Comma-separated_values" target="_blank" rel="noopener noreferrer">CSV (Comma-separated Values)</a></em>, o qual poderá ser utilizado em outros visualizadores de gráficos, como o Excel ou o <a href="http://www.gnuplot.info/" target="_blank" rel="noopener noreferrer">gnuplot</a>.</p><p>A área de plotagem permite a navegação utilizando ferramentas de zoom, por área de seleção e comandos de mouse, e arrasto das curvas impressas.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Ao clicar com o botão direito sobre a área de plotagem as opções de manipulação são exibidas por meio de um menu de contexto.</p></div></div><p>Os seguintes <strong>comandos de mouse</strong> são utilizados para navegação das curvas:</p><table><thead><tr><th>Comando</th><th>Ação</th></tr></thead><tbody><tr><td>Botão esquerdo pressionado + Selecionar área</td><td>Seleciona a área de zoom</td></tr><tr><td>Botão direito pressionado + Mover mouse</td><td>Arrasta a curva</td></tr><tr><td>Roda do mouse</td><td>Arrasto vertical</td></tr><tr><td>Roda do mouse + Shift</td><td>Arrasto horizontal</td></tr><tr><td>Roda do mouse + Ctrl</td><td>Mais/menos zoom</td></tr></tbody></table></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/graphViewer.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/tabularReport"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Relatórios Tabulares</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/heatmap"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Mapa de Tensão »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#criação-manipulação-e-navegação-de-curvas" class="table-of-contents__link">Criação, manipulação e navegação de curvas</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Fonte de Corrente Harmônica</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><p>(<strong>Fonte de perturbação harmônica</strong>) Parte do sistema elétrico de potência ou das instalações a ele conectadas, que causa distorção harmônica da forma de onda da corrente e / ou tensão. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=614-01-16" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="fonte-de-corrente-harmônica-no-psp-ufu"></a>Fonte de corrente harmônica no PSP-UFU<a class="hash-link" href="#fonte-de-corrente-harmônica-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As fontes de corrente harmônica são responsáveis pela injeção de correntes harmônicas e são utilizadas pela ferramenta de cálculo de <a href="/PSP/pt/docs/harmonics">distorções harmônicas</a>. Uma lista de correntes harmônicas pode ser inserida em um mesmo elemento, conforme é exibido na o formulário de edição de dados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A presença da fonte de corrente harmônica <strong>não é considerada</strong> nos estudos de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>, <a href="/PSP/pt/docs/fault">curto-circuito</a> e <a href="/PSP/pt/docs/stability">estabilidade</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-das-fontes-de-corrente-harmônica"></a>Formulário de edição das fontes de corrente harmônica<a class="hash-link" href="#formulário-de-edição-das-fontes-de-corrente-harmônica" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados da fonte de corrente harmônica:</p><div><center><img src="/PSP/pt/images/harmSourceForm.png" alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU"></center></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="ordem"></a>Ordem<a class="hash-link" href="#ordem" title="Direct link to heading">#</a></h4><p>Especifica a ordem harmônica a ser inserida na lista de correntes harmônicas do elemento. Caso a ordem harmônica já exista na lista, ela será substituída pelos novos parâmetros.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="corrente"></a>Corrente<a class="hash-link" href="#corrente" title="Direct link to heading">#</a></h4><p>Módulo da corrente harmônica.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="unidade"></a>Unidade<a class="hash-link" href="#unidade" title="Direct link to heading">#</a></h4><p>Unidade do módulo de corrente harmônica, em A ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="ângulo"></a>Ângulo<a class="hash-link" href="#ângulo" title="Direct link to heading">#</a></h4><p>Ângulo da corrente harmônica, em graus.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="adicionar--remover"></a>Adicionar / Remover<a class="hash-link" href="#adicionar--remover" title="Direct link to heading">#</a></h4><p>Insere ou modifica a nova ordem harmônica ao clicar em “Adicionar”. Retira a ordem harmônica selecionada da lista ao clicar em “Remover”.</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/harmSource.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/inductor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Indutor</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/syncGenerator"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Gerador Síncrono »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#fonte-de-corrente-harmônica-no-psp-ufu" class="table-of-contents__link">Fonte de corrente harmônica no PSP-UFU</a></li><li><a href="#formulário-de-edição-das-fontes-de-corrente-harmônica" class="table-of-contents__link">Formulário de edição das fontes de corrente harmônica</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Harmônicos</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Duas ferramentas relacionadas a harmônicos foram desenvolvidas no PSP-UFU:</p><ul><li>Distorção Harmônica Total de Tensão (THD, do inglês <em>Total Harmonic Distortion</em>);</li><li>Resposta na Frequência.</li></ul><p>A ferramenta de <strong><a href="https://en.wikipedia.org/wiki/Total_harmonic_distortion" target="_blank" rel="noopener noreferrer">Distorção Harmônica Total</a></strong> calcula as tensões harmônicas causadas por <a href="/PSP/pt/docs/harmSource">fontes de corrente harmônicas</a>, assim como o THD de todos os barramentos do sistema.</p><p>A ferramenta de <strong>Resposta na Frequência</strong> (ou análise de varredura de frequência) envolve a variação da impedância da rede em um espectro de frequências observado a partir de um certo barramento.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A análise de varredura de frequência é amplamente usada no projeto de filtros harmônicos.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/pt/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/pt/docs/bus">barramento</a> e um <a href="/PSP/pt/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/pt/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Na versão atual do programa não são consideradas as alterações das resistências do sistema causado pelo efeito pelicular. Versões futuras irão contemplar tal característica.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="execução-do-cálculo-de-harmônicos-no-psp-ufu"></a>Execução do cálculo de harmônicos no PSP-UFU<a class="hash-link" href="#execução-do-cálculo-de-harmônicos-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/pt/docs/powerEditor">editor de potência</a>, a execução do cálculo das distorções harmônicas é realizada no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Distorções Harmônicas</strong>. Para acessar a ferramenta de de varredura de frequência, basta clicar no botão <strong>Resposta na Frequência</strong>.</p><div><center><img src="/PSP/pt/images/menuSimulationHamonics.svg" alt="Acesso às ferramentas de estudo harmônico" title="Acesso às ferramentas de estudo harmônico"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="distorções-harmônicas"></a>Distorções Harmônicas<a class="hash-link" href="#distorções-harmônicas" title="Direct link to heading">#</a></h3><p>Ao clicar sobre o botão &quot;Distorções Harmônicas&quot; as distorções causadas pelas <a href="/PSP/pt/docs/harmSource">fontes de corrente harmônica</a> são calculadas em todos os barramentos do sistema.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso não forem inseridas <a href="/PSP/pt/docs/harmSource">fontes de corrente harmônica</a> no sistema de potência, a distorção de tensão de todas as barras será <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">0{,}0~\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.94444em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord">%</span></span></span></span></span>.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Os cálculos dos níveis de THD <strong>não são habilitados por padrão</strong> no cálculo contínuo e devem ser inseridos nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a>.</p></div></div><p>Os resultados das distorções harmônicas são exibidos nos <a href="/PSP/pt/docs/text">elementos de texto vinculados</a> e ao posicionar o mouse sobre um barramento.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resposta-na-frequência"></a>Resposta na Frequência<a class="hash-link" href="#resposta-na-frequência" title="Direct link to heading">#</a></h3><p>Ao clicar sobre o botão &quot;Resposta na Frequência&quot; será exibido um formulário para inserção dos parâmetros da ferramenta:</p><div><center><img src="/PSP/pt/images/injHarmCurrent.png" alt="Acesso à ferramentas resposta na frequência" title="Acesso à ferramentas resposta na frequência"></center></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="frequência-inicial"></a>Frequência inicial<a class="hash-link" href="#frequência-inicial" title="Direct link to heading">#</a></h4><p>Define a frequência inicial da varredura.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="frequência-final"></a>Frequência final<a class="hash-link" href="#frequência-final" title="Direct link to heading">#</a></h4><p>Define a frequência final da varredura.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="passo-de-frequência"></a>Passo de frequência<a class="hash-link" href="#passo-de-frequência" title="Direct link to heading">#</a></h4><p>Define o passo de incremento da frequência. Passos menores irão gerar gráficos mais precisos, porém podem aumentar muito o tempo de execução.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="corrente-injetada"></a>Corrente injetada<a class="hash-link" href="#corrente-injetada" title="Direct link to heading">#</a></h4><p>Define o barramento o qual será injetada a corrente harmônica para análise.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Esse campo pode ser entendido como: &quot;<em>A resposta das impedâncias harmônicas vistas pelos barramentos do sistema caso tenha uma fonte de corrente harmônica na barra selecionada</em>&quot;.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="botão-executar"></a>Botão Executar<a class="hash-link" href="#botão-executar" title="Direct link to heading">#</a></h4><p>Ao clicar no botão executar a varredura será executada e os resultados exibidos no <a href="/PSP/pt/docs/graphViewer">vizualizador de gráficos</a>, como mostra a imagem abaixo.</p><div><center><img src="/PSP/pt/images/freqScanPlot.png" alt="Exemplo de resposta na frequência" title="Exemplo de resposta na frequência"></center></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Para visualizar a varredura da impedância harmônica de um barramento, essa opção deve ser habilitada em seu <a href="/PSP/pt/docs/bus#imprimir-imped%C3%A2ncia-harm%C3%B4nica-da-barra">formulário de edição de dados</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="cálculos-das-ferramentas-harmônicas"></a>Cálculos das ferramentas harmônicas<a class="hash-link" href="#cálculos-das-ferramentas-harmônicas" title="Direct link to heading">#</a></h2><p>Ambos cálculos são realizados a partir da seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><msup><mo stretchy="false">]</mo><mi>h</mi></msup><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{I}]^h= [Y_{bus}]^h [\dot{V}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{I}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span>	Vetor das correntes harmônicas injetadas nas barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[Y_{bus}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span>	Matriz admitância harmônica de barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{V}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span>	Vetor das tensões harmônicas nas barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	Ordem harmônica</li></ul><p>O vetor das correntes harmônicas injetadas é definido no programa utilizando um elemento de potência, chamado “fonte de corrente harmônica”, em que o usuário pode criar uma lista de correntes injetadas (em A e/ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) em um barramento.</p><p>O programa define automaticamente as ordens harmônicas na simulação analisando todas as listas de fontes de corrente harmônicas previamente aos cálculos. Com isso são calculadas as admitâncias harmônicas necessárias de cada elemento pela multiplicação das reatâncias indutivas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>l</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">x_{l}^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.097216em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>) e susceptâncias capacitivas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>b</mi><mi>c</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">b_{c}^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.061108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span>) fundamentais de cada elemento pelas ordens harmônicas:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>l</mi><mi>h</mi></msubsup><mo>=</mo><mi>h</mi><mo>×</mo><msubsup><mi>x</mi><mi>l</mi><mn>1</mn></msubsup><mspace linebreak="newline"></mspace><msubsup><mi>b</mi><mi>c</mi><mi>h</mi></msubsup><mo>=</mo><mi>h</mi><mo>×</mo><msubsup><mi>b</mi><mi>c</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">x_{l}^h=h×x_l^1\\ b_{c}^h=h×b_c^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1111079999999998em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1111079999999998em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Uma vez calculadas as admitâncias harmônicas, são utilizados os mesmos modelos e algoritmos convencionais para construção da <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow><mi>h</mi></msubsup></mrow><annotation encoding="application/x-tex">Y_{bus}^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.132216em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.4168920000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>A equação matricial é resolvida usando o método de eliminação gaussiana para evitar a inversão de cada matriz de admitância harmônica. Este procedimento torna o cálculo das tensões harmônicas computacionalmente eficiente.</p><p>Uma vez calculadas as tensões harmônicas, a THD em uma barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span> pode ser definida por:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>THD</mtext><mi>i</mi></msub><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>h</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></munderover><msubsup><mi>V</mi><mi>i</mi><mi>h</mi></msubsup></mrow><msubsup><mi>V</mi><mi>i</mi><mn>1</mn></msubsup></mfrac></mrow><annotation encoding="application/x-tex">\text{THD}_i=\dfrac{\sum_{h=2}^{n} V_{i}^h}{V_i^1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord text"><span class="mord">THD</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.5016819999999997em;vertical-align:-0.9628639999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.538818em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="mrel mtight">=</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9628639999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Na ferramenta de Resposta na Frequência uma corrente senoidal é injetada na barra em uma faixa de frequências e o conjunto de equações matriciais é usado para calcular a resposta da tensão. Este cálculo é repetido em etapas discretas cobrindo o espectro de frequência especificado.</p><p>O programa usa a matriz admitância de sequência positiva e uma corrente injetada de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">1{,}0 p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> também de sequência positiva na equação matricial para calcular diretamente as impedâncias em p.u.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>ARRILLAGA, J.; WATSON, N. R. Power System Harmonics. John Wiley &amp; Sons; Chichester, 2003. doi: <a href="https://doi.org/10.1002/0470871229" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/0470871229</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/harmonics.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/fault"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Curto-Circuito</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/stability"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Estabilidade »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#execução-do-cálculo-de-harmônicos-no-psp-ufu" class="table-of-contents__link">Execução do cálculo de harmônicos no PSP-UFU</a><ul><li><a href="#distorções-harmônicas" class="table-of-contents__link">Distorções Harmônicas</a></li><li><a href="#resposta-na-frequência" class="table-of-contents__link">Resposta na Frequência</a></li></ul></li><li><a href="#cálculos-das-ferramentas-harmônicas" class="table-of-contents__link">Cálculos das ferramentas harmônicas</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Mapa de Tensão</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>A ferramenta Mapa de Tensão utiliza o concento de <a href="https://en.wikipedia.org/wiki/Heat_map" target="_blank" rel="noopener noreferrer">mapa de calor</a> para exibir os valores de tensão de todas as barras do sistema por meio de cores.</p><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/pt/videos/heatmap.mp4" type="video/mp4"><source src="/PSP/pt/videos/heatmap.webm" type="video/webm"></video><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Essa ferramenta é útil para identificação de barras com sub e sobretensões no sistema de potência, assim como o efeito da inserção / remoção de elementos do sistema.</p></div></div><p>A legenda que correlaciona os valores do módulo da tensão com as cores são exibidas à esquerda da área de trabalho.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="habilitando-a-ferramenta-mapa-de-tensão"></a>Habilitando a ferramenta Mapa de Tensão<a class="hash-link" href="#habilitando-a-ferramenta-mapa-de-tensão" title="Direct link to heading">#</a></h2><p>A ferramenta pode ser habilitada ou desabilitada no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">submenu Ribbon <strong>Ferramentas</strong></a> clicando no botão <strong>Mapa de tensão</strong> ou pela tecla de atalho &quot;Ctrl + Shift + H&quot;.</p><p>As configurações da ferramenta podem ser também acessadas no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">submenu Ribbon <strong>Ferramentas</strong></a> clicando no botão <strong>Configurações do projeto</strong></p><div><center><img src="/PSP/pt/images/heatmapForm.png" alt="Configurações da ferramenta Mapa de Tensão" title="Configurações da ferramenta Mapa de Tensão"></center></div><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Na versão atual do programa, somente as opções do mapa de tensão são exibidas nas configurações de projeto.</p></div></div><p>Os limites superiores e inferiores, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> são editados nesse formulário.</p><p>Alternativamente, o programa pode calcular automaticamente os limites. Nesse caso, o limite superior será a maior tensão do circuito e o limite inferior será a menor tensão do circuito.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O cálculo automático dos limites pode prejudicar a identificação de barras com sub e sobretensões no sistema.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/heatmap.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/graphViewer"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Visualizador de Gráficos</div></a></div><div class="pagination-nav__item pagination-nav__item--next"></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#habilitando-a-ferramenta-mapa-de-tensão" class="table-of-contents__link">Habilitando a ferramenta Mapa de Tensão</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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href="/PSP/pt/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Motor de Indução</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><p>Uma máquina assíncrona da qual apenas um enrolamento é energizado. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=411-31-10" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="motor-de-indução-trifásico-no-psp-ufu"></a>Motor de indução trifásico no PSP-UFU<a class="hash-link" href="#motor-de-indução-trifásico-no-psp-ufu" title="Direct link to heading">#</a></h2><p>No PSP-UFU, os motores de indução são contemplados nos estudos de <a href="/PSP/pt/docs/powerFLow">fluxo de carga</a> e no estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os dados de estabilidade da máquina de indução são utilizados em conjunto com o fluxo de carga, <strong>calculando de forma correta a potência reativa das máquinas</strong> e consequentemente a tensão no barramento conectado.</p></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao marcar a opção &quot;Calcular a potência reativa no fluxo de carga&quot;, devem ser inseridos os dados corretos na aba &quot;Estabilidade&quot;, caso contrário, será atribuída uma potência reativa incorreta no motor.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga"></a>Motor de indução trifásico no cálculo do fluxo de carga<a class="hash-link" href="#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>A figura abaixo apresenta o modelo do motor de indução trifásico (MIT) de gaiola simples.</p><div><center><img src="/PSP/pt/images/indMotorModel.svg" alt="Circuito equivalente do motor de indução" title="Circuito equivalente do motor de indução"></center></div><p>As potências ativa e reativa podem ser calculadas em relação às variáveis e parâmetros do motor em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> como:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mfrac><mrow><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">{</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mo fence="true">}</mo></mrow></mrow><mrow><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"></mspace><mi>Q</mi><mo>=</mo><mfrac><mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">{</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mo>−</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mo fence="true">}</mo></mrow></mrow><mrow><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">P = \frac{V^2 \left\{ \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]+K_1 \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\ Q = \frac{-V^2 \left\{ K_1 \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]- \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\ ~\\ K_1=x_2+x_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.884028em;vertical-align:-1.2940180000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em"><span style="top:-2.11em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.284008em"><span class="pstrut" style="height:3.054008em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7940180000000003em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">{</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">}</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2940180000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.884028em;vertical-align:-1.2940180000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em"><span style="top:-2.11em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.284008em"><span class="pstrut" style="height:3.054008em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7940180000000003em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">{</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">}</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2940180000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Como pode ser observado nas equações acima, existem quatro variáveis e somente duas equações. Na prática, as variáveis podem ser reduzidas a três, uma vez que o módulo da tensão (V) é obtido nos resultados do <a href="/PSP/pt/docs/powerFlow">fluxo de potência</a>. Para resolver as equações é necessário definir uma variável adicional. A variável escolhida como fixa no PSP-UFU é a potência ativa (P), por fornecer resultados numericamente corretos e adequados para motores em situações de estabilidade.</p><p>Portanto, nesse modelo estático, a potência ativa é mantida constante durante o cálculo do fluxo de carga e o escorregamento (s) é atualizado em cada iteração. A equação da potência ativa pode ser reescrita em relação ao escorregamento:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mi>A</mi><mo>+</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mi>B</mi><mo>+</mo><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)^2 A + \left(\frac{r_2}{s} \right) B + C = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.040008em;vertical-align:-0.686em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">s</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3540079999999999em"><span style="top:-3.6029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.8359999999999999em;vertical-align:-0.686em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">s</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></div><p>Em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>P</mi><mrow><mo fence="true">(</mo><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>K</mi><mn>3</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><msub><mi>r</mi><mn>1</mn></msub><mspace linebreak="newline"></mspace><mi>B</mi><mo>=</mo><mn>2</mn><mi>P</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>2</mn></msub><mo>+</mo><msub><mi>K</mi><mn>3</mn></msub><msub><mi>K</mi><mn>4</mn></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">(</mo><msub><mi>K</mi><mn>2</mn></msub><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><msub><mi>K</mi><mn>3</mn></msub><mo fence="true">)</mo></mrow><mspace linebreak="newline"></mspace><mi>C</mi><mo>=</mo><mi>P</mi><mrow><mo fence="true">(</mo><msubsup><mi>K</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>K</mi><mn>4</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><msub><mi>K</mi><mn>1</mn></msub><msub><mi>K</mi><mn>4</mn></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>3</mn></msub><mo>=</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>4</mn></msub><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A = P \left( r_1^2 + K_3^2 \right) - V^2 r_1\\ B = 2P(r_1 K_2 + K_3 K_4) - V^2 \left(K_2 + K_1 K_3 \right)\\ C = P \left( K_2^2 + K_4^2 \right) - V^2 K_1 K_4\\ K_2 = -x_1 K_1 -x_2 x_m\\ K_3 = x_m + x_1\\ K_4 = r_1 K_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.2141179999999998em;vertical-align:-0.35001em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.0141079999999998em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.2141179999999998em;vertical-align:-0.35001em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.0141079999999998em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Esse modelo pode ser inserido na solução do <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a> seguindo os seguintes passos:</p><ol><li>As constantes <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">K_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">K_4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são inicialmente calculadas . Esses valores são mantidos constantes durante toda a solução;</li><li>Em cada iteração são calculados os coeficientes <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span></span></span></span></span> utilizando o valor atualizado de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>;</li><li>A equação quadrática é resolvida e dois valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span> são obtidos, em que o maior deles é escolhido por estar na região estável da característica toque-escorregamento do motor;</li><li>Utilizando o novo valor de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span>, a potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span>) é obtida. O vetor de potências é então atualizado e os procedimentos convencionais de solução do fluxo de potência são realizados.</li></ol><p>Os passos de 2 a 4 são repetidos até que se obtenha a convergência.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU, os motores de indução não são considerados no cálculo de curto-circuito.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="motor-de-indução-trifásico-no-estudo-de-estabilidade"></a>Motor de indução trifásico no estudo de estabilidade<a class="hash-link" href="#motor-de-indução-trifásico-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>Uma importante carga dinâmica são os motores de indução, uma vez que correspondem a uma parcela significativa das cargas presentes no sistema elétrico. O <a href="/PSP/pt/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">modelo da máquina de indução apresentado anteriormente</a>, a qual pode ser utilizada tanto como motor quanto como gerador, é bem estabelecida na literatura.</p><p>Como descrito na <a href="/PSP/pt/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">seção anterior</a>, a inicialização dessa máquina é realizada em conjunto com o fluxo de potência, visto que a potência reativa exigida pela máquina de indução é dependente dos parâmetros do motor, assim como a tensão do seu barramento. Essa abordagem é necessária, pois métodos convencionais conduzem a resultados errôneos em sistemas altamente carregados.</p><p>É necessário expressar a equação de movimento da máquina de indução em termos de torque e não potência, como é realizado com as máquinas síncronas. A simetria do rotor também faz com que a posição angular não seja importante e o escorregamento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">s</span></span></span></span></span>) é utilizado no lugar da velocidade (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span></span></span></span>), em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>=</mo><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><msub><mi mathvariant="normal">Ω</mi><mn>0</mn></msub><mo>−</mo><mi>ω</mi></mrow><msub><mi mathvariant="normal">Ω</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">s = frac{\Omega_0 - \omega}{\Omega_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mord mathdefault">a</span><span class="mord mathdefault">c</span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Desprezando as perdas por atrito e ventilação e a potência no eixo suave, as equações mecânicas do motor são expressas da seguinte forma:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>=</mo><mi>A</mi><mo>−</mo><mi>B</mi><mi>s</mi><mo>+</mo><mi>C</mi><msup><mi>s</mi><mn>2</mn></msup><mspace linebreak="newline"></mspace><msub><mi>T</mi><mi>e</mi></msub><mo>=</mo><mfrac><mrow><mi>R</mi><mi>e</mi><mrow><mo fence="true">{</mo><mover accent="true"><mi>E</mi><mo>˙</mo></mover><msup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mo>∗</mo></msup><mo fence="true">}</mo></mrow></mrow><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>s</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo fence="true">(</mo><msub><mi>T</mi><mi>m</mi></msub><mo>−</mo><msub><mi>T</mi><mi>e</mi></msub><mo fence="true">)</mo></mrow><mrow><mn>2</mn><mi>H</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">T_m = A -Bs + Cs^2\\ T_e = \frac{Re\left\{ \dot{E}\dot{I}^* \right\}}{\Omega_b}\\ \frac{ds}{dt} = \frac{\left( T_m - T_e \right)}{2H}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.02602em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.19002em"><span style="top:-2.464em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.38em"><span class="pstrut" style="height:3.15em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-4.1900200000000005em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">{</span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">}</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">T_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o torque mecânico;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">T_e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o torque elétrico;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span></span></span> é a inércia do conjunto motor - carga mecânica</li></ul><p>Os termos <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span></span></span></span></span> são termos que definem o comportamento do torque mecânico da carga de acordo com o escorregamento. O torque mecânico normalmente varia com a <strong>velocidade</strong>, podendo ser expressa proporcionalmente com a seguinte equação quadrática:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>∝</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>ω</mi><mo>+</mo><mi>c</mi><msup><mi>ω</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">T_m \propto a + b\omega + c\omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">b</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord mathdefault">c</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mo>∝</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>B</mi><mo>∝</mo><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>C</mi><mo>∝</mo><mi>c</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} A \propto a + b + c \\ B \propto b + 2c \\ C \propto c \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">c</span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">2</span><span class="mord mathdefault">c</span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>As equações elétricas do motor de indução de gaiola simples são baseadas no circuito equivalente da <a href="/PSP/pt/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">figura anterior</a>. De forma semelhante ao modelo transitório da máquina síncrona, o motor de indução pode ser modelado pelo circuito equivalente de Thevenin de tensão transitória <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>E</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">E&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> atrás de uma resistência do estator <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">r_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e uma reatância transitória <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">x&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>. A reatância transitória aparente de rotor bloqueado é dada por:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">x&#x27; = x_1 + \frac{x_2 x_m}{x_2 + x_m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.801892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.94356em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A constante de tempo de circuito aberto do rotor (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">T_0&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.24810799999999997em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4518920000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em"><span></span></span></span></span></span></span></span></span></span></span>) é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><mrow><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><msub><mi>r</mi><mn>2</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">T_0&#x27; = \frac{x_2 + x_m}{\Omega_b r_2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.09633em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603300000000002em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>E a reatância de circuito aberto é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">x_0 = x_1 + x_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Uma vez que as reatâncias não são afetadas pela posição do rotor, as EADs do motor de indução podem ser expressar diretamente por componentes reais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>) e imaginárias (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">m</span></span></span></span></span>). Portanto, a descrição completa desse modelo é representada pelo seguinte sistema de equações algébrico-diferenciais:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>r</mi></msub><mo>−</mo><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>I</mi><mi>r</mi></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>I</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><msub><mi>V</mi><mi>m</mi></msub><mo>−</mo><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>I</mi><mi>m</mi></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>I</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mi>s</mi><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><mfrac><mrow><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>m</mi></msub></mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mi>s</mi><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><mfrac><mrow><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>i</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>r</mi></msub></mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac></mrow><annotation encoding="application/x-tex">V_r - E_r&#x27; = r_1 I_r - x&#x27; I_m\\ V_m - E_m&#x27; = r_1 I_m - x&#x27; I_m\\ \frac{dE_r&#x27;}{dt} = \Omega_b s E_m&#x27; - \frac{E_r&#x27; + \left( x_0 - x&#x27; \right) I_m}{T_0&#x27;}\\ \frac{dE_m&#x27;}{dt} = \Omega_b s E_r&#x27; - \frac{E_m&#x27; i \left( x_0 - x&#x27; \right) I_r}{T_0&#x27;}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.9518920000000001em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.9518920000000001em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.3811999999999998em;vertical-align:-0.9523079999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.433692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9523079999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.3811999999999998em;vertical-align:-0.9523079999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.433692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9523079999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-geradores-síncronos"></a>Formulário de edição dos geradores síncronos<a class="hash-link" href="#formulário-de-edição-dos-geradores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos motores de indução:</p><div><center><img src="/PSP/pt/images/indMotorForm.png" alt="Formulário dos motores de indução no PSP-UFU" title="Formulário dos motores de indução no PSP-UFU"></center></div><p>No formulário pode ser observado o botão &quot;Chaveamento&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do motor durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/pt/images/indMotorSw.png" alt="Formulário de chaveamento do motores de indução" title="Formulário de chaveamento do motores de indução"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Chaveamento</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do gerador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) e reativa (inserida em var, kvar, Mvar ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) do motor.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso a opção “Calcular a potência reativa no fluxo de carga” esteja ativada, o campo de potência reativa é desativado para edição.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="calcular-a-potência-reativa-no-fluxo-de-carga"></a>Calcular a potência reativa no fluxo de carga<a class="hash-link" href="#calcular-a-potência-reativa-no-fluxo-de-carga" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar os dados fornecidos no formulário de estabilidade para calcular a potência reativa do motor durante o processo iterativo do fluxo de carga.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Caso essa opção não seja utilizada o motor será considerado uma <a href="/PSP/pt/docs/load">carga de potência constante</a> no estudo de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>.</p><p>A não utilização dessa opção poderá gerar erros de regime permamente no estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a potência base do sistema.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-dados-da-máquina-de-indução"></a>Imprimir dados da máquina de indução<a class="hash-link" href="#imprimir-dados-da-máquina-de-indução" title="Direct link to heading">#</a></h4><p>Exibe os dados do gerador síncrono nos gráficos no tempo. Os seguintes dados são exibidos:</p><ul><li>tensão terminal</li><li>potências ativa e reativas</li><li>corrente, torque elétrico</li><li>torque mecânico</li><li>velocidade</li><li>escorregamento</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inércia"></a>Inércia<a class="hash-link" href="#inércia" title="Direct link to heading">#</a></h4><p>Inércia do conjunto motor-carga, em s.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-e-reatância-do-estator"></a>Resistência e reatância do estator<a class="hash-link" href="#resistência-e-reatância-do-estator" title="Direct link to heading">#</a></h4><p>Resistência e reatância do estator do motor de indução, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-e-reatância-do-rotor"></a>Resistência e reatância do rotor<a class="hash-link" href="#resistência-e-reatância-do-rotor" title="Direct link to heading">#</a></h4><p>Resistência e reatância do rotor do motor de indução, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatância-de-magnetização"></a>Reatância de magnetização<a class="hash-link" href="#reatância-de-magnetização" title="Direct link to heading">#</a></h4><p>Reatância do ramo de magnetização do motor de indução, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-fator-de-gaiola"></a>Utilizar fator de gaiola<a class="hash-link" href="#utilizar-fator-de-gaiola" title="Direct link to heading">#</a></h4><p>Permite a inserção de um fator de gaiola (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">K_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) no motor de indução. Tal fator eleva o valor da resistência do rotor em condições de grande escorregamento, como ocorre em motores de gaiola dupla ou gaiola profunda. A resistência do rotor segue a seguinte equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>2</mn></msub><mo>=</mo><msub><mi>r</mi><msub><mn>2</mn><mn>0</mn></msub></msub><mrow><mo fence="true">(</mo><mn>1</mn><mo>+</mo><msub><mi>K</mi><mi>g</mi></msub><mi>s</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">r_2 = r_{2_0} \left( 1+K_g s \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><msub><mn>2</mn><mn>0</mn></msub></msub></mrow><annotation encoding="application/x-tex">r_{2_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68066em;vertical-align:-0.2501em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em"><span></span></span></span></span></span></span></span></span></span></span> é a resistência do rotor inserida no formulário.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="característica-da-carga"></a>Característica da carga<a class="hash-link" href="#característica-da-carga" title="Direct link to heading">#</a></h4><p>Descreve a característica do torque da carga mecânica no eixo do motor seguindo a equação do torque mecânico.</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>∝</mo><mi mathvariant="bold">a</mi><mo>+</mo><mi mathvariant="bold">b</mi><mi>ω</mi><mo>+</mo><mi mathvariant="bold">c</mi><msup><mi>ω</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">T_m \propto \bold{a} + \bold{b}\omega + \bold{c}\omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord"><span class="mord mathbf">a</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord"><span class="mord mathbf">b</span></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">c</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A soma dos coeficientes (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a + b + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">c</span></span></span></span></span>) deve ser unitária.</p></div></div></div><div role="tabpanel" hidden=""><p>O botão &quot;Chaveamento&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do gerador durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>SÁNCHEZ, J. C.; OLIVARES, T. I. A.; ORTIZ, G. R.; VEGA, D. R. Induction Motor Static Models for Power Flow and Voltage Stability Studies. In: IEEE Power and Energy Society General Meeting, 2012, San Diego. doi: <a href="https://doi.org/10.1109/PESGM.2012.6345618" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/PESGM.2012.6345618</a></li><li>IEEE Std 399-1997. IEEE Recommended Practice for Industrial and Commercial Power Systems Analysis (Brown Book). IEEE, New York, ago. 1998. doi: <a href="https://doi.org/10.1109/IEEESTD.1998.88568" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/IEEESTD.1998.88568</a></li><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/indMotor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/syncMotor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Compensador Síncrono</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/controlEditor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Editor de Controle »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#motor-de-indução-trifásico-no-psp-ufu" class="table-of-contents__link">Motor de indução trifásico no PSP-UFU</a><ul><li><a href="#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga" class="table-of-contents__link">Motor de indução trifásico no cálculo do fluxo de carga</a></li><li><a href="#motor-de-indução-trifásico-no-estudo-de-estabilidade" class="table-of-contents__link">Motor de indução trifásico no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-dos-geradores-síncronos" class="table-of-contents__link">Formulário de edição dos geradores síncronos</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">PSP-UFU</h1></header><div class="markdown"><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="sobre-o-psp-ufu"></a>Sobre o PSP-UFU<a class="hash-link" href="#sobre-o-psp-ufu" title="Direct link to heading">#</a></h2><p>O PSP-UFU (Plataforma de Sistemas de Potência da Universidade Federal de Uberlândia) é um software <strong>multi-plataforma</strong>, <strong>multilíngue</strong>, <strong>livre e de código aberto (FOSS)</strong> com <strong>recursos avançados de GUI (<em>Graphical User Interface</em>)</strong> e <strong>ferramentas CAD (<em>Computer Aided Design</em>)</strong> para estudos de sistemas elétricos de potência.</p><p>O software permite a construção de qualquer rede de transmissão elétrica e sistemas de controle através da inserção de elementos visuais. Para a visualização dos resultados, o programa oferece <strong>elementos de texto vinculados</strong> na tela principal e também <strong>editores de tabelas e gráficos</strong>.</p><p>O PSP-UFU tem como objetivo fornecer ferramentas eficientes de simulação para fins de <strong>pesquisa</strong> e <strong>educação</strong>, além de <strong>aplicações industriais</strong> em sistemas elétricos de potência.</p><p>Resumidamente, o <em>software</em> pode executar os seguintes estudos:</p><ul><li>Fluxo de potência</li><li>Cálculo de curto-circuito</li><li>Harmônicos</li><li>Estabilidade transitória e dinâmica</li></ul><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="publicações"></a>Publicações<a class="hash-link" href="#publicações" title="Direct link to heading">#</a></h3><p>Os artigos abaixo possuem maiores detalhes a respeito do PSP-UFU:</p><blockquote><p>Oliveira, T. L., Guimarães, G. C., &amp; Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. <em>International Transactions on Electrical Energy Systems</em>, e12185. doi: <a href="https://doi.org/10.1002/2050-7038.12185" target="_blank" rel="noopener noreferrer">10.1002/2050-7038.12185</a></p></blockquote><blockquote><p>Oliveira, T. L., Guimarães, G. C., Silva, L. R., &amp; Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. <em>The International Journal of Electrical Engineering &amp; Education</em>, 0020720919879058. doi: <a href="https://doi.org/10.1177/0020720919879058" target="_blank" rel="noopener noreferrer">10.1177/0020720919879058</a></p></blockquote><p>A Tese de Doutorado abaixo detalha em profundidade os cálculos envolvidos, arquitetura de software e ferramentas implementadas no PSP-UFU:</p><blockquote><p>Oliveira, T. L. (2019). Desenvolvimento de um programa computacional livre, gráfico, e multiplataforma para analisar sistemas elétricos de potência em regime permanente e dinâmico. doi: <a href="https://dx.doi.org/10.14393/ufu.te.2019.2444" target="_blank" rel="noopener noreferrer">10.14393/ufu.te.2019.2444</a></p></blockquote><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="contribuindo-com-o-projeto"></a>Contribuindo com o projeto<a class="hash-link" href="#contribuindo-com-o-projeto" title="Direct link to heading">#</a></h3><p>Se você está interessado em contribuir com o PSP-UFU de alguma forma (desenvolvimento, críticas, sugestões, etc.), entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="desenvolvedores"></a>Desenvolvedor<del>es</del><a class="hash-link" href="#desenvolvedores" title="Direct link to heading">#</a></h3><p><a href="https://github.com/Thales1330" target="_blank" rel="noopener noreferrer">Thales Lima Oliveira</a></p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/aboutPSP.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/installation"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Instalação »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#sobre-o-psp-ufu" class="table-of-contents__link">Sobre o PSP-UFU</a><ul><li><a href="#publicações" class="table-of-contents__link">Publicações</a></li><li><a href="#contribuindo-com-o-projeto" class="table-of-contents__link">Contribuindo com o projeto</a></li><li><a href="#desenvolvedores" class="table-of-contents__link">Desenvolvedor<del>es</del></a></li></ul></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Indutor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><p>Um reator destinado à conexão de derivação em uma rede para compensar a corrente capacitiva. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=421-01-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="indutor-no-psp-ufu"></a>Indutor no PSP-UFU<a class="hash-link" href="#indutor-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento indutor (ou reator) representa, geralmente, reator <em>shunt</em> no circuito do PSP-UFU.</p><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-indutores"></a>Formulário de edição dos indutores<a class="hash-link" href="#formulário-de-edição-dos-indutores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos indutores:</p><div><center><img src="/PSP/pt/images/indForm.png" alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU"></center></div><p>Além do único contexto geral, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do indutor durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/pt/images/indSw.png" alt="Formulário de chaveamento do indutor" title="Formulário de chaveamento do indutor"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-reativa"></a>Potência reativa<a class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base de potência do sistema). </p></div><div role="tabpanel" hidden=""><p>O botão &quot;Estabilidade&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/inductor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/capacitor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Capacitor</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/harmSource"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Fonte de Corrente Harmônica »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#indutor-no-psp-ufu" class="table-of-contents__link">Indutor no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-indutores" class="table-of-contents__link">Formulário de edição dos indutores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Instalação</h1></header><div class="markdown"><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="download"></a>Download<a class="hash-link" href="#download" title="Direct link to heading">#</a></h2><p>O download do PSP-UFU pode ser realizado pelo <a href="https://github.com/Thales1330/PSP/releases/latest" target="_blank" rel="noopener noreferrer">&quot;<em>last release</em>&quot; do repositório do PSP-UFU no GitHub</a>.</p><p>Ao acessar ao link, role a página até encontrar os <em>Assets</em>:</p><div><center><img src="/PSP/pt/images/download.png" alt="Assets download" title="Informações da Figura"></center></div><p>Como o PSP-UFU é um <em>software</em> multiplataforma, você deve baixar o arquivo correto:</p><ul><li>Para o sistema operacional <a href="/PSP/pt/docs/installation#windows">Windows</a> baixe os arquivos <em>.exe</em> (recomendado) ou <em>.zip</em>.</li><li>Para os sistemas <a href="/PSP/pt/docs/installation#linux">Linux</a> baseados no Debian (por exemplo, Ubuntu, Mint) <em>64 bits</em> baixe o pacote de instalação <em>.deb</em>.</li></ul><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Se você está interessado em contribuir com o PSP-UFU construindo-o para outros sistemas operacionais (principalmente macOS!) entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><p>Na sequência são apresentados o passo-a-passo para instalação dos SOs <a href="/PSP/pt/docs/installation#windows">Windows</a> e <a href="/PSP/pt/docs/installation#linux">Linux</a>. Também são apresentadas as soluções para os erros mais recorrentes.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="windows"></a>Windows<a class="hash-link" href="#windows" title="Direct link to heading">#</a></h2><p>Após baixar o arquivo <em>.exe</em> avance em cada passo do instalador. O executável irá solicitar automaticamente a instalação do <strong>Microsoft Visual C++ Redistribuível</strong> e, caso não esteja instalado no seu computador, realize os procedimentos do programa.</p><p>Caso o executável falhe em abrir, baixe o arquivo <em>.zip</em> e descompacte em alguma pasta do seu sistema. Nesse caso é necessário baixar e instalar manualmente o <a href="https://aka.ms/vs/16/release/vc_redist.x86.exe" target="_blank" rel="noopener noreferrer"><strong>Microsoft Visual C++ Redistribuível 32 bits</strong></a>.</p><p>Após a conclusão da instalação abra o PSP-UFU para verificar a correta instalação. O executável está presente na pasta &quot;<em>&lt;local de instalação&gt;/PSP-UFU/bin/PSP-UFU.exe</em>&quot;.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="problemas-recorrentes-no-windows"></a>Problemas recorrentes no Windows<a class="hash-link" href="#problemas-recorrentes-no-windows" title="Direct link to heading">#</a></h3><ul><li>Ao baixar o navegador pode <strong>falsamente</strong> sinalizar que é um arquivo perigoso, basta clicar em &quot;Manter arquivo&quot;;</li><li>Ao abrir o instalador, novamente o Windows sinaliza que o arquivo é perigoso, basta clicar em &quot;Executar assim mesmo&quot;;</li><li>Ao abrir o programa e alterar o seu idioma, uma mensagem de erro é exibida. Para resolver esse problema basta executar o PSP-UFU como administrador (clicar com botão direito no atalho do programa e posteriormente em executar como administrador). Altere novamente o idioma e reinicie o programa;</li><li>Ao criar um novo projeto a tela de trabalho não é exibida ou é exibida uma tela preta. Vá nas opções gerais e altere o renderizador para &quot;Device Context&quot;. Caso apareça uma mensagem de erro, execute o programa como administrador e repita o processo. Reinicie o programa.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="linux"></a>Linux<a class="hash-link" href="#linux" title="Direct link to heading">#</a></h2><p>A instalação no sistema Linux é facilmente realizado pelo pacote de instalação automático <em>.deb</em>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Note que essa versão para Linux está defasada e algumas funcionalidades do programa podem não estar presentes.</p><p>Caso queira contribuir com o projeto e construir a versão atual no sistema Linux baseado em Debian ou em outras distribuições entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="problemas-recorrentes-no-linux"></a>Problemas recorrentes no Linux<a class="hash-link" href="#problemas-recorrentes-no-linux" title="Direct link to heading">#</a></h3><p>Pode ser necessário adicionar o <a href="https://wiki.wxwidgets.org/Compiling_and_getting_started#Running_wxWidgets_projects" target="_blank" rel="noopener noreferrer">diretório das bibliotecas do wxWidgets</a> nas variáveis de ambiente.
+Para isso, siga <strong>UM</strong> dos métodos apresentados abaixo:</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="método-i-recomendado"></a><em>Método I (recomendado)</em><a class="hash-link" href="#método-i-recomendado" title="Direct link to heading">#</a></h4><p>Insira o seguinte comando no arquivo <strong>~/.bashrc</strong>:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly shell"><div tabindex="0" class="prism-code language-shell codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">echo</span><span class="token plain"> </span><span class="token string" style="color:rgb(195, 232, 141)">&quot;export LD_LIBRARY_PATH=</span><span class="token string variable" style="color:rgb(191, 199, 213)">$LD_LIBRARY_PATH</span><span class="token string" style="color:rgb(195, 232, 141)">:/usr/local/lib&quot;</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;&gt;</span><span class="token plain"> ~/.bashrc</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="método-ii-não-recomendado"></a><em>Método II (não recomendado)</em><a class="hash-link" href="#método-ii-não-recomendado" title="Direct link to heading">#</a></h4><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Utilizando esse método os passos abaixo devem ser executados <strong>todas as vezes</strong> que você quiser abrir o PSP-UFU.</p></div></div><ol><li>Abra o terminal e acesse o diretório onde está instalado o PSP-UFU:</li></ol><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly shell"><div tabindex="0" class="prism-code language-shell codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">cd</span><span class="token plain"> /usr/local/bin</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><ol start="2"><li>Insira as bibliotecas compartilhadas do wxWidgets nas variáveis de ambiente:</li></ol><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly shell"><div tabindex="0" class="prism-code language-shell codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">export</span><span class="token plain"> </span><span class="token assign-left variable" style="color:rgb(191, 199, 213)">LD_LIBRARY_PATH</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token variable" style="color:rgb(191, 199, 213)">$LD_LIBRARY_PATH</span><span class="token plain">:/usr/local/lib</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><ol start="3"><li>Então, execute o PSP-UFU na <strong>mesma seção</strong> do terminal:</li></ol><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly shell"><div tabindex="0" class="prism-code language-shell codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">./PSP-UFU</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Verifique se o computador possui as unidades de vídeo instaladas corretamente, caso contrário, o programa não exibirá a área de trabalho para criar e editar diagramas de linha única ou diagramas de blocos de controle.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/installation.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav 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href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Entrada / Saída</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>As entradas e saída são variáveis relacionadas às máquinas síncronas. Cada tipo de entrada e saída está relacionada a um regulador da máquina: <strong>Regulador de Tensão (AVR, do inglês <em>Automatic Voltage Regulator</em>)</strong> e <strong>Regulador de Velocidade (RV)</strong>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O sistema de controle deve ter <strong>ao menos uma entrada e uma saída</strong>.</p></div></div><p>As entradas e saídas do sistema de controle são definidas por esses blocos, os quais são distintos para cada tipo de escopo. A figura abaixo apresenta um exemplo de um formulário de edição das entradas e saídas de um AVR:</p><div><center><img src="/PSP/pt/images/ioForm.png" alt="Formulário de edição de entradas e saídas de sistemas de controle de um AVR" title="Formulário de edição de entradas e saídas de sistemas de controle de um AVR"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="variáveis-de-entrada-e-saída"></a>Variáveis de entrada e saída<a class="hash-link" href="#variáveis-de-entrada-e-saída" title="Direct link to heading">#</a></h2><p>O usuário deve sinalizar o tipo de bloco entre entrada e saída e, então, selecionar em uma lista a variável desejada. As seguintes variáveis de controle estão atualmente disponíveis no programa:</p><ul><li><p><strong>Tensão terminal</strong> (<em>entrada: AVR</em>): Módulo da tensão no barramento da máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, variável no tempo. Essa variável é normalmente utilizada no cálculo do erro da tensão de referência do AVR;</p></li><li><p><strong>Velocidade</strong> (<em>entrada: AVR e RV</em>): Velocidade da máquina síncrona, em rad/s, variável no tempo. Normalmente utilizada no cálculo do erro de velocidade nos reguladores de velocidade, além de entrada do PSS em AVRs;</p></li><li><p><strong>Potência ativa e reativa</strong> (<em>entrada: AVR</em>): Potência ativa fornecida pela máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, variável no tempo. Normalmente utilizada como entrada do PSS (potência ativa) e controle de sub e sobrecorrente de excitação nos AVRs;</p></li><li><p><strong>Tensão terminal inicial</strong> (<em>entrada: AVR</em>): Módulo da tensão no barramento da máquina síncrona prévia ao estudo dinâmico originado do fluxo de carga, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, fixo no tempo. Essa variável está normalmente associada à referência de tensão do AVR;</p></li><li><p><strong>Velocidade inicial</strong> (<em>entrada: AVR e RV</em>): Velocidade do sistema (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>π</mi><msub><mi>f</mi><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">2 \pi f_{ref}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03588em">π</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>), definida nas <a href="/PSP/pt/docs/simulationConfig">opções de simulação</a>, em rad/s, fixa no tempo. Normalmente utiliza-se essa variável como referência de velocidade em RVs e normalização da velocidade;</p></li><li><p><strong>Potência mecânica inicial</strong> (<em>entrada: RV</em>): Potência mecânica inicial, calculada após a inicialização das máquinas síncronas com os dados originados do fluxo de carga, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, fixa no tempo. Normalmente é utilizada como referência de potência mecânica nos reguladores de velocidade;</p></li><li><p><strong>Variação de velocidade e potência ativa</strong> (<em>entrada: AVR</em>): Cálculo da variação dessas entradas entre os passos de integração normalizada pelo passo de integração, conforme a equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>−</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">\Delta x = \frac{x_n-x_{n-1}}{h}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.9463300000000001em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">h</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span>	é a variação da entrada (velocidade ou potência ativa);</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>	é a variável no passo atual e anterior, respectivamente;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	é o passo de integração.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A normalização é necessária para a correta utilização da razão de passo de controle, definida nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a>.</p></div></div></li><li><p><strong>Tensão de campo</strong> (<em>saída: AVR</em>): Define a tensão aplicada ao campo na máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> Utilizada como saída dos AVRs, controlando principalmente tensão no barramento conectado e/ou fator de potência da máquina;</p></li><li><p><strong>Potência mecânica</strong> (<em>saída: RV</em>): Define qual a potência mecânica aplicada no eixo da máquina síncrona. Utilizada como saída dos reguladores de velocidade, controlando principalmente a potência ativa injetada pela máquina e sua frequência.</p></li></ul></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/io.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/controlEditor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Editor de Controle</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/transferFunction"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Função Transferência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#variáveis-de-entrada-e-saída" class="table-of-contents__link">Variáveis de entrada e saída</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Limitador absoluto</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Os blocos de limite absoluto cumprem a função de restringir os valores de entrada de acordo com os limites máximos e mínimos inseridos pelo usuário. Caso a entrada ultrapasse algum dos limites impostos, a saída será igual a esse valor limite até que a entrada retorne para a faixa permitida.</p><p>As restrições máximas e mínimas de valores dos sistemas de controle são modeladas utilizando um Limitador, cuja implementação é bastante simples e segue a seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>u</mi><mi>n</mi></msub><mo>&gt;</mo><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>u</mi><mi>n</mi></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub><mo>≤</mo><msub><mi>u</mi><mi>n</mi></msub><mo>≤</mo><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>u</mi><mi>n</mi></msub><mo>&lt;</mo><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">y_n = \begin{cases} L_{sup}{,} &amp; \text{se } u_n &gt; L_{sup} \\ u_n{,} &amp; \text{se } L_{inf} \le u_n \le L_{sup}\\ L_{inf}{,} &amp; \text{se } u_n &lt; L_{inf} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	Éé o valor de saída atual do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{sup}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{inf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>	são os limites superior e inferior, respectivamente</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor de entrada atual do bloco</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-do-bloco-limitador"></a>Formulário de edição do bloco Limitador<a class="hash-link" href="#formulário-de-edição-do-bloco-limitador" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco limitador.</p><div><center><img src="/PSP/pt/images/limiterForm.png" alt="Formulário de edição de dados do bloco limitador no PSP-UFU" title="Formulário de edição de dados do bloco limitador no PSP-UFU"></center></div><p>Esse bloco não linear é definido pelo <strong>limite superior e inferior</strong>, inseridos pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tais blocos são bastante encontrados nos sistemas de controle para limitar a atuação do 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class="menu__link" tabindex="-1" href="/PSP/pt/docs/installation">Instalação</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mainScreen">Tela Principal</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/cadTools">Ferramentas CAD</a></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Editor de Elementos de Potência</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/powerEditor">Editor de Potência</a></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!" tabindex="0">Eelementos de Potência</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/bus">Barramento</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" 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href="/PSP/pt/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Linha</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><p>Um meio de transmissão fabricado usado para transmitir energia eletromagnética entre dois pontos com um mínimo de radiação. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=704-02-02" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="linha-no-psp-ufu"></a>Linha no PSP-UFU<a class="hash-link" href="#linha-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As linhas no PSP-UFU são modelos <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> equilibradas. Podem ser utilizadas como linhas de transmissão e distribuição de sistemas elétricos de potência.</p><p>A figura abaixo mostra o modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> implementado no PSP-UFU:</p><div><center><img src="/PSP/pt/images/lineModel.svg" alt="Modelo PI da linha" title="Modelo PI da linha"></center></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mi mathvariant="bold">L</mi></msub></mrow><annotation encoding="application/x-tex">\bold{r_L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a resistência da linha;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">L</mi></msub></mrow><annotation encoding="application/x-tex">\bold{x_L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a reatância indutiva da linha;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">b</mi><mi mathvariant="bold">L</mi><mi mathvariant="bold">d</mi></msubsup></mrow><annotation encoding="application/x-tex">\bold{b_{L}^{d}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.126385em;vertical-align:-0.277277em"></span><span class="mord"><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.4227230000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.277277em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a susceptância capacitiva <em>shunt</em> da linha.</li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O modelo utilizado no PSP-UFU é <strong>equilibrado</strong> e <strong>não possui impedâncias mútuas</strong> entre as fases.</p></div></div><p>A linha pode ser inseridas com pontos de ancoragem, ou &quot;nós&quot;, para maior personalização gráfica do elemento, como apresentado nas <a href="/PSP/pt/docs/cadTools#linha">Ferramentas CAD</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A linha deve ser inserida entre duas barras de <strong>mesma tensão nominal</strong>. Caso você tente inseri-la entre barras de tensão diferentes, uma mensagem de erro será exibida.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/pt/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/pt/docs/bus">barramento</a> e um <a href="/PSP/pt/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/pt/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-das-linhas"></a>Formulário de edição das linhas<a class="hash-link" href="#formulário-de-edição-das-linhas" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das linhas de transmissão:</p><div><center><img src="/PSP/pt/images/lineForm.png" alt="Formulário das linhas no PSP-UFU" title="Formulário das linhas no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais da linha e informações do fluxo de carga;</li><li><strong>Falta</strong>: local onde as impedâncias de sequência zero são inseridas.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os parâmetros necessários para construção da linha segundo seu modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> são inseridos na aba Geral, utilizados para construção da matriz admitância de sequência positiva e negativa.</p><p>Dados adicionais de impedâncias de sequência zero necessário para o cálculo de curtos-circuitos desbalanceados são editados na aba Falta, utilizados na construção da matriz admitância de sequência zero.</p></div></div><p>Além desses dois contextos, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da linha durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/pt/images/lineSw.png" alt="Formulário de chaveamento da linha" title="Formulário de chaveamento da linha"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-nominal"></a>Tensão nominal<a class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Campo de informação <em>não editável</em> que apresenta a tensão nominal da linha. Para alterar esse campo é necessário editar o campo correspondente do <a href="/PSP/pt/docs/bus#geral">barramento</a> conectado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal da linha, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="parâmetros-do-modelo-pi"></a>Parâmetros do modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span><a class="hash-link" href="#parâmetros-do-modelo-pi" title="Direct link to heading">#</a></h4><p>Resistência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">r_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), Reatância indutiva (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">x_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) série e Susceptância capacitiva shunt <strong>total</strong> (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><msubsup><mi>b</mi><mi>L</mi><mi>d</mi></msubsup></mrow><annotation encoding="application/x-tex">2b_{L}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.124439em;vertical-align:-0.275331em"></span><span class="mord">2</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.424669em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em"><span></span></span></span></span></span></span></span></span></span></span>) presentes no modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> da linha.</p><p>Esses parâmetros são inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">\Omega/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">Ω</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">S/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>b</mi><mi>L</mi><mi>d</mi></msubsup></mrow><annotation encoding="application/x-tex">b_{L}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.124439em;vertical-align:-0.275331em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.424669em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em"><span></span></span></span></span></span></span></span></span></span></span>).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="comprimento-da-linha"></a>Comprimento da linha<a class="hash-link" href="#comprimento-da-linha" title="Direct link to heading">#</a></h4><p>Utilizado para calcular o valor dos parâmetros da linha inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">\Omega/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">Ω</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> (ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">S/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span>).</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O comprimento da linha é ignorado caso não sejam utilizadas as unidades por quilometro.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal da linha como base para a conversão das unidades, caso contrário será usada a <a href="/PSP/pt/docs/simulationConfig">potência base do sistema</a>.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="parâmetros-do-modelo-pi-de-sequência-zero"></a>Parâmetros do modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> de sequência zero<a class="hash-link" href="#parâmetros-do-modelo-pi-de-sequência-zero" title="Direct link to heading">#</a></h4><p>Esses parâmetros são necessários para o correto cálculo das correntes de <a href="/PSP/pt/docs/fault">falta desbalanceadas</a> (fase-fase, fase-fase-terra e fase-terra) e devem ser inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A não inserção desses dados acarretarão em resultados incorretos para <a href="/PSP/pt/docs/fault">faltas desbalanceadas</a>.</p></div></div></div><div role="tabpanel" hidden=""><p>O botão &quot;Estabilidade&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção da linha durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/line.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/bus"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Barramento</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/transformer"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Transformador »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#linha-no-psp-ufu" class="table-of-contents__link">Linha no PSP-UFU</a></li><li><a href="#formulário-de-edição-das-linhas" class="table-of-contents__link">Formulário de edição das linhas</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Carga</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><ol><li>a potência ativa, reativa ou aparente gerada, transmitida ou distribuída dentro de um sistema;</li><li>a potência demandada por um grupo de consumidores classificados de acordo com suas particularidades e características, por exemplo, carga de aquecimento, carga reativa diurna, etc. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=601-01-15" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</li></ol></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="carga-no-psp-ufu"></a>Carga no PSP-UFU<a class="hash-link" href="#carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As cargas são os elementos de potência consumidores de potência genéricos. Nos estudos de fluxo de carga é possível modelar a carga como <strong>potência ou impedância constante</strong> e em estabilidade pode-se compor a carga na proporção desejada em três tipos (<strong>carga ZIP</strong>): potência constante, impedância constante e corrente constante.</p><p>As cargas de potência constante nos estudos de fluxo de carga são inseridos no vetor de potências e permanecem fixos durante toda a simulação.</p><p>Os elementos shunt, como reatores, banco de capacitores e cargas de impedância constante, são representados simplesmente por suas admitâncias em derivação. Como os parâmetros de entrada geralmente se constituem de suas potências ativa (P) e reativa (Q) nominais (no caso de reatores e capacitores, P=0), deve-se, portanto, encontrar sua admitância com a equação abaixo para inseri-la na matriz admitância:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mfrac><mrow><mi>P</mi><mo>−</mo><mi>j</mi><mi>Q</mi></mrow><msup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\overline{y} = \frac{P-jQ}{\dot{V}^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.825em;vertical-align:-0.19444em"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.17052em;vertical-align:-0.8101900000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em"><span style="top:-2.18981em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord mathdefault">Q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8101900000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Para os estudos de <strong>curto-circuito</strong> e <strong>harmônicos</strong> as cargas são modeladas como impedância constante, independentemente da sinalização indicada em seu formulário de edição de dados.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="carga-no-estudo-de-estabilidade"></a>Carga no estudo de estabilidade<a class="hash-link" href="#carga-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A modelagem das cargas do sistema de forma exata se torna impraticável em um problemas de estabilidade, visto à grande quantidade e variedade dos componentes envolvidos, além de apresentar modificações em suas composições causado por vários fatores, como tempo, condições climáticas e economia. Portanto várias aproximações devem ser utilizadas na formulação de um modelo de carga nos estudos aqui postos.</p><p>Uma forma interessante de se representar cargas dependentes da tensão são as cargas ZIP, as quais são funções quadráticas, possuindo três parcelas: impedância constante (Z), corrente constante (I) e potência constante (P). As potências ativa e reativa são obtidas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>L</mi></msub><mo>=</mo><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><mrow><msub><mi>V</mi><mi>i</mi></msub><mn>0</mn></mrow></mfrac><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_L=P_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+P_{I0} \left(\frac{V_i}{V_i0} \right)+P_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.604038em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.6540080000000001em"><span style="top:-3.9029000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>L</mi></msub><mo>=</mo><msub><mi>Q</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>Q</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><mrow><msub><mi>V</mi><mi>i</mi></msub><mn>0</mn></mrow></mfrac><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>Q</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_L=Q_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+Q_{I0} \left(\frac{V_i}{V_i0} \right)+Q_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.604038em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.6540080000000001em"><span style="top:-3.9029000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">P_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">Q_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	são as potências ativa e reativa da carga ZIP, respectivamente, conectadas em uma barra genérica i;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão atual de uma barra genérica i;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">V_{i0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão inicial de uma barra genérica i, obtida do estudo de fluxo de carga;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{Z0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{Z0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de impedância constante para potências ativa e reativa, respectivamente;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{I0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{I0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de corrente constante para potências ativa e reativa, respectivamente;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de potência constante para potências ativa e reativa, respectivamente.</p><p>Os valores das parcelas de potência da carga ZIP são obtidos utilizando as potências da carga após a convergência do cálculo do fluxo de carga (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{L0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, para a potência ativa e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{L0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, para a potência reativa), utilizando as seguintes expressões:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>Z</mi></mrow></msub><mn>100</mn></mfrac><mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub><msubsup><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow><mn>2</mn></msubsup></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>I</mi></mrow></msub><mn>100</mn></mfrac><mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>P</mi></mrow></msub><mn>100</mn></mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mstyle></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} P_{Z0}=\displaystyle \frac{k_{PZ}}{100} \frac{P_{L0}}{V_{i0}^2}\\ P_{I0}=\displaystyle \frac{k_{PI}}{100} \frac{P_{L0}}{V_{i0}}\\ P_{P0}=\displaystyle \frac{k_{PP}}{100} P_{L0} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.60004em;vertical-align:-3.05002em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5500200000000004em"><span style="top:-0.9999899999999999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-0.99499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.2899900000000002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.5849900000000003em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.8799900000000005em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.1749900000000006em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.2049900000000004em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.1500100000000004em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.29501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.59001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.885010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.180010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.475010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.50501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.80002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.05002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5495919999999996em"><span style="top:-5.549592em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9628639999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.215288em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.007848em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.0495919999999996em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PZ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PP}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> representam a composição em impedância, corrente e potência constantes, respectivamente, da parcela ativa da carga, os quais são inseridos pelo usuário na forma de porcentagem, sendo a soma desses três valores necessariamente igual a 100%. Os parâmetros <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QZ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QP}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> podem ser interpretados de forma análoga, porém compõe a parcela reativa da carga.</p><p>O comportamento das potências, retiradas de simulações no PSP-UFU, de cada parcela que compõe a carga ZIP são evidenciadas na figura abaixo.</p><div><center><img src="/PSP/pt/images/loadModel.svg" alt="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente" title="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente"></center></div><p>Cargas de corrente e potência constantes possuem problemas em tensões muito baixas. À medida que a tensão diminui as correntes dessas cargas não reduz, como pode ser observado na figura acima, resultando em perda de precisão e problemas na convergência de processos iterativos. Para contornar esse problema utiliza-se uma tensão pré-definida pelo usuário (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>l</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow><annotation encoding="application/x-tex">V_{low}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02691em">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), a qual as cargas (ou parcelas) de corrente e potência constantes são modeladas como impedância constante, resultando no comportamento de potência e corrente apresentados na figura acima.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-das-cargas"></a>Formulário de edição das cargas<a class="hash-link" href="#formulário-de-edição-das-cargas" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das cargas:</p><div><center><img src="/PSP/pt/images/loadForm.png" alt="Formulário das cargas no PSP-UFU" title="Formulário das cargas no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais das cargas, informações e o tipo de carga no fluxo de carga;</li><li><strong>Estabilidade</strong>: contendo opções de visualização de dados da carga em gráficos no tempo e opções de parametrização da carga ZIP.</li></ul><p>Além desses dois contextos, pode ser observado o botão &quot;Chaveamento&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da carga durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/pt/images/loadSw.png" alt="Formulário de chaveamento da carga" title="Formulário de chaveamento da carga"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Chaveamento</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-ativa"></a>Potência ativa<a class="hash-link" href="#potência-ativa" title="Direct link to heading">#</a></h4><p>Parcela de potência ativa da carga. Pode ser inserido em MW, kW W ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base do sistema).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-reativa"></a>Potência reativa<a class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Parcela de potência reativa da carga. Pode ser inserido em Mvar, kvar var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base do sistema).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tipo-de-carga-fluxo-de-carga"></a>Tipo de carga (fluxo de carga)<a class="hash-link" href="#tipo-de-carga-fluxo-de-carga" title="Direct link to heading">#</a></h4><p>Tipo da carga para o estudo de fluxo de carga, podendo ser selecionado dois tipos: Potência constante e Impedância constante. As cargas de potência constantes são inseridas nos vetores de potência e permanecem invariantes no cálculo, enquanto que para as cargas de impedância constante determina-se o valor da impedância utilizando a potência e tensão nominal, a qual é inserida na matriz admitância.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-dados"></a>Imprimir dados<a class="hash-link" href="#imprimir-dados" title="Direct link to heading">#</a></h4><p>Exibe os dados da carga nos gráficos no tempo.
+Os seguintes dados são exibidos:</p><ul><li>módulo da tensão</li><li>corrente</li><li>potências ativa e reativa.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-composição-de-carga-zip"></a>Utilizar composição de carga ZIP<a class="hash-link" href="#utilizar-composição-de-carga-zip" title="Direct link to heading">#</a></h4><p>Permite compor a carga ZIP para estudos de estabilidade. Caso essa opção não esteja selecionada é usada a composição global definida nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>Por padrão, todas as cargas no PSP-UFU são definidas como impedância constante no estudo de estabilidade.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso todas as barras possuírem o mesmo comportamento de carga ZIP, a parametrização poderá ser realizada nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a> para todas as cargas, sem a necessidade de parametrização individual.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="composição-de-potência-ativa-e-reativa"></a>Composição de potência ativa e reativa<a class="hash-link" href="#composição-de-potência-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Valores da composição da carga ZIP, inseridas para potência ativa e reativa separadamente, em porcentagem. A soma da composição deve ser igual a 100%, caso contrário será exibida uma mensagem de erro ao usuário.</p></div><div role="tabpanel" hidden=""><p>O botão &quot;Chaveamento&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/load.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/transformer"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Transformador</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/capacitor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Capacitor »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#carga-no-psp-ufu" class="table-of-contents__link">Carga no PSP-UFU</a><ul><li><a href="#carga-no-estudo-de-estabilidade" class="table-of-contents__link">Carga no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-das-cargas" class="table-of-contents__link">Formulário de edição das cargas</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="0" href="/PSP/pt/docs/installation">Instalação</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/pt/docs/mainScreen">Tela Principal</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/cadTools">Ferramentas CAD</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Editor de Elementos de Potência</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/powerEditor">Editor de Potência</a></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!" tabindex="-1">Eelementos de Potência</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/bus">Barramento</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" 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href="/PSP/pt/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Tela Principal</h1></header><div class="markdown"><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="organização-da-tela-principal"></a>Organização da tela principal<a class="hash-link" href="#organização-da-tela-principal" title="Direct link to heading">#</a></h2><p>A tela principal do PSP-UFU é dividida em três seções: <em><a href="/PSP/pt/docs/mainScreen#menu-ribbon">Menu &quot;Ribbon&quot;</a></em>, <em><a href="/PSP/pt/docs/mainScreen#%C3%A1rea-de-trabalho">Área de Trabalho</a></em> e <em><a href="/PSP/pt/docs/mainScreen#barra-de-status">Barra de Status</a></em>.</p><div><center><img src="/PSP/pt/images/mainScreen.svg" alt="Tela principal" title="Tela principal do PSP-UFU no Windows"></center></div><p>Resumidamente, no <strong>Menu Ribbon</strong> são acessadas as ferramentas do PSP-UFU; na <strong>Área de Trabalho</strong> são criados os diagramas unifilares de potência; e na <strong>Barra de Status</strong> são exibidas importantes informações acerca do estado da Área de Trabalho.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="menu-ribbon"></a>Menu Ribbon<a class="hash-link" href="#menu-ribbon" title="Direct link to heading">#</a></h3><blockquote><p>Em informática o ribbon (faixa, em inglês) é um formato de apresentação de interface baseada na GUI onde a barra de ferramentas é mostrada através de uma barra mais larga com icones maiores possibilitando o uso dos aplicativos por dispositivos <em>touch screen</em>.
+<a href="https://pt.wikipedia.org/wiki/Ribbon_(inform%C3%A1tica)" target="_blank" rel="noopener noreferrer">Wikipedia</a></p></blockquote><p>O menu Ribbon é composto de três submenus: “Arquivo”, “Ferramentas” e “Simulação”.</p><div><center><img src="/PSP/pt/images/menuFile.png" alt="Menu Arquivo" title="Menu Arquivo"></center></div><p>Em <strong>Arquivo</strong> estão as ferramentas gerais do programa, as quais permitem ao usuário criar e salvar novos projetos, além daquelas que possibilitam carregar os projetos já existentes e importar arquivos de outros programas. Esse submenu também possui opções gerais do programa e acesso a informações sobre o programa.</p><div><center><img src="/PSP/pt/images/menuTools.png" alt="Menu Ferramentas" title="Menu Ferramentas"></center></div><p>Em <strong>Ferramentas</strong> estão presentes os itens específicos, os quais estão relacionados à criação e manipulação da rede elétrica de potência, relatórios tabulares de saída de dados e acesso ao formulário de criação de gráficos no tempo. Algumas configurações específicas do projeto ativo são acessadas por esse menu.</p><div><center><img src="/PSP/pt/images/menuSimulation.png" alt="Menu Simulação" title="Menu Simulação"></center></div><p>Finalmente, o submenu <strong>Simulação</strong> proporciona o acesso do usuário aos cálculos realizados pelo <em>software</em> e suas configurações, além da ferramenta de “solução contínua”, a qual resolve os cálculos estáticos (fluxo de potência, curto-circuito e harmônicos) após quaisquer mudanças na rede de potência, como remoção de componente ou alteração de seus dados. Todas as configurações de simulação do projeto ativo são acessadas por esse menu.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="área-de-trabalho"></a>Área de Trabalho<a class="hash-link" href="#área-de-trabalho" title="Direct link to heading">#</a></h3><p>Na área de trabalho estão presentes os projetos em execução, os quais são divididos por abas identificadas pelo nome do projeto (caso já esteja gravado no disco). É nessa área que são inseridos e excluídos os elementos elétricos, cujas ferramentas são acessadas por meio do <a href="/PSP/pt/docs/mainScreen#menu-ribbon">Menu Ribbon</a>.</p><p>As edições gráficas dos elementos, como posição, conexão e tamanho dos barramentos, são realizadas por meio da ação <em>drag-and-drop</em> (arrastar e soltar) com a utilização do mouse. A edição dos dados elétricos é feita por meio de formulários de dados acessados clicando duas vezes sobre os elementos.</p><div><center><img src="/PSP/pt/images/workspace.png" alt="Área de Trabalho" title="Área de Trabalho"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barra-de-status"></a>Barra de Status<a class="hash-link" href="#barra-de-status" title="Direct link to heading">#</a></h3><p>A barra de status é responsável por informações interessantes sobre as circunstâncias atuais do programa: modo de operação do mouse (editar, mover, arrastar), zoom aplicado e posição do mouse na área de trabalho, além de dados sobre as ações dos usuários, como: inserção de elementos, informação sobre copiar e colar, etc.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p><strong>Sempre fique atendo à barra de status</strong>, pois são fornecidas informações importantes acerca da operação do programa. Caso tenha alguma dúvida siga as instruções apresentadas nesse componente.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="configurações-gerais"></a>Configurações gerais<a class="hash-link" href="#configurações-gerais" title="Direct link to heading">#</a></h2><p>As configurações gerais do programa são acessadas no submenu Ribbon <strong>Arquivo</strong>. Essas configurações são aplicadas para todos os projetos e permanecem gravadas no disco.</p><div><center><img src="/PSP/pt/images/generalSettings.png" alt="Configurações gerais" title="Configurações gerais"></center></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado</h5></div><div class="admonition-content"><p>Algumas configurações de segurança do seu computador (principalmente em sistemas Windows) podem gerar uma mensagem de erro ao confirmar as alterações das configurações gerais.</p><p>Para resolver esse problema basta <strong>executar o PSP-UFU como administrador</strong> (clicar com botão direito no atalho do programa e posteriormente em executar como administrador). Altere novamente as configurações e reinicie o programa.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="idioma"></a>Idioma<a class="hash-link" href="#idioma" title="Direct link to heading">#</a></h3><p>Atualmente os seguintes idiomas estão disponíveis no PSP-UFU:</p><ul><li>Inglês</li><li>Português</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O programa deve ser reiniciado para surtir efeito da alteração do idioma.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="renderização"></a>Renderização<a class="hash-link" href="#renderização" title="Direct link to heading">#</a></h3><p>Define como os elementos gráficos do <a href="/PSP/pt/docs/powerEditor">editor de potência</a> e do <a href="/PSP/pt/docs/controlEditor">editor de controle</a> são desenhados na tela. Atualmente duas opções estão disponíveis:</p><ul><li><a href="https://www.opengl.org/about/" target="_blank" rel="noopener noreferrer">OpenGL</a></li><li><a href="https://docs.wxwidgets.org/3.0/classwx_graphics_context.html" target="_blank" rel="noopener noreferrer">Device Context</a></li></ul><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>o OpenGL pode não ser suportado pelo seu computador. Nesse caso, ao criar um novo projeto a tela de trabalho não é exibida ou é exibida uma tela preta. A solução é alterar o renderizador para &quot;Device Context&quot;. </p><p>Caso apareça uma mensagem de erro, execute o programa como administrador e repita o processo.</p></div></div><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O programa deve ser reiniciado para surtir efeito de alteração do renderizador.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="arquivos-de-projeto-do-psp-ufu"></a>Arquivos de projeto do PSP-UFU<a class="hash-link" href="#arquivos-de-projeto-do-psp-ufu" title="Direct link to heading">#</a></h2><p>As opções de criação, gravação e abertura de projetos no disco, assim como importação de arquivos de outros programas estão presentes no submenu Ribbon <strong>Arquivo</strong>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="novo-projeto"></a>Novo projeto<a class="hash-link" href="#novo-projeto" title="Direct link to heading">#</a></h3><p>A criação de um novo projeto é realizada clicando no botão <strong>Novo projeto</strong> no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">submenu <strong>Arquivo</strong></a>. Essa ação cria um sistema em branco na área de trabalho, local onde é possível inserir os elementos elétricos por meio do <a href="/PSP/pt/docs/mainScreen#menu-ribbon">submenu <strong>Ferramentas</strong></a> ou pelas <a href="/PSP/pt/docs/powerEditor#teclas-de-atalho">teclas de atalho</a>.</p><p>Esse sistema em branco pode ser ciado utilizando o <a href="/PSP/pt/docs/powerEditor">Editor de Potência</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="salvar-e-salvar-como"></a>Salvar e Salvar como...<a class="hash-link" href="#salvar-e-salvar-como" title="Direct link to heading">#</a></h3><p>A opção &quot;Salvar&quot; sobrepõe as alterações realizadas no projeto aberto e grava no disco. A opção &quot;Salvar como...&quot; cria um novo arquivo e grava o projeto no disco com o auxílio de uma janela de seleção de pasta (e definição do nome do arquivo).</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Para projetos que estão sendo gravados pela primeira vez a opção &quot;Salvar&quot; se comporta de forma idêntica à opção &quot;Salvar como...&quot;.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>O PSP-UFU grava os arquivos com a extensão <em>.psp</em>. Esses arquivos nada mais são que arquivos de texto utilizando a <a href="https://en.wikipedia.org/wiki/Markup_language" target="_blank" rel="noopener noreferrer">linguagem de marcação</a> XML (<a href="https://en.wikipedia.org/wiki/XML" target="_blank" rel="noopener noreferrer">eXtensible Markup Language</a>), que define uma série de regras de formatação dos dados de forma que eles são tanto legíveis por humanos quanto por máquinas.</p><p>Portanto, os dados elétricos contidos neles podem ser facilmente identificados e alterados, caso necessário.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="abrir-projeto"></a>Abrir projeto<a class="hash-link" href="#abrir-projeto" title="Direct link to heading">#</a></h3><p>Essa opção abre os projetos gravados no disco por meio com o auxílio de uma janela de seleção de arquivos. </p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="importar-projeto"></a>Importar projeto<a class="hash-link" href="#importar-projeto" title="Direct link to heading">#</a></h3><p>O PSP-UFU permite a importação de arquivos dos seguintes programas:</p><ul><li><a href="http://www.cepel.br/pt_br/produtos/programas-computacionais-por-categoria/menu/anarede-analise-de-redes-eletricas.htm" target="_blank" rel="noopener noreferrer">ANAREDE</a></li><li><a href="https://matpower.org/" target="_blank" rel="noopener noreferrer">Matpower</a></li></ul><p>Para importação dos arquivos do <strong>ANAREDE</strong> são utilizados tanto o arquivo de dados elétricos (<em>.pwf</em>) quanto de dados gráficos dos elementos (<em>.lst</em>).</p><p>Uma vez que o arquivo do <strong>Matpower</strong> (<em>.m</em>) não possui dados gráficos dos elementos elétricos, o diagrama unifilar é automaticamente gerado pelo PSP-UFU ao importá-lo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="geração-automática-do-layout-de-diagramas-unifilares"></a>Geração automática do <em>layout</em> de diagramas unifilares<a class="hash-link" href="#geração-automática-do-layout-de-diagramas-unifilares" title="Direct link to heading">#</a></h4><p>Para criar o <em>layout</em> automático, utilizou-se a teoria dos grafos baseada no posicionamento direcionado à força e aplicação de grafos com arestas ponderadas. De acordo com a teoria dos grafos, o sistema é modelado como vértices e arestas. Nesse contexto, os barramentos são os vértices e os ramos do sistema (linhas e transformadores) são as arestas. As localizações de elementos de derivação, assim como os nós dos elementos, são automaticamente controladas pelo PSP-UFU.</p><p>A ferramenta de <em>layout</em> automático é composta por um processo iterativo e os resultados são mais refinados com um número maior de iterações. Os melhores resultados são obtidos utilizando o número de iterações igual ou superior a cinco vezes o número de barramentos do sistema.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O <em>layout</em> automático ainda está em desenvolvimento e algumas melhorias podem ser feitas para evitar cruzamentos e sobreposições, porém, em seu estado atual, é totalmente funcional e satisfatório.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Uma vez que o Matpower pode converter os formatos CDF (<em>Common Data Format</em>) e PSS/E RAW para arquivos <em>.m</em>, estes podem ser importados de forma indireta para o PSP-UFU.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a 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class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Expressão Matemática</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>O bloco de expressão matemática utiliza a biblioteca externa <a href="http://warp.povusers.org/FunctionParser/" target="_blank" rel="noopener noreferrer">fparser</a> e permite ao usuário inserir expressões matemáticas genéricas de qualquer complexidade combinadas a estruturas condicionais em um diagrama de controle, aumentando significativamente a capacidade de generalização do programa.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="sintaxe-das-expressões-matemáticas"></a>Sintaxe das expressões matemáticas<a class="hash-link" href="#sintaxe-das-expressões-matemáticas" title="Direct link to heading">#</a></h2><p>As sintaxes das operações (para expressões <code>A</code> e <code>B</code>) são apresentadas na tabela abaixo:</p><table><thead><tr><th>Operador</th><th>Expressão</th></tr></thead><tbody><tr><td><code>( )</code></td><td>Expressão em parênteses primeiro</td></tr><tr><td><code>A unit</code></td><td>Um multiplicador de unidade (se um estiver sido adicionado)</td></tr><tr><td><code>A^B</code></td><td>Exponenciação (<code>A</code> elevado à potência <code>B</code>)</td></tr><tr><td><code>-A</code></td><td>Operação unária de oposto</td></tr><tr><td><code>!A</code></td><td>Operação lógica unária de negação (resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> se <code>int(A)</code> é <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, senão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>)</td></tr><tr><td><code>A*B</code> <code>A/B</code> <code>A%B</code></td><td>Multiplicação, divisão e módulo</td></tr><tr><td><code>A+B</code> <code>A-B</code></td><td>Adição e subtração</td></tr><tr><td><code>A=B</code> <code>A&lt;B</code> <code>A&lt;=B</code> <code>A!=B</code> <code>A&gt;B</code> <code>A&gt;=B</code></td><td>Comparação entre <code>A</code> e <code>B</code> (resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>)</td></tr><tr><td><code>A&amp;B</code></td><td>Resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> se <code>int(A)</code> e <code>int(B)</code> forem diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, senão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></td></tr><tr><td><code>A</code>|<code>B</code></td><td>Resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> se <code>int(A)</code> ou <code>int(B)</code> forem diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, senão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></td></tr></tbody></table><p>As <strong>funções matemáticas</strong> suportadas pela biblioteca, as quais podem ser utilizadas na construção de diagrama de blocos, são descritas na tabela abaixo:</p><table><thead><tr><th>Expressão</th><th>Descrição</th></tr></thead><tbody><tr><td><code>abs(A)</code></td><td>Valor absoluto de <code>A</code>. Para números reais, se <code>A</code> é negativo, retorna <code>–A</code>, senão retorna <code>A</code>. Para números complexos a expressão é equivalente a <code>hypot(real(x),imag(x))</code>.</td></tr><tr><td><code>acos(A)</code></td><td>Arco cosseno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>acosh(A)</code></td><td>O mesmo que <code>acos()</code> mas para cosseno hiperbólico</td></tr><tr><td><code>arg(A)</code></td><td>Ângulo de fase de um número complexo <code>A</code>.</td></tr><tr><td><code>asin(A)</code></td><td>Arco seno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>asinh(A)</code></td><td>Mesmo que <code>asin()</code>, mas para seno hiperbólico</td></tr><tr><td><code>atan(A)</code></td><td>Arco tangente de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>atan2(A,B)</code></td><td>Arco tangente de <code>A/B</code>, a qual os sinais dos dois argumentos para determinar o quadrante do resultado. Retorna a solução de do seguinte sistema: <code>hypot(A,B)*sin(x)=A, hypot(A,B)*cos(x)=B</code>. O valor retornado varia de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>π</mi></mrow><annotation encoding="application/x-tex">-\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span>.</td></tr><tr><td><code>atanh(A)</code></td><td>Mesmo que <code>atan()</code>, mas para tangente hiperbólica.</td></tr><tr><td><code>cbrt(A)</code></td><td>Raiz cúbica de <code>A</code>.</td></tr><tr><td><code>conj(A)</code></td><td>Conjugado complexo de <code>A</code>.</td></tr><tr><td><code>ceil(A)</code></td><td>Teto de <code>A</code>. Arredonda para o próximo maior inteiro.</td></tr><tr><td><code>cos(A)</code></td><td>Cosseno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>cosh(A)</code></td><td>Mesmo que <code>cos()</code>, mas para cosseno hiperbólico</td></tr><tr><td><code>cot(A)</code></td><td>Cotangente de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>csc(A)</code></td><td>Cossecante de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>exp(A)</code></td><td>Exponencial de <code>A</code>. Retorna o valor de e elevado a potência A.</td></tr><tr><td><code>exp2(A)</code></td><td>Exponencial de <code>A</code> na base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span>.</td></tr><tr><td><code>floor(A)</code></td><td>Piso de <code>A</code>. Arredonda para o próximo menor inteiro.</td></tr><tr><td><code>hypot(A,B)</code></td><td>Função de distância Euclidiana.</td></tr><tr><td><code>if(A,B,C)</code></td><td>Se <code>int(A)</code> é diferente de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span>, retorna o valor de <code>B</code>, senão retorna <code>C</code>.</td></tr><tr><td><code>imag(A)</code></td><td>Retorna a parte imaginária do número complexo <code>A</code>.</td></tr><tr><td><code>int(A)</code></td><td>Arredonda <code>A</code> para o inteiro mais próximo.</td></tr><tr><td><code>log(A)</code></td><td>Logaritmo natural (base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">e</span></span></span></span></span>) de <code>A</code>.</td></tr><tr><td><code>log2(A)</code></td><td>Logaritmo na base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span> de <code>A</code>.</td></tr><tr><td><code>log10(A)</code></td><td>Logaritmo na base <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">0</span></span></span></span></span> de A.</td></tr><tr><td><code>max(A,B)</code></td><td>Se <code>A&gt;B</code>, o resultado é <code>A</code>, senão é <code>B</code>.</td></tr><tr><td><code>min(A,B)</code></td><td>Se <code>A&lt;B</code>, o resultado é <code>A</code>, senão é <code>B</code>.</td></tr><tr><td><code>polar(A,B)</code></td><td>Retorna o número complexo de magnitude <code>A</code> e ângulo de fase <code>B</code> (em radianos).</td></tr><tr><td><code>pow(A,B)</code></td><td>Exponenciação (<code>A</code> elevado à potência <code>B</code>)</td></tr><tr><td><code>real(A,B)</code></td><td>Retorna a parte real do número complexo <code>A</code>.</td></tr><tr><td><code>sec(A)</code></td><td>Secante de <code>A</code>.</td></tr><tr><td><code>sin(A)</code></td><td>Seno de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>sinh(A)</code></td><td>Mesmo que <code>sin()</code>, mas para seno hiperbólico.</td></tr><tr><td><code>sqrt(A)</code></td><td>Raiz quadrada de <code>A</code>.</td></tr><tr><td><code>tan(A)</code></td><td>Tangente de <code>A</code>. Retorna o ângulo em radianos.</td></tr><tr><td><code>tanh(A)</code></td><td>Mesmo que <code>tan()</code>, mas para tangente hiperbólica.</td></tr><tr><td><code>trunc(A)</code></td><td>Valor truncado de <code>A</code>. Retorna o número inteiro de <code>A</code> sem a parcela fracionada.</td></tr></tbody></table><p>É possível atribuir novas variáveis utilizando a seguinte sintaxe: </p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token operator" style="color:rgb(137, 221, 255)">&lt;</span><span class="token plain">nome da variável</span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">&lt;</span><span class="token plain">expressão</span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">&lt;</span><span class="token plain">função</span><span class="token operator" style="color:rgb(137, 221, 255)">&gt;</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Por exemplo:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">comprimento </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain">y</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">y</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">comprimento</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token function" style="color:rgb(130, 170, 255)">sin</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">comprimento</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>O caractere de espaço e de nova linha são ignorados na interpretação da expressão, portanto para maior organização, o código anterior pode ser escrito da seguinte forma:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">comprimento </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">x</span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain">y</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">y</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain"></span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain">comprimento</span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token function" style="color:rgb(130, 170, 255)">sin</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">comprimento</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Note que a expressão que determina o valor de saída do bloco <strong>não possui</strong> o caractere <code>;</code> em seu final.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-bloco-de-expressão-matemática"></a>Formulário de edição de dados do bloco de Expressão Matemática<a class="hash-link" href="#formulário-de-edição-de-dados-do-bloco-de-expressão-matemática" title="Direct link to heading">#</a></h2><p>O formulário de inserção e edição das entradas do bloco e da expressão matemática genérica, assim como ferramentas de auxílio de sua construção é apresentado na figura abaixo.</p><div><center><img src="/PSP/pt/images/mathExpressionForm.png" alt="Formulário de edição de dados do bloco de expressão matemática no PSP-UFU" title="Formulário de edição de dados do bloco de expressão matemática no PSP-UFU"></center></div><p>No campo de “Variáveis de entrada” é inserida uma lista com os nomes das entradas separados por espaços. Um número qualquer de entrada pode ser definido nessa lista esses nomes são apresentados no ícone gráfico presente no editor de controle, além de serem destacados na expressão inserida pelo usuário. O número de entradas e saídas se comporta de forma semelhante aos <a href="/PSP/pt/docs/sum">blocos somadores</a>.</p><p>Abaixo do campo das variáveis de entrada está presente o local para inserção da expressão matemática. A sintaxe da expressão inserida pelo usuário possui realce (por meio de diferentes formas e cores da fonte) para números, operadores, variáveis de entrada, funções e constantes, facilitando a criação, manipulação e identificação de erros de digitação e lógica.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Como ferramenta de auxílio ao usuário foi desenvolvida uma verificação da expressão inserida.</p></div></div><p>Tal ferramenta irá encontrar erros e indicará ao usuário qual o tipo do erro, além da sua localização, destacando-o. A figura anterior exemplifica a identificação automática do erro pelo PSP-UFU, assim como sua posição na expressão inserida.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="exemplo-de-expressão-matemática"></a>Exemplo de Expressão Matemática<a class="hash-link" href="#exemplo-de-expressão-matemática" title="Direct link to heading">#</a></h2><p>A corrente de campo em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> pode ser estimada de forma aproximada utilizando as potências ativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span>) e reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span>), assim como as reatâncias transitórias de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">x_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">x_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) e o módulo da tensão terminal (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>) da máquina:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>i</mi><mi>f</mi></msub><mo>≈</mo><msqrt><mrow><msup><mrow><mo fence="true">(</mo><mi>V</mi><mo>+</mo><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>P</mi><mn>2</mn></msup></mrow></msqrt><mo>+</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>x</mi><mi>d</mi></msub><msub><mi>x</mi><mi>q</mi></msub></mfrac><mo>+</mo><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mo fence="true">)</mo></mrow><mo>×</mo><mfrac><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mrow><mo fence="true">(</mo><mi>V</mi><mo>+</mo><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mo>+</mo><msup><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mn>2</mn></msup></mrow><msqrt><mrow><msup><mrow><mo fence="true">(</mo><mi>V</mi><mo>+</mo><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>P</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mrow><annotation encoding="application/x-tex">i_f \approx \sqrt{ \left( V + Q&#x27; \right)^2 + P^2} + \left(\frac{x_d}{x_q} +1{,}0 \right) \times \frac{Q&#x27; \left( V + Q&#x27; \right) + {P&#x27;}^2}{\sqrt{\left( V + Q&#x27;\right)^2 + P^2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9456279999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.454121em"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.385879em"><span class="svg-align" style="top:-3.8em"><span class="pstrut" style="height:3.8em"></span><span class="mord" style="padding-left:1em"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6778919999999999em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954008em"><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.345879em"><span class="pstrut" style="height:3.8em"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em"><svg width="400em" height="1.8800000000000001em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:3.3629em;vertical-align:-1.73em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6329em"><span style="top:-2.1100000000000003em"><span class="pstrut" style="height:3.3370040000000003em"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.337004em"><span class="svg-align" style="top:-3.8em"><span class="pstrut" style="height:3.8em"></span><span class="mord" style="padding-left:1em"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6778919999999999em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954008em"><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.297004em"><span class="pstrut" style="height:3.8em"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em"><svg width="400em" height="1.8800000000000001em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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+M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.502996em"><span></span></span></span></span></span></span></span><span style="top:-3.5670040000000003em"><span class="pstrut" style="height:3.3370040000000003em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-4.014004em"><span class="pstrut" style="height:3.3370040000000003em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9559em"><span style="top:-3.2047920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.73em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mi>q</mi></msub><mo>×</mo><mfrac><mi>P</mi><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">P&#x27; = x_q \times \frac{P}{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.217331em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.872331em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mi>q</mi></msub><mo>×</mo><mfrac><mi>Q</mi><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">Q&#x27; = x_q \times \frac{Q}{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.269439em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.924439em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.446108em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li></ul><p>Primeiramente, deve-se inserir no campo &quot;variáveis de entrada&quot; as grandezas fornecidas pelo PSP-UFU. Nesse caso, como pode ser observado no bloco de <a href="/PSP/pt/docs/io">entrada / saída</a>, todas as variáveis necessárias são fornecidas: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>. Essas variáveis devem ser inseridas separadas por espaço: <code>p q v</code>.</p><p>Utilizando três blocos de <a href="/PSP/pt/docs/io">entrada / saída</a> pode-se fornecer tais dados ao conectá-los ao bloco de expressão matemática.</p><p>As reatâncias transitórias de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">x_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">x_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) podem ser definidas diretamente na expressão matemática:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.146</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">xq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.0969</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">P&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">Q&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> podem ser também calculados:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">yp </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> p </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> q </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Com todos os dados necessários pode-se calcular a corrente de campo:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">1.0</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">       </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yp</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Finalmente, pode-se definir a saída do bloco (não inserindo <code>;</code>):</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><p>Portanto, para as entradas definidas como <code>p q v</code>, a expressão matemática total para cálculo da corrente de campo será:</p><div class="codeBlockContainer_K1bP"><div class="codeBlockContent_hGly cpp"><div tabindex="0" class="prism-code language-cpp codeBlock_23N8 thin-scrollbar"><div class="codeBlockLines_39YC" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.146</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">xq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.0969</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yp </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> p </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> q </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">1.0</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">       </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yp</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f</span></div></div></div><button type="button" aria-label="Copy code to clipboard" class="copyButton_Ue-o">Copy</button></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ul><li>NIEMINEN, J.; YLILUOMA, J. Function Parser for C++ v4.5.2. 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href="/PSP/pt/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Editor de Potência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>O Editor de Elementos de Potência no PSP-UFU permite a inserção, manipulação e edição de elementos elétricos de potência por meio de um diagrama unifilar.</p><blockquote><p>(<strong>Diagrama unifilar</strong>) Um diagrama do sistema no qual as linhas polifásicas são representados por sua única linha equivalente. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=601-02-04" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Todos os elementos inseridos pode ser manipulados e editados utilizando as <a href="/PSP/pt/docs/cadTools">Ferramentas CAD</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="novo-projeto"></a>Novo projeto<a class="hash-link" href="#novo-projeto" title="Direct link to heading">#</a></h2><p>A criação de um novo projeto é realizada clicando no item <strong>Novo projeto</strong> no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">submenu <strong>Arquivo</strong></a>. Essa ação cria um sistema em branco na área de trabalho, local onde é possível inserir os elementos elétricos por meio do <a href="/PSP/pt/docs/mainScreen#menu-ribbon">submenu <strong>Ferramentas</strong></a> ou pelas <a href="/PSP/pt/docs/powerEditor#teclas-de-atalho">teclas de atalho</a>.</p><p>Abaixo são apresentados os elementos elétricos de potência que podem ser inseridos no PSP-UFU:</p><ul><li><a href="/PSP/pt/docs/bus">Barramento</a></li><li>Elementos de ramo<ul><li><a href="/PSP/pt/docs/line">Linha</a></li><li><a href="/PSP/pt/docs/transformer">Transformador</a></li></ul></li><li>Elementos Shunt<ul><li><a href="/PSP/pt/docs/load">Carga</a></li><li><a href="/PSP/pt/docs/capacitor">Capacitor</a></li><li><a href="/PSP/pt/docs/inductor">Indutor</a></li><li><a href="/PSP/pt/docs/harmSource">Fonte de Corrente Harmônica</a></li></ul></li><li>Máquinas<ul><li><a href="/PSP/pt/docs/syncGenerator">Gerador Síncrono</a></li><li><a href="/PSP/pt/docs/syncMotor">Compensador Síncrono</a></li><li><a href="/PSP/pt/docs/indMotor">Motor de Indução</a></li></ul></li></ul><div><center><img src="/PSP/pt/images/powerElements.svg" alt="Elementos de potência" title="Elementos de potência"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserindo-elementos"></a>Inserindo elementos<a class="hash-link" href="#inserindo-elementos" title="Direct link to heading">#</a></h2><p>Por meio do submenu <strong>Ferramentas</strong> pode-se inserir os elementos de potência, como é apresentado na imagem abaixo.</p><div><center><img src="/PSP/pt/images/addPowerElements.png" alt="Adicionar elementos de potência" title="Adicionar elementos de potência"></center></div><p>No caso de inserção de um barramento, o usuário deve apenas clicar na posição desejada, já para os outros componentes será solicitado que o usuário selecione em uma ou duas barras, o qual o novo componente será conectado.</p><p>Sempre que são inseridos novos elementos, um formulário de dados é exibido e pode-se editar os dados elétricos e acessar formulários adicionais, como dados de estabilidade.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O primeiro elemento que a ser inserido no projeto <strong>deve</strong> ser um barramento. Os demais elementos elétricos devem ser conectados em uma ou mais barras.</p></div></div><p>Além dos elementos de potência pode-se inserir os elementos de <em><a href="/PSP/pt/docs/text">texto vinculado</a></em>, os quais estão associados a uma grandeza que se deseja vizualizar diretamente na <a href="/PSP/pt/docs/mainScreen#%C3%A1rea-de-trabalho">Área de Trabalho</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao inserir um barramento, a mesma irá acompanhar o ponteiro do mouse até que você clique na posição desejada. Para os demais elementos <strong>deve-se clicar em uma mais barras</strong> para exibir o elemento inserido.</p><p><strong>Sempre fique atento às instruções da <a href="/PSP/pt/docs/mainScreen#barra-de-status">Barra de Status</a>.</strong></p></div></div><video autoplay="" loop="" muted="" playsinline="" controls=""><source src="/PSP/videos/timelapseBuild.mp4" type="video/mp4"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="teclas-de-atalho"></a>Teclas de atalho<a class="hash-link" href="#teclas-de-atalho" title="Direct link to heading">#</a></h3><p>É possível inserir todos os elementos de potência por meio de teclas de atalho, facilitando e agilizando a criação dos diagramas unifilares no PSP-UFU. A tabela abaixo apresenta tais atalhos:</p><table><thead><tr><th>Elemento</th><th align="center">Atalho</th><th>Elemento</th><th align="center">Atalho</th></tr></thead><tbody><tr><td>Barramento</td><td align="center">B</td><td>Carga</td><td align="center">Shift + L</td></tr><tr><td>Linha</td><td align="center">L</td><td>Capacitor</td><td align="center">Shift + C</td></tr><tr><td>Transformador</td><td align="center">T</td><td>Indutor</td><td align="center">Shift + I</td></tr><tr><td>Gerador</td><td align="center">G</td><td>Corrente harmônica</td><td align="center">Shift + H</td></tr><tr><td>Motor de indução</td><td align="center">I</td><td>Texto</td><td align="center">A</td></tr><tr><td>Compensador síncrono</td><td align="center">K</td><td></td><td align="center"></td></tr></tbody></table><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="editando-dados-elétricos"></a>Editando dados elétricos<a class="hash-link" href="#editando-dados-elétricos" title="Direct link to heading">#</a></h2><p>Ao inserir um elemento, seu respectivo formulário de edição é exibido para inserção dos dados elétricos d elemento. Todos os dados podem ser <strong>alterados</strong> ao <strong>clicar duas vezes sobre o elemento</strong> ou por opção do <strong>menu de contexto</strong> exibido ao clicar com o botão direito sobre o elemento.</p><p>Cada formulário de edição de dados elétricos dos elementos possuem suas particularidades e são descritos em documentação específica: <a href="/PSP/pt/docs/bus">Barramento</a>, <a href="/PSP/pt/docs/line">Linha</a>, <a href="/PSP/pt/docs/transformer">Transformador</a>, <a href="/PSP/pt/docs/load">Carga</a>, <a href="/PSP/pt/docs/capacitor">Capacitor</a>, <a href="/PSP/pt/docs/inductor">Indutor</a>, <a href="/PSP/pt/docs/harmSource">Fonte de Corrente Harmônica</a>, <a href="/PSP/pt/docs/syncGenerator">Gerador Síncrono</a>, <a href="/PSP/pt/docs/syncMotor">Compensador Síncrono</a>, <a href="/PSP/pt/docs/indMotor">Motor de Indução</a>.</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/powerEditor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/cadTools"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Ferramentas CAD</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/bus"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Barramento »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#novo-projeto" class="table-of-contents__link">Novo projeto</a></li><li><a href="#inserindo-elementos" class="table-of-contents__link">Inserindo elementos</a><ul><li><a href="#teclas-de-atalho" class="table-of-contents__link">Teclas de atalho</a></li></ul></li><li><a href="#editando-dados-elétricos" class="table-of-contents__link">Editando dados elétricos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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+<link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/pt/opensearch.xml"><title data-react-helmet="true">Fluxo de Potência | PSP-UFU</title><meta data-react-helmet="true" property="og:url" content="https://thales1330.github.io/PSP/pt/docs/powerFlow"><meta data-react-helmet="true" name="docsearch:language" content="pt"><meta data-react-helmet="true" name="docsearch:version" content="current"><meta data-react-helmet="true" name="docsearch:docusaurus_tag" content="docs-default-current"><meta data-react-helmet="true" property="og:title" content="Fluxo de Potência | PSP-UFU"><meta data-react-helmet="true" name="description" content="Um estudo fundamental no planejamento da expansão e operação de um sistema elétrico é o fluxo de potência (ou fluxo de carga) uma vez que a operação satisfatória desse sistema depende do conhecimento dos efeitos da interligação, de novas cargas, de novas centrais geradoras e de novas linhas antes que elas sejam instaladas. Esse estudo tem como objetivo encontrar o fluxo de potência nos ramos e as tensões nodais do Sistema Elétrico de Potência (SEP) impostas pela geração e a carga."><meta data-react-helmet="true" property="og:description" content="Um estudo fundamental no planejamento da expansão e operação de um sistema elétrico é o fluxo de potência (ou fluxo de carga) uma vez que a operação satisfatória desse sistema depende do conhecimento dos efeitos da interligação, de novas cargas, de novas centrais geradoras e de novas linhas antes que elas sejam instaladas. Esse estudo tem como objetivo encontrar o fluxo de potência nos ramos e as tensões nodais do Sistema Elétrico de Potência (SEP) impostas pela geração e a carga."><link data-react-helmet="true" rel="shortcut icon" href="/PSP/pt/img/favicon.ico"><link data-react-helmet="true" rel="canonical" href="https://thales1330.github.io/PSP/pt/docs/powerFlow"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/docs/powerFlow" hreflang="en"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/pt/docs/powerFlow" hreflang="pt"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/docs/powerFlow" hreflang="x-default"><link data-react-helmet="true" rel="preconnect" href="https://BH4D9OD16A-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="/PSP/pt/assets/css/styles.73f64b8b.css">
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tabindex="0" href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Fluxo de Potência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Um estudo fundamental no planejamento da expansão e operação de um sistema elétrico é o <strong>fluxo de potência (ou fluxo de carga)</strong> uma vez que a operação satisfatória desse sistema depende do conhecimento dos efeitos da interligação, de novas cargas, de novas centrais geradoras e de novas linhas antes que elas sejam instaladas. Esse estudo tem como objetivo encontrar o fluxo de potência nos ramos e as tensões nodais do Sistema Elétrico de Potência (SEP) impostas pela geração e a carga.</p><p>As equações formuladas a partir dos modelos dos elementos presentes no estudo de fluxo de carga são não lineares e não possuem solução analítica explícita. Portanto é necessária a utilização de métodos numéricos iterativos para solução do problema de fluxo de carga.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulação-do-problema-de-fluxo-de-carga"></a>Formulação do problema de fluxo de carga<a class="hash-link" href="#formulação-do-problema-de-fluxo-de-carga" title="Direct link to heading">#</a></h2><p>O problema de fluxo de carga pode ser representado por um sistema de equações e inequações algébricas não-lineares que correspondem às leis de Kirchhoff e a um conjunto de restrições de operação impostos pelos componentes de uma rede elétrica.
+Na formulação do problema a cada barra da rede são associadas quatro variáveis, sendo que duas delas surgem como dados e duas como incógnitas (em uma barra de índice <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>):</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a magnitude da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é o ângulo da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a injeção líquida de potência ativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a injeção líquida de potência reativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</li></ul><p>Relativo às variáveis que são incógnitas e os dados do sistema, podem ser definidos <strong>três tipos de barras</strong>:</p><ul><li><strong>Barra PQ</strong>:	<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra PV</strong>:	<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra de Referência</strong>	<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As <strong>barras PQ</strong> geralmente são representadas pelos barramentos de carga, os quais não há geração e controle de tensão.</p><p>As <strong>barras PV</strong> se caracterizam pelo controle de tensão mediante a injeção ou absorção de potência reativa por meio do controle da excitação de uma máquina síncrona.</p><p>A <strong>barra de Referência</strong> (ou de folga, de oscilação) tem como função, assim como o próprio nome diz, servir de referência de tensão e ângulo do sistema. Essa barra é necessariamente geradora, uma vez que ela é responsável pelo equilíbrio do balanço de potência do sistema.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O tipo de barra deve ser definido no <a href="/PSP/pt/docs/bus">elemento barramento</a>.</p><p>Note que <strong>o sistema deve possuir somente uma barra de referência</strong>.</p></div></div><p>Como mencionado anteriormente, as equações são não-lineares e a solução analítica não é prática. As soluções dessas equações seguem <strong><a href="https://en.wikipedia.org/wiki/Iterative_method" target="_blank" rel="noopener noreferrer">processos iterativos</a></strong>, em que são atribuídos valores estimados (ou iniciais) para as barras com tensões desconhecidas e, baseado na potência ativa e reativa e módulo da tensão especificados, calcula-se por meio das equações previamente apresentadas as novas tensões complexas em cada nó do sistema.</p><p>Na sequência, esse conjunto de valores para as tensões em cada barra é utilizado para novamente calcular outro grupo de tensões. Cada cálculo de um novo conjunto de tensões é chamado iteração. O processo iterativo é repetido até que as mudanças em todas as barras sejam menores do que um valor pré-estipulado, obtendo assim a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="execução-do-fluxo-de-carga-no-psp-ufu"></a>Execução do fluxo de carga no PSP-UFU<a class="hash-link" href="#execução-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/pt/docs/powerEditor">editor de potência</a>, a execução do fluxo de carga é realizada no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Fluxo de carga</strong>.</p><div><center><img src="/PSP/pt/images/menuSimulationPF.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso o fluxo de carga tenha sido executado com sucesso, as setas de potência serão exibidas, a barra de status indicará sucesso na operação e os <a href="/PSP/pt/docs/text">elementos de texto</a> serão atualizados.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Evite construir o circuito com o cálculo contínuo habilitado, uma vez que configurações temporárias podem levar a erros de execução da simulação.</p><p>Para desabilitar o cálculo contínuo clique no botão <strong>Desabilitar solução</strong>.</p></div></div><p>Os resultados do fluxo de carga são exibidos nos <a href="/PSP/pt/docs/text">elementos de texto vinculado</a>, ao posicionar o mouse sobre os elementos e em <a href="/PSP/pt/docs/tabularReport">relatórios tabulares</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="erros-comuns-na-execução-do-fluxo-de-carga"></a>Erros comuns na execução do fluxo de carga<a class="hash-link" href="#erros-comuns-na-execução-do-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao fluxo de carga.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado"></a>A seguinte mensagem de erro é exibida: &quot;O número máximo de iterações foi alcançado&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando o método de solução numérica selecionado nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a> atinge o número máximo de iterações inserido. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li><li><strong>O número máximo de iterações está muito baixo</strong>. Alguns circuitos exigem um número maior de iterações, portanto altere o valor do máximo de iterações nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a>. Também tente alternar entre os métodos numéricos de solução disponíveis.</li><li><strong>Os parâmetros de simulação estão inadequados</strong>. Caso um parâmetro do método de solução esteja inadequado, como fator de aceleração ou tolerância, o cálculo pode não alcançar a convergência. Altere esses parâmetros nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="os-dados-de-saída-são-exibidos-como-nan-ou-nan"></a>Os dados de saída são exibidos como &quot;NaN&quot; ou &quot;nan&quot;<a class="hash-link" href="#os-dados-de-saída-são-exibidos-como-nan-ou-nan" title="Direct link to heading">#</a></h4><p>Isso ocorre devido a erros de operações matemáticas nos cálculos de fluxo de carga. &quot;NaN&quot; significa <em>Not a Number</em>.</p><ul><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu"></a>Métodos de solução numérica do fluxo de carga no PSP-UFU<a class="hash-link" href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Os métodos implementados no programa para solução do problema de fluxo de carga no PSP-UFU são <strong>Gauss-Seidel (GS)</strong> e <strong>Newton-Raphson (NR)</strong>. Além desses métodos clássicos, um método híbrido pode ser utilizado (definido nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a>), em que é utilizado inicialmente o GS e na sequência o NR, aumentando a chance de convergência do NR.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gauss-seidel"></a>Gauss-Seidel<a class="hash-link" href="#gauss-seidel" title="Direct link to heading">#</a></h3><p>O método de Gauss-Seidel tem sido bastante utilizado nas últimas décadas para solução do problema de fluxo de carga, uma vez que não há a necessidade de fatorar a matrizes, reduzindo o esforço computacional. Atualmente, restrições computacionais são menos problemáticas e outros métodos são normalmente escolhidos, porém o Gauss-Seidel ainda possui valor didático e, visto que o PSP-UFU também possui fins educacionais, optou-se pela implementação desse método.</p><p>Para iniciar as iterações do método são necessários valores iniciais para as tensões (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_i^0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) que devem ser calculadas. Normalmente para barras do tipo PQ tem-se <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=1{,}0+j0{,}0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> e para barras do tipo PV <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=V_i^{esp}+j0{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> [p.u.], em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">V_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> é o módulo da tensão especificada para a barra PV. A barra de referência tem o módulo e ângulo de tensão fixos e não participam no processo iterativo.</p><p>O método de Gauss-Seidel mostra um <strong>número excessivo de iterações</strong> e, com o intuito reduzi-los, multiplicam-se as correções de tensões por uma constante. Essa operação amplia o valor da correção, trazendo a tensão para mais perto do valor do valor final. Os multiplicadores que realizam essa convergência melhorada são chamados de fatores de aceleração. Para qualquer sistema existem valores ótimos para os fatores de aceleração e uma escolha inadequada pode resultar em uma convergência mais lenta ou torná-la impossível. Normalmente é utilizado um fator de aceleração igual a 1,6, valor definido como padrão no programa. A utilização do fator de aceleração é realizada por meio da equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mi>α</mi><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow><mo>+</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)} = \alpha \left( \dot{V}_{i}^{(v + 1)}- \dot{V}_{i_{AC}}^{(v)} \right) + \dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.27686399999999994em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração anterior com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span> é o fator de aceleração</li></ul><p>O fluxograma abaixo demonstra como o método de Gauss-Seidel foi implementado no PSP-UFU:</p><div><center><img src="/PSP/pt/images/gaussSeidel.svg" alt="Método numérico de Gauss-Seidel para fluxo de carga" title="Método numérico de Gauss-Seidel para fluxo de carga"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="newton-raphson"></a>Newton-Raphson<a class="hash-link" href="#newton-raphson" title="Direct link to heading">#</a></h3><p>O método de Newton-Raphson (também conhecido como método de Newton ou Newton-Fourier) para solução do fluxo de carga é descrito em vários livros e artigos. Atualmente é o algoritmo mais utilizado para solução do fluxo de carga. Para casos bem condicionados, esse método geralmente converge em 4 a 5 iterações, porém existe a possibilidade da técnica contornar o ponto da solução sem nunca atingi-la, o que também justifica implementação do método de Gauss-Seidel no <em>software</em>.</p><p>A expansão da <a href="https://en.wikipedia.org/wiki/Taylor_series" target="_blank" rel="noopener noreferrer">série de Taylor</a> para uma função de duas ou mais variáveis é a base do método de Newton-Raphson para resolver o problema do fluxo de carga. Fazendo a expansão em série de Taylor, para duas equações e duas incógnitas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_1\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_2\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span>), sem listar as derivadas parciais maiores que 1 na forma matricial, obtêm-se:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>2</mn></msub><mo>−</mo><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} K_1 - f_1\left( x_{1}^{(0)}, x_{2}^{(0)} \right)\\ K_2 - f_2\left( x_{1}^{(0)}, x_{2}^{(0)} \right) \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x_1} &amp; \displaystyle\frac{\partial f_1}{\partial x_2}\\ \displaystyle\frac{\partial f_2}{\partial x_1} &amp; \displaystyle\frac{\partial f_2}{\partial x_2} \end{bmatrix} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6010299999999997em;vertical-align:-1.55002em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em"><span style="top:-4.05002em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.41488em;vertical-align:-1.95744em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span> é o resultado da equação <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> é a estimativa iniciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span> é o valor acrescido de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> que resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>, ou seja: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow><mo>=</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right) = f_1\left( x_{1}^{(0)} + \Delta x_1, x_{2}^{(0)} + \Delta x_2 \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span></span></li></ul><p>Essa expressão pode ser resumida em:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>J</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \Delta K_1^{(0)}\\ \Delta K_2^{(0)} \end{bmatrix} = \begin{bmatrix} J \end{bmatrix}^{(0)} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.09618em">J</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1279000000000001em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>J</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[J]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.09618em">J</span><span class="mclose">]</span></span></span></span></span> é a <strong>Matriz Jacobiana</strong></p><p>Com a equação acima é possível calcular os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>. Entretanto, esses valores somados às estimativas iniciais não determinam a solução correta, sendo necessário repetir o processo de determinação das constantes, formação da matriz jacobiana e solução da equação acima, o qual será refeito determinando novas estimativas <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>Esse processo é repetido até que as correções se tornem tão pequenas que satisfaçam uma precisão escolhida.</p><p>Para aplicar o método de Newton-Raphson à solução das equações do fluxo de carga, utiliza-se as equações que representam as potências ativa e reativa injetadas em uma barra.</p><p>Assim como no método de Gauss-Seidel, a barra de referência é omitida da solução iterativa para determinar as tensões, pois a tensão complexa dessa barra é especificada. Como é conhecido o valor da potência ativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras do tipo PQ e PV, além da potência reativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras PQ pode-se definir <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> como os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span>. Os valores estimados do módulo e ângulo da tensão correspondem aos valores estimados para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>O jacobiano consiste nas derivadas parciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> em relação a cada uma das variáveis das equações injeção de potência líquida na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>. A matriz coluna formada por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> corresponde às correções de ângulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta \theta_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e módulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) das tensões de barra.</p><p>Com isso pode-se escrever a equação matricial de um sistema de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">n</span></span></span></span></span> barras, em que a barra número 1 corresponde à barra de referência e as barras de número 2 a n são barras do tipo PQ:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtable rowspacing="0.15999999999999992em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mo>+</mo><mtext>NPV</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{array}{r} \text{NPQ} + \text{NPV} \begin{cases} ~\\ ~ \end{cases}\\ ~\\ \text{NPQ} \begin{cases} ~\\ ~ \end{cases} \end{array} \begin{bmatrix} \Delta P_2\\ \vdots\\ \Delta P_n\\ \Delta Q_2\\ \vdots\\ \Delta Q_n\\ \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial P_2}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial P_2}{\partial \theta_n} &amp; \displaystyle\frac{\partial P_2}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial P_2}{\partial V_n}\\ \vdots &amp; \ddots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ \displaystyle\frac{\partial P_n}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial P_n}{\partial \theta_n} &amp; \displaystyle\frac{\partial P_n}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial P_n}{\partial V_n}\\ \displaystyle\frac{\partial Q_2}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_2}{\partial \theta_n} &amp; \displaystyle\frac{\partial Q_2}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_2}{\partial V_n}\\ \vdots &amp; \ddots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ \displaystyle\frac{\partial Q_n}{\partial \theta_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_n}{\partial \theta_n} &amp; \displaystyle\frac{\partial Q_n}{\partial V_2} &amp; \dots &amp; \displaystyle\frac{\partial Q_n}{\partial V_n} \end{bmatrix} \begin{bmatrix} \Delta \theta_2\\ \vdots\\ \Delta \theta_2\\ \Delta V_2\\ \vdots\\ \Delta V_n\\ \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.520000000000003em;vertical-align:-4.010000000000001em"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.8500299999999994em"><span style="top:-5.850029999999999em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord text"><span class="mord">NPV</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.7599999999999993em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-1.6499999999999997em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.3500300000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:12.61603em;vertical-align:-6.050095em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPQ</mtext></mrow><annotation encoding="application/x-tex">\text{NPQ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord text"><span class="mord">NPQ</span></span></span></span></span></span> é o número de barras PQ</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPV</mtext></mrow><annotation encoding="application/x-tex">\text{NPV}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord text"><span class="mord">NPV</span></span></span></span></span></span> é o número de barras PV</li></ul><p>O processo iterativo se inicia calculando as potências ativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ e PV e as potências reativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ, ambas as equações utilizando as estimativas iniciais das tensões complexas. Calcula-se, então, as correções de potência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">\Delta P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">\Delta Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span></span></span></span></span>):</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi><mo>=</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup><mspace linebreak="newline"></mspace><mi mathvariant="normal">Δ</mi><mi>Q</mi><mo>=</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta P = P_i^{esp} - P_i^{calc}\\ \Delta Q = Q_i^{esp} - Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O passo seguinte é a formação da matriz jacobiana. Com isso é possível calcular as correções de módulo e ângulo das tensões de todas as barras (com exceção da barra de referência). Para tanto, no PSP-UFU utiliza-se o método de <a href="https://en.wikipedia.org/wiki/Gaussian_elimination" target="_blank" rel="noopener noreferrer">Eliminação Gaussiana</a> e em sequência a substituição regressiva. Esse procedimento diminui o esforço computacional, uma vez que a inversão da matriz jacobiana em todas as iterações é evitada.</p><p>Com as correções de módulo e ângulo das tensões das barras calculados, aplicam-se as seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mspace linebreak="newline"></mspace><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\theta_i^{(v+1)}=\theta_i^{(v)}+\Delta \theta_i^{(v)}\\ V_i^{(v+1)}=V_i^{(v)}+\Delta V_i^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O processo é então reiniciado e será repetido até que se obtenha a convergência, quando as correções se tornam tão pequenas que satisfaçam uma tolerância pré-estipulada.</p><p>O fluxograma abaixo mostra o método de Newton-Raphson para solução do fluxo de carga implementado.</p><div><center><img src="/PSP/pt/images/newtonRaphson.svg" alt="Método numérico de Newton-Raphson para fluxo de carga" title="Método numérico de Newton-Raphson para fluxo de carga"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="controles-e-limites-em-um-problema-de-fluxo-de-carga"></a>Controles e limites em um problema de fluxo de carga<a class="hash-link" href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" title="Direct link to heading">#</a></h2><p>Nas barras de <a href="/PSP/pt/docs/syncGenerator">geração</a> e também naquelas em que se encontra um <a href="/PSP/pt/docs/syncMotor">compensador síncrono</a> conectado, o controle da magnitude da tensão no barramento é realizado por meio do ajuste da corrente de campo das máquinas síncronas, as quais podem operar sobrexcitadas (injetando reativos) ou subexcitadas (absorvendo reativos). Os valores limites de potência reativa que podem ser injetadas ou absorvidas dependem da máquina síncrona em estudo. Esses limites são incluídos no fluxo de carga com a criação de dois novos parâmetros, a potência reativa máxima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e potência reativa mínima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>), sendo esses valores a soma dos limites individuais das máquinas em uma mesma barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</p><p>A manutenção da potência reativa dentro dos limites é realizada pela troca do tipo de barra, ou seja, as barras violadoras que controlam a tensão (PV), passam a ser barras de carga (PQ), cuja potência é fixada como o limite que seria ultrapassado (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e a tensão deixa de ser controlada partir de então.</p><p>A verificação de violação e troca de tipo de barra pode ser realizada a cada iteração ou ao final da convergência do cálculo. No PSP-UFU foi implementada a última estratégia, uma vez que separa os conceitos de cálculo e de verificação de limites, tornando mais fácil o desenvolvimento de novos métodos numéricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situação não violadora o cálculo iterativo deve ser retomado até que obtenha novamente a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MONTICELLI, A. J. Fluxo de Carga em Redes de Energia Elétrica. São Paulo: Edgar Blücher, 1983.</li><li>STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.</li><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton’s Method. IEEE Transaction on Power Apparatus and Systems, v. PAS-86, n. 11, nov. 1967. doi: <a href="https://doi.org/10.1109/TPAS.1967.291823" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPAS.1967.291823</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/powerFlow.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/mathExpression"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Expressão Matemática</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/fault"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Curto-Circuito »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#formulação-do-problema-de-fluxo-de-carga" class="table-of-contents__link">Formulação do problema de fluxo de carga</a></li><li><a href="#execução-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Execução do fluxo de carga no PSP-UFU</a><ul><li><a href="#erros-comuns-na-execução-do-fluxo-de-carga" class="table-of-contents__link">Erros comuns na execução do fluxo de carga</a></li></ul></li><li><a href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Métodos de solução numérica do fluxo de carga no PSP-UFU</a><ul><li><a href="#gauss-seidel" class="table-of-contents__link">Gauss-Seidel</a></li><li><a href="#newton-raphson" class="table-of-contents__link">Newton-Raphson</a></li></ul></li><li><a href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" class="table-of-contents__link">Controles e limites em um problema de fluxo de carga</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Limitador de taxa</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Os blocos de limite de taxa têm como objetivo conter a taxa de crescimento ou decrescimento da sua entrada a valores inseridos pelo usuário. A taxa de variação é calculada a partir dos valores de entrada atual e passado e, caso seja violada, será definida uma saída que respeite essas taxas (de crescimento ou decrescimento).</p><p>A taxa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>) é calculada de acordo com a equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mi>n</mi></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">r = \frac{u_n - y_{n-1}}{h}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.9463300000000001em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">h</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é a entrada atual do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor de saída anterior do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	é o passo de integração</li></ul><p>Obtida a taxa, o bloco irá resultar em um valor calculado pelas seguintes condições:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>h</mi><mo>×</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><mi>r</mi><mo>&gt;</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>u</mi><mi>n</mi></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub><mo>≤</mo><mi>r</mi><mo>≤</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>h</mi><mo>×</mo><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><mi>r</mi><mo>&lt;</mo><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} h \times R_{cre}{,} &amp; \text{se } r &gt; R_{cre} \\ u_n{,} &amp; \text{se } R_{dec} \le r \le R_{cre}\\ h \times R_{dec}{,} &amp; \text{se } r &lt; R_{dec} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{cre}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{dec}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	são as taxas de crescimento e decaimento, respectivamente</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-limitador-de-taxa"></a>Formulário de edição de dados do Limitador de taxa<a class="hash-link" href="#formulário-de-edição-de-dados-do-limitador-de-taxa" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco limitador de taxa.</p><div><center><img src="/PSP/pt/images/rateLimitForm.png" alt="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU" title="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU"></center></div><p>Esse bloco é definido pelo <strong>limite superior (taxa de crescimento máxima) e inferior (taxa de decrescimento máxima)</strong>, inseridos pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Esses blocos são úteis para limitar variações bruscas na entrada que possam comprometer o controle ou limites físicos de modelos de turbina, por exemplo.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a 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Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Configurações da Simulação</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>As parametrizações e configurações das simulações são acessadas no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no ícone do botão <strong>Configurações de simulação</strong>.</p><div><center><img src="/PSP/pt/images/menuSimulationSettings.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-das-configurações-de-simulação"></a>Formulário de edição das configurações de simulação<a class="hash-link" href="#formulário-de-edição-das-configurações-de-simulação" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de edição das configurações de simulação:</p><div><center><img src="/PSP/pt/images/simulationConfigForm.png" alt="Configurações de simulação" title="Configurações de simulação"></center></div><p>Esse formulário é subdividido em quatro contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais e comuns a várias simulações;</li><li><strong>Fluxo de carga</strong>: local onde é selecionado o <a href="/PSP/pt/docs/powerFlow#m%C3%A9todos-de-solu%C3%A7%C3%A3o-num%C3%A9rica-do-fluxo-de-carga-no-psp-ufu">método de solução numérica</a>, assim como seus parâmetros;</li><li><strong>Estabilidade</strong>: contendo opções de simulação no tempo e parâmetros do <a href="/PSP/pt/docs/stability#integra%C3%A7%C3%A3o-num%C3%A9rica">método de integração</a>;</li><li><strong>Carga ZIP</strong>: contém as opções gerais das <a href="/PSP/pt/docs/load#carga-no-estudo-de-estabilidade">cargas do tipo ZIP</a> no estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</li></ul><hr><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Fluxo de carga</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Carga ZIP</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-base"></a>Potência base<a class="hash-link" href="#potência-base" title="Direct link to heading">#</a></h4><p>Potência base do sistema utilizada para conversão dos dados reais em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> e vice-versa. Pode ser inserido em VA, kVA ou MVA.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>A potência base de cada elemento pode ser distinta da potência base do sistema. Para isso, basta marcar a opção &quot;Utilizar potência nominal [<em>do elemento</em>] como base&quot;, presente em seus respectivos <a href="/PSP/pt/docs/powerEditor">formulários de edição da dados</a>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="frequência-do-sistema"></a>Frequência do sistema<a class="hash-link" href="#frequência-do-sistema" title="Direct link to heading">#</a></h4><p>Define a frequência nominal do sistema.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao alterar a frequência nominal atente-se ao campo &quot;<a href="/PSP/pt/docs/syncGenerator#frequ%C3%AAncia-de-circuito-aberto">Frequência de circuito aberto</a>&quot; das máquinas síncronas.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="cálculo-contínuo"></a>Cálculo contínuo<a class="hash-link" href="#cálculo-contínuo" title="Direct link to heading">#</a></h4><p>Habilita ou desabilita o cálculo contínuo para os cálculos de <a href="/PSP/pt/docs/fault">curto-circuito</a>, <a href="/PSP/pt/docs/fault">nível de curto-circuito</a> e <a href="/PSP/pt/docs/harmonics">distorções harmônicas</a>. O cálculo contínuo para o <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a> é sempre habilitado.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Para habilitar o cálculo contínuo aperte o botão <strong>Habilitar solução</strong> presente no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">menu Simulação</a>.</p><p>Com essa opção, os cálculos estáticos selecionados são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p></div></div></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="método-de-solução"></a>Método de solução<a class="hash-link" href="#método-de-solução" title="Direct link to heading">#</a></h4><p>Define o método de solução numérica para o estudo de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>. Na versão atual, está presente no PSP-UFU os seguintes métodos:</p><ul><li><a href="/PSP/pt/docs/powerFlow#gauss-seidel">Gauss-Seidel</a></li><li><a href="/PSP/pt/docs/powerFlow#newton-raphson">Newton-Raphson</a></li><li><a href="/PSP/pt/docs/powerFlow#m%C3%A9todos-de-solu%C3%A7%C3%A3o-num%C3%A9rica-do-fluxo-de-carga-no-psp-ufu">Gauss-Newton híbrido</a></li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="fator-de-aceleração"></a>Fator de aceleração<a class="hash-link" href="#fator-de-aceleração" title="Direct link to heading">#</a></h4><p>Fator utilizado para acelerar a convergência nos métodos de Gauss. A utilização do fator de aceleração é realizada por meio da equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mi>α</mi><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow><mo>+</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)} = \alpha \left( \dot{V}_{i}^{(v + 1)}- \dot{V}_{i_{AC}}^{(v)} \right) + \dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.27686399999999994em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração anterior com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span> é o fator de aceleração</li></ul><p>Essa opção é habilitada somente para os métodos de <a href="/PSP/pt/docs/powerFlow#gauss-seidel">Gauss-Seidel</a> e <a href="/PSP/pt/docs/powerFlow#m%C3%A9todos-de-solu%C3%A7%C3%A3o-num%C3%A9rica-do-fluxo-de-carga-no-psp-ufu">Gauss-Newton híbrido</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tolerância"></a>Tolerância<a class="hash-link" href="#tolerância" title="Direct link to heading">#</a></h4><p>É o valor de erro entre as iterações (em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) no qual é obtida a convergência do método numérico.</p><p>Para o método de <a href="/PSP/pt/docs/powerFlow#gauss-seidel">Gauss-Seidel</a> o erro é calculado pelo maior valor do módulo da diferença entre a tensão na iteração anterior e atual (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo>=</mo><mrow><mo fence="true">∣</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">∣</mo></mrow></mrow><annotation encoding="application/x-tex">e = \left| V_{i}^{(v + 1)} - V_{i}^{(v)} \right|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.8239899999999998em;vertical-align:-0.6500149999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.173975em"><span style="top:-1.955985em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-2.556985em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-2.572975em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-3.173975em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6500149999999999em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.173975em"><span style="top:-1.955985em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-2.556985em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-2.572975em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span><span style="top:-3.173975em"><span class="pstrut" style="height:2.606em"></span><span class="delimsizinginner delim-size1"><span>∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6500149999999999em"><span></span></span></span></span></span></span></span></span></span></span></span>); enquanto que no método de <a href="/PSP/pt/docs/powerFlow#newton-raphson">Newton-Raphson</a> esse erro é o módulo do maior valor de ajuste de potência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">Δ</mi><mi>P</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\Delta P|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">∣</span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mord">∣</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">Δ</mi><mi>Q</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\Delta Q|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">∣</span><span class="mord">Δ</span><span class="mord mathdefault">Q</span><span class="mord">∣</span></span></span></span></span>).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="iterações-máx"></a>Iterações máx<a class="hash-link" href="#iterações-máx" title="Direct link to heading">#</a></h4><p>É o número máximo de iterações que um método poderá alcançar. Caso esse valor seja alcançado, o cálculo de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a> é interrompido com erro.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="ângulo-da-barra-de-referência"></a>Ângulo da barra de referência<a class="hash-link" href="#ângulo-da-barra-de-referência" title="Direct link to heading">#</a></h4><p>É o valor do ângulo da <a href="/PSP/pt/docs/powerFlow">barra de referência</a>, em graus.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inércia-do-newton"></a>Inércia do Newton<a class="hash-link" href="#inércia-do-newton" title="Direct link to heading">#</a></h4><p>É o valor que multiplica as correções de potência. Valores maiores que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> incrementam a correção e menores que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> diminuem a correção.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Valores menores que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> <em>podem</em> auxiliar na convergência, porém aumentam o número de iterações e consequentemente o tempo de processamento do método.</p></div></div><p>Essa opção é habilitada somente para o método de <a href="/PSP/pt/docs/powerFlow#newton-raphson">Newton-Raphson</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tolerância-do-gauss"></a>Tolerância do Gauss<a class="hash-link" href="#tolerância-do-gauss" title="Direct link to heading">#</a></h4><p>Define a tolerância do Gauss-Seidel para o método híbrido. Essa opção é habilitada somente para o método de <a href="/PSP/pt/docs/powerFlow#m%C3%A9todos-de-solu%C3%A7%C3%A3o-num%C3%A9rica-do-fluxo-de-carga-no-psp-ufu">Gauss-Newton híbrido</a>.</p><p>Uma vez que o Newton-Raphson é consideravelmente sensível às condições iniciais, as primeiras iterações são calculadas utilizado o Gauss-Seidel até que o erro fique menor que a tolerância especificada no campo “Tolerância do Gauss”. A partir desse ponto o cálculo é realizado utilizando o método de Newton-Raphson até que se obtenha a convergência com um erro menor que a tolerância estipulada no campo “Tolerância”.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tal opção auxilia na convergência de sistemas impossíveis de resolver utilizando Newton-Raphson convencional.</p></div></div></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="passo-de-integração"></a>Passo de integração<a class="hash-link" href="#passo-de-integração" title="Direct link to heading">#</a></h4><p>Define o passo de integração para o método de integração <a href="/PSP/pt/docs/stability#integra%C3%A7%C3%A3o-num%C3%A9rica">Trapezoidal Implícito</a>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Valores muito elevados poderão gerar erros de simulação, enquanto valores muito pequenos aumentarão significativamente o tempo de processamento do cálculo de estabilidade.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tempo-de-simulação"></a>Tempo de simulação<a class="hash-link" href="#tempo-de-simulação" title="Direct link to heading">#</a></h4><p>Define o tempo total de simulação, em segundos.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tolerância-1"></a>Tolerância<a class="hash-link" href="#tolerância-1" title="Direct link to heading">#</a></h4><p>Tolerância do processo iterativo (em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) do método de integração <a href="/PSP/pt/docs/stability#integra%C3%A7%C3%A3o-num%C3%A9rica">Trapezoidal Implícito</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="iterações-máx-1"></a>Iterações máx<a class="hash-link" href="#iterações-máx-1" title="Direct link to heading">#</a></h4><p>Número máximo de iterações do processo iterativo do método de integração <a href="/PSP/pt/docs/stability#integra%C3%A7%C3%A3o-num%C3%A9rica">Trapezoidal Implícito</a>. Caso o número de iterações ultrapasse esse valor, o cálculo de estabilidade será encerrado com erro.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="razão-de-passo-dos-controles"></a>Razão de passo dos controles<a class="hash-link" href="#razão-de-passo-dos-controles" title="Direct link to heading">#</a></h4><p>Define a quantidade passos de integração que o sistema de controle será submetido para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> passo de integração do sistema de potência.</p><p>Por exemplo, se o passo de integração for definido em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>01</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">0{,}01~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord">1</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span> e a razão de passo como <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">0</span></span></span></span></span>, o valor efetivo do passo de integração para os sistemas de controle será de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>001</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">0{,}001~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord">0</span><span class="mord">1</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span> (10 vezes menor).</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A utilização desse campo se justifica em situações de grandes diferenças nas constantes de tempo das máquinas síncronas (geralmente maiores) e dos sistemas de controle (geralmente menores).</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um valor de razão de passo de controles igual a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> não irá alterar o passo de integração do sistema de controle.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tempo-de-impressão"></a>Tempo de impressão<a class="hash-link" href="#tempo-de-impressão" title="Direct link to heading">#</a></h4><p>É o intervalo de tempo o qual o programa irá armazenar os dados para impressão em <a href="/PSP/pt/docs/graphViewer">gráficos no tempo</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Valores muito pequenos (menores que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>001</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">0{,}001~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord">0</span><span class="mord">1</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span>) podem aumentar significativamente o tempo de processamento do cálculo de estabilidade.</p><p>O tempo de impressão <strong>deve ser maior ou igual</strong> ao passo de integração.</p></div></div></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-composição-geral-para-todas-as-cargas-do-sistema"></a>Utilizar composição geral para todas as cargas do sistema<a class="hash-link" href="#utilizar-composição-geral-para-todas-as-cargas-do-sistema" title="Direct link to heading">#</a></h4><p>Habilita a composição da <a href="/PSP/pt/docs/load#carga-no-estudo-de-estabilidade">carga ZIP</a> para todas as cargas que <strong>não</strong> possuem parametrização individual.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="composição-da-carga-potências-ativa-e-reativa"></a>Composição da carga (potências ativa e reativa)<a class="hash-link" href="#composição-da-carga-potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Define a composição, em porcentagem, das parcelas de Impedância, Corrente e Potência constantes da carga. Essa composição pode ser implementada individualmente para a potência ativa e reativa.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A soma das parcelas da carga ZIP para uma potência deve ser 100%, caso contrário será exibida uma mensagem de erro.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="subtensão-a-qual-as-cargas-serão-modeladas-como-impedância-constante"></a>Subtensão a qual as cargas serão modeladas como impedância constante<a class="hash-link" href="#subtensão-a-qual-as-cargas-serão-modeladas-como-impedância-constante" title="Direct link to heading">#</a></h4><p>Define o valor percentual da tensão (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>l</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow><annotation encoding="application/x-tex">V_{low}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02691em">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) que, para valores inferiores a esse, a carga passa a ser considerada como impedância constante pura.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Cargas de corrente e potência constantes possuem problemas em tensões muito baixas. À medida que a tensão diminui as correntes dessas cargas não reduz, resultando em perda de precisão e problemas na convergência de processos iterativos. Para contornar esse problema utiliza-se uma tensão pré-definida (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>l</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow><annotation encoding="application/x-tex">V_{low}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02691em">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), a qual as cargas (ou parcelas) de corrente e potência constantes são modeladas como impedância constante.</p></div></div></div></div></div><hr></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/simulationConfig.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/stability"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Estabilidade</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/text"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Texto Vinculado »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#formulário-de-edição-das-configurações-de-simulação" class="table-of-contents__link">Formulário de edição das configurações de simulação</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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href="/PSP/pt/docs/line">Linha</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/transformer">Transformador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/load">Carga</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/capacitor">Capacitor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/inductor">Indutor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmSource">Fonte de Corrente Harmônica</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/syncGenerator">Gerador Síncrono</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/syncMotor">Compensador Síncrono</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/indMotor">Motor de Indução</a></li></ul></li></ul></li><li 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href="/PSP/pt/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Estabilidade</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Estabilidade de sistemas de potência é a habilidade de um Sistema Elétrico de Potência (SEP), para uma certa condição de operação, de reestabelecer um estado de operação de equilíbrio após ser submetido a um distúrbio físico. A estabilidade é reconhecida como um importante problema para a segurança da operação desde a década de 1920. Muitos <em>blackouts</em> de grande porte causados pela instabilidade do SEP têm ilustrado a importância desse fenômeno.</p><div><center><img src="/PSP/pt/images/stability.svg" alt="Estabilidade" title="Estabilidade"></center></div><p>Normalmente são realizados dois tipos de estudos de estabilidade (uma classificação mais aprofundada pode ser observada <a href="https://doi.org/10.1109/TPWRS.2004.825981" target="_blank" rel="noopener noreferrer">nesse artigo</a>). A recuperação de uma rápida e brusca perturbação é chamada de <strong>estabilidade transitória</strong>, e sua solução é geralmente obtida no domínio do tempo. Outro estudo é denominado <strong>estabilidade dinâmica</strong>, utilizado para descrever a resposta do sistema frente a pequenas perturbações ao longo de muito tempo, o qual pode ser resolvido tanto no domínio da frequência quanto no domínio do tempo.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>No PSP-UFU, a estabilidade dinâmica é tratada como uma extensão da estabilidade transitória, em que os efeitos de pequenas constantes de tempo devem ser ignorados.</p></div></div><p>O modelo para representação do SEP em um estudo de estabilidade transitória é feito por meio do sistema de equações algébrico-diferenciais (EADs), em que o estado inicial é considerado estável e originado de um estudo de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>. A solução das equações se dá por meio de um <a href="https://en.wikipedia.org/wiki/Numerical_integration" target="_blank" rel="noopener noreferrer">método de integração numérica</a> e, então, é observada a resposta do sistema de potência.</p><p>Caso a solução no domínio do tempo, após alguma perturbação, leve um ou mais parâmetros do sistema a valores fisicamente impossíveis ou, até mesmo, a solução das equações diferenciais não obtenha convergência em processos iterativos, o sistema é dito <strong>instável</strong>. Em contrapartida, caso as variáveis de estado dos elementos do sistema oscilem após uma perturbação e se estabeleçam em um novo ponto de operação, mesmo que distinto dos valores iniciais, o sistema é considerado <strong>estável</strong>.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="execução-do-estudo-de-estabilidade-no-psp-ufu"></a>Execução do estudo de estabilidade no PSP-UFU<a class="hash-link" href="#execução-do-estudo-de-estabilidade-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/pt/docs/powerEditor">editor de potência</a>, assim como os <a href="/PSP/pt/docs/controlEditor">controles</a> das <a href="/PSP/pt/docs/syncGenerator">máquinas síncronas</a>, a execução da estabilidade é realizada no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no ícone do botão <strong>Executar Estabilidade</strong>. Ao clicar na parte inferior do botão &quot;Executar Estabilidade&quot; será exibido um menu suspenso com a opção <strong>Lista de eventos de estabilidade</strong>.</p><div><center><img src="/PSP/pt/images/menuSimulationStability.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><p>Ao executar o cálculo de estabilidade, uma caixa com o status do processo de simulação será exibida, indicando primeiramente a inicialização do estudo e posteriormente o tempo de simulação calculado.</p><p>Ao clicar na lista de eventos de estabilidade, será exibida uma janela com a descrição dos eventos de estabilidade inseridos.</p><div><center><img src="/PSP/pt/images/stabList.png" alt="Lista de eventos de estabilidade" title="Lista de eventos de estabilidade"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserindo-um-evento-de-estabilidade"></a>Inserindo um evento de estabilidade<a class="hash-link" href="#inserindo-um-evento-de-estabilidade" title="Direct link to heading">#</a></h3><p>Os distúrbios mais comuns aplicados em estudos de estabilidade transitória são faltas e chaveamentos. Tais operações são facilmente realizadas no PSP-UFU:</p><ul><li><p><strong>Faltas</strong>: Curtos-circuitos trifásicos podem ser inseridos nos <a href="/PSP/pt/docs/bus">barramentos</a> do sistema por meio da inclusão da impedância de falta na matriz admitância de barras da mesma maneira realizada em um elemento <em>shunt</em>. Tal valor é definido pelo usuário e caso seja um curto-circuito franco, um valor não nulo, mas suficientemente próximo de zero, é aplicado, de forma que a tensão no barramento é levada a zero durante o distúrbio.</p></li><li><p><strong>Chaveamento de ramo</strong>: Da mesma forma que as faltas, o chaveamento de ramos é realizado por meio da alteração na matriz admitância, removendo ou inserindo os parâmetros do elemento a ser chaveado.
+Cada <a href="/PSP/pt/docs/powerEditor">elemento de potência</a> possui um botão de &quot;Chaveamento&quot; ou &quot;Estabilidade&quot;, em que pode ser inserido os tempos de remoção e/ou inserção do componente.</p></li><li><p><strong>Chaveamento de máquinas</strong>: A remoção de uma máquina síncrona é efetivada com a retirada de sua participação no vetor de correntes, além da remoção de sua admitância fictícia.
+Assim como os <a href="/PSP/pt/docs/powerEditor">elementos de potência</a>, as <a href="/PSP/pt/docs/syncGenerator">máquinas síncronas</a> possui um botão &quot;Chaveamento&quot;, em que pode ser inserido os tempos de remoção e/ou inserção do componente.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Mesmo que removida da barra, os parâmetros das <a href="/PSP/pt/docs/syncGenerator">máquinas síncronas</a> continuam a ser calculados com a corrente do estator nula, podendo fornecer resultados em uma eventual reconexão.</p></div></div></li></ul><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Eventos nos <strong>sistemas de controle</strong> podem ser facilmente introduzidos com o bloco de <a href="/PSP/pt/docs/mathExpression">expressão matemática</a>. Nesse caso, tais eventos <strong>não</strong> serão exibidos na lista de eventos de estabilidade.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="erros-comuns-na-execução-do-estudo-de-estabilidade"></a>Erros comuns na execução do estudo de estabilidade<a class="hash-link" href="#erros-comuns-na-execução-do-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao estudo de estabilidade.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-não-foi-possível-construir-a-matriz-admitância"></a>A seguinte mensagem de erro é exibida: &quot;Não foi possível construir a matriz admitância&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-não-foi-possível-construir-a-matriz-admitância" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível construir a matriz admitância de barras. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros do circuito estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-os-valores-de-saturação-do"></a>A seguinte mensagem de erro é exibida: &quot;Erro ao inicializar os valores de saturação do...&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-os-valores-de-saturação-do" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível calcular os fatores de saturação da <a href="/PSP/pt/docs/syncGenerator">máquina síncrona</a>. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>O fator de saturação é menor que 1,2</strong>. Esse valor deve ser maior que 1,2, ou irá gerar erros na simulação. Caso não seja informado, a saturação da máquina não é considerada nos cálculos.</li><li><strong>Os parâmetros da máquina síncrona estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão das máquinas síncronas sejam inseridos, o cálculo dos fatores de saturação pode divergir. Verifique se os dados foram inseridos corretamente.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-avr--regulador-de-velocidade"></a>A seguinte mensagem de erro é exibida: &quot;Erro ao inicializar o AVR / regulador de velocidade...&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-avr--regulador-de-velocidade" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível <a href="/PSP/pt/docs/controlEditor#inicializa%C3%A7%C3%A3o-do-sistema-de-controle">inicializar o sistema de controle</a> de uma <a href="/PSP/pt/docs/syncGenerator">máquina síncrona</a>. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>O sistema de controle está vazio</strong>. Caso esteja habilitado o AVR e/ou o regulador de velocidade e o controle não foi inserido, esse erro pode ser acionado. Insira o controle da máquina ou desmarque a opção de utilização do AVR e/ou regulador de velocidade.</li><li><strong>O sistema de controle não possui ao menos uma entrada e uma saída</strong>. O sistema de controle deve ter ao menos uma <a href="/PSP/pt/docs/io">entrada e uma saída</a>, caso contrário apresentará erro de execução.</li><li><strong>O passo de integração está muito pequeno</strong>. Caso o passo de integração esteja muito pequeno, os cálculos gerarão erros e irão divergir. Reduza o passo de integração nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-impossível-resolver-as-máquinas-do-sistema"></a>A seguinte mensagem de erro é exibida: &quot;Impossível resolver as máquinas do sistema&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-impossível-resolver-as-máquinas-do-sistema" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível resolver as EADs das máquinas síncronas inseridas no sistemas. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros da máquina síncrona estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão das máquinas síncronas sejam inseridos, o cálculo das EADs pode se tornar impossível. Verifique se os dados foram inseridos corretamente.</li><li><strong>O passo de integração está muito pequeno</strong>. Caso o passo de integração esteja muito pequeno, os cálculos gerarão erros e irão divergir. Reduza o passo de integração nas <a href="/PSP/pt/docs/simulationConfig">configurações de simulação</a>.</li><li><strong>Os parâmetros do circuito estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-escorregamento-do-motor"></a>A seguinte mensagem de erro é exibida: &quot;Erro ao inicializar o escorregamento do motor...&quot;<a class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-erro-ao-inicializar-o-escorregamento-do-motor" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando não é possível calcular o valor de escorregamento inicial do motor de indução. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros de estabilidade do motor estão incorretos</strong>. Caso parâmetros muito fora dos valores padrão dos motores de indução sejam inseridos, o cálculo do escorregamento pode se tornar impossível. Verifique se os dados foram inseridos corretamente.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="estrutura-da-ferramenta-de-estabilidade"></a>Estrutura da ferramenta de estabilidade<a class="hash-link" href="#estrutura-da-ferramenta-de-estabilidade" title="Direct link to heading">#</a></h2><p>A estabilidade de um SEP é um problema dinâmico e necessita de modelos mais elaborados de elementos de potência comparados àqueles apresentados nos outros estudos. Esses modelos são descritos individualmente, com destaque às <a href="/PSP/pt/docs/syncGenerator#gerador-s%C3%ADncrono-no-estudo-de-estabilidade">máquinas síncronas</a>, <a href="/PSP/pt/docs/load#carga-no-estudo-de-estabilidade">cargas ZIP</a> e <a href="/PSP/pt/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-estudo-de-estabilidade">motores de indução</a>.</p><p>A representação dos demais componentes do sistema elétrico: <a href="/PSP/pt/docs/line">linhas de transmissão</a>, <a href="/PSP/pt/docs/transformer">transformadores</a> e elementos <em>shunt</em> (com exceção de cargas ZIP), que formam a rede de transmissão ou distribuição balanceada, é realizada utilizando os mesmos modelos do <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>.</p><p>Os modelos dinâmicos para a análise no domínio do tempo são na forma de um sistema de equações algébrico-diferenciais (EADs), descritas a seguir:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mi>g</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\dfrac{dx}{dt}= f(x,y,u)\\ g = f(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault">u</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.03588em">g</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>	são as variáveis de estado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span></span></span>	são as variáveis de algébricas</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">u</span></span></span></span></span>	são as variáveis de entrada</li></ul><p>As variáveis de entrada são inseridas pelo usuário e permanecem constantes durante todo o processo de cálculo. Para o problema de estabilidade, as variáveis algébricas iniciais correspondem às tensões e ângulos das barras calculadas no <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>. As variáveis de estado iniciais são estimadas após a convergência do fluxo de potência (inicialização dos elementos dinâmicos).</p><p>Na sequência é apresentado, na forma de fluxogramas, a estrutura do módulo de estabilidade implementado no PSP-UFU. O fluxograma abaixo mostra a estrutura geral da ferramenta de estabilidade.</p><div><center><img src="/PSP/pt/images/stabGeneral.svg" alt="Estrutura geral da ferramenta de estabilidade" title="Estrutura geral da ferramenta de estabilidade"></center></div><p>A inicialização da rede elétrica e dos elementos dinâmicos, assim como o precesso iterativo para solução das EADs são apresentados nos dois fluxogramas conseguintes.</p><div><center><img src="/PSP/pt/images/stabInit.svg" alt="Inicialização da ferramenta de estabilidade" title="Inicialização da ferramenta de estabilidade"></center></div><div><center><img src="/PSP/pt/images/stabIterative.svg" alt="Processo iterativo para cálculo dos elementos dinâmicos" title="Estrutura geral da ferramenta de estabilidade"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="integração-numérica"></a>Integração numérica<a class="hash-link" href="#integração-numérica" title="Direct link to heading">#</a></h3><p>Para resolver as equações diferenciais da máquina é necessário um método de integração numérica. Tais métodos são classificados em dois grupos: métodos explícitos e métodos implícitos. Os métodos explícitos, devido à sua formulação, calculam diretamente o estado do sistema em um instante de tempo posterior, enquanto métodos implícitos envolvem estados atuais e posteriores em suas equações, exigindo, portanto, um processo iterativo.</p><p>As constantes de tempo presentes no estudo de estabilidade têm uma grande variação em seu valor (podem variar de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">10^{-3}~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">10~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span>). Isso torna o sistema de equações diferenciais da máquina síncrona como um sistema rígido (<em><a href="https://en.wikipedia.org/wiki/Stiff_equation" target="_blank" rel="noopener noreferrer">stiff equation</a></em>). Caso a análise da estabilidade numérica tanto das equações diferenciais rígidas quanto do método de integração obtenham o mesmo comportamento, o método é chamado de absolutamente estável, ou A-estável.</p><p>Métodos de integração numérica explícitos, como por exemplo o <a href="https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods" target="_blank" rel="noopener noreferrer">Runge-Kutta</a> de quarta ordem, não podem ser A-estáveis e, portanto, normalmente possuem comportamento ruim em problemas com equações diferenciais rígidas. Por outro lado, métodos implícitos podem ser A-estáveis. Um método implícito adequado para solução do comportamento dinâmico de sistemas elétricos é o <strong>Trapezoidal Implícito</strong>, por possuir as seguintes vantagens:</p><ul><li>É numericamente estável (A-estável);</li><li>É bastante rápida;</li><li>Possui boa precisão (dependendo somente do passo de integração utilizado).</li></ul><p>Tal método foi implementado no PSP-UFU tanto para solução das equações diferenciais da máquina síncrona quanto nas <a href="/PSP/pt/docs/transferFunction">funções transferência</a> do <a href="/PSP/pt/docs/controlEditor">sistema de controle</a>. Sua formulação é dada pela seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><mi>h</mi><mrow><mo fence="true">(</mo><msubsup><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><msubsup><mi>y</mi><mi>n</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y_{n+1}=y_n + 0{,}5 h \left(y_{n+1}&#x27; + y_{n}&#x27; \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>	é o passo de integração</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor do estado no passo anterior (instante <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">t_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>)</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_{n+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>	é o valor do estado no passo atual (instante <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">t_{n+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8234109999999999em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span>)</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>y</mi><mi>n</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">y_{n}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.998892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span>	é a equação diferencial no passo anterior</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">y_{n+1}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.058331em;vertical-align:-0.30643899999999996em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.451892em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.30643899999999996em"><span></span></span></span></span></span></span></span></span></span></span>	é a equação diferencial no passo atual</li></ul><p>Nota-se nessa equação que o método é implícito devido ao termo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> aparecer em ambos os lados da equação, por isso é necessário um processo iterativo para sua solução.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>KUNDUR, P.; et al. Definition and classification of power system stability. IEEE Transactions Power Systems, v. 19, n. 2, mai 2004, p. 1387-1401. doi: <a href="https://doi.org/10.1109/TPWRS.2004.825981" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPWRS.2004.825981</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>PARK, R. H. Two-reactions Theory of Synchronous Machine. Generalized Method of Analysis – Part I. AIEE Transactions, v. 48, n. 3, jul 1929. doi: <a href="https://doi.org/10.1109/T-AIEE.1929.5055275" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/T-AIEE.1929.5055275</a></li><li>ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: <a href="https://doi.org/10.1109/9780470545577" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/9780470545577</a></li><li>SAUER, P. W.; PAI, M. A. Power System Dynamics and Stability. Pretience Hall, Upper Saddle River, 1998.</li><li>KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994.</li><li>DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: <a href="https://doi.org/10.1109/TPAS.1972.293341" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPAS.1972.293341</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/stability.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/harmonics"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Harmônicos</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/simulationConfig"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Configurações da Simulação »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#execução-do-estudo-de-estabilidade-no-psp-ufu" class="table-of-contents__link">Execução do estudo de estabilidade no PSP-UFU</a><ul><li><a href="#inserindo-um-evento-de-estabilidade" class="table-of-contents__link">Inserindo um evento de estabilidade</a></li><li><a href="#erros-comuns-na-execução-do-estudo-de-estabilidade" class="table-of-contents__link">Erros comuns na execução do estudo de estabilidade</a></li></ul></li><li><a href="#estrutura-da-ferramenta-de-estabilidade" class="table-of-contents__link">Estrutura da ferramenta de estabilidade</a><ul><li><a href="#integração-numérica" class="table-of-contents__link">Integração numérica</a></li></ul></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Somador</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Como o próprio nome sugere, os somadores são elementos de controle capazes de somar qualquer quantidade de valores reais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-somador"></a>Formulário de edição de dados do somador<a class="hash-link" href="#formulário-de-edição-de-dados-do-somador" title="Direct link to heading">#</a></h2><p>A quantidade de entradas, assim como seus sinais são definidas pelo usuário, como mostra a figura abaixo.</p><div><center><img src="/PSP/pt/images/sumForm.png" alt="Formulário de edição de dados do somador no PSP-UFU" title="Formulário de edição de dados do somador no PSP-UFU"></center></div><p>No seu único campo “<strong>Sinais</strong>” é inserida uma lista de sinais <strong>separados por espaços</strong> contendo os símbolos “+” para uma entrada positiva e “-” para negativa.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O usuário pode inserir duas ou mais entradas de sinais no 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href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Gerador Síncrono</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><p>Uma máquina de corrente alternada na qual a frequência das tensões geradas e a velocidade da máquina estão em uma proporção constante. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=411-31-08" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gerador-síncrono-no-psp-ufu"></a>Gerador Síncrono no PSP-UFU<a class="hash-link" href="#gerador-síncrono-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Os geradores síncronos são a fonte de potência para o cálculo do fluxo de carga e estabilidade, além de serem um dos principais contribuintes para as correntes de falta.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Somente barramentos que possuem esse elemento conectado podem ser considerados <a href="/PSP/pt/docs/powerFlow">barras de referência</a>. <a href="/PSP/pt/docs/powerFlow">Barras do tipo PV</a> devem conter um gerador síncrono ou um <a href="/PSP/pt/docs/syncMotor">motor síncrono (compensador síncrono)</a>.</p></div></div><p>O formulário de dados é dividido em dois, sendo o primeiro responsável pelos dados gerais, de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a> e de <a href="/PSP/pt/docs/fault">falta</a> e um segundo pelos dados de <a href="/PSP/pt/docs/stability">estabilidade</a> . Esse último também permite o acesso aos controles das máquinas síncronas manipulados pelo editor de controle.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os dados referentes às <strong>impedâncias de sequência positiva do gerador síncrono</strong> inseridas no <a href="/PSP/pt/docs/syncGenerator#falta">contexto (aba) &quot;Falta&quot;</a> são utilizados <strong>tanto para os estudos de <a href="/PSP/pt/docs/fault">curto-circuito</a> quanto para os estudos de <a href="/PSP/pt/docs/harmonics">harmônicos</a></strong>.</p><p>Esses dados são ignorados nos estudos de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a> (<em>não são utilizado nesse estudo</em>) e <a href="/PSP/pt/docs/stability">estabilidade</a> (<em>são utilizados dados inseridos em formulário específico</em>).</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gerador-síncrono-no-fluxo-de-carga"></a>Gerador Síncrono no fluxo de carga<a class="hash-link" href="#gerador-síncrono-no-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>O gerador síncrono é a fonte de potência do PSP-UFU no estudo de fluxo de carga. Seu comportamento difere de acordo com o tipo de barra conectada:</p><ul><li><a href="/PSP/pt/docs/powerFlow">Barra de referência</a>: Os dados de potência ativa e reativa inseridos são desprezados, uma vez que esse elemento será utilizado para completar o balanço de potência do estudo de fluxo d carga;</li><li><a href="/PSP/pt/docs/powerFlow">Barra PV</a>: O dado de potência ativa é considerado, porém o dados de potência reativa são desprezados. O valor da potência reativa é utilizado para manter o módulo da tensão constante no barramento conectado;</li><li><a href="/PSP/pt/docs/powerFlow">Barra PQ</a>: Para os geradores conectados a essa barra tanto a potência ativa quanto reativa inseridas são consideradas.</li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>Caso o limite de potência reativa seja excedido, o programa automaticamente transforma a <a href="/PSP/pt/docs/powerFlow">Barra PV</a> conectada em uma <a href="/PSP/pt/docs/powerFlow">Barra PQ</a>, utilizando o valor limite de potência reativa que seria ultrapassado.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gerador-síncrono-no-estudo-de-curto-circuito"></a>Gerador Síncrono no estudo de curto-circuito<a class="hash-link" href="#gerador-síncrono-no-estudo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>Enquanto os geradores no estudo de fluxo de carga são modelados somente por correntes injetadas nas barras, para o curto-circuito utiliza-se uma tensão atrás de uma impedância. A figura abaixo mostra o caminho da corrente e o circuito equivalente de cada sequência nos geradores.</p><div><center><img src="/PSP/pt/images/sycGeneratorSeq.svg" alt="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero" title="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero"></center></div><p>As tensões geradas são somente de sequência positiva, uma vez que o gerador fornece sempre tensões trifásicas equilibradas. Portanto a rede de sequência positiva é composta de uma tensão pré-falta atrás de uma impedância de sequência positiva. As redes de sequência negativa e zero não contêm forças eletromotrizes, porém incluem as impedâncias do gerador de sequência negativa e zero.</p><p>A corrente que circula na impedância <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> entre o neutro e a terra é <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">3\dot{I}_{a0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>. Pela figura acima (c), observa-se que a queda de tensão de sequência zero do ponto a para terra (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\dot{V}_{a0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>3</mn><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>g</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\dot{V}_{a0} = -3\dot{I}_{a0}\overline{z}_n - \dot{I}_{a0}\overline{z}_{g0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord">−</span><span class="mord">3</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span></div><p>A rede de sequência zero, que é um circuito monofásico pelo qual se supõe que circule apenas corrente de sequência zero e deve, portanto, ter uma impedância definida pela seguinte equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>0</mn></msub><mo>=</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>g</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{0} = 3\overline{z}_n + \overline{z}_{g0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.916668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a seguinte mensagem de erro pode ser exibida:</p><p><center>&quot;<i>Falha ao inverter a matriz admitância de sequência zero</i>&quot;</center></p><p>Isso ocorre porque a matriz admitância de sequência zero é singular. Para contornar esse problema escolha <strong>uma</strong> das duas soluções abaixo:</p><ol><li>Marque a opção &quot;Neutro aterrado&quot; e insira um alto valor de reatância de aterramento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>9999</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j9999~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo);</li><li><strong>Ou</strong>, na barra do gerador, insira um <a href="/PSP/pt/docs/inductor">reator</a> de baixo valor de potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">1,0~var</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault" style="margin-right:0.03588em">v</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>, por exemplo).</li></ol></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="gerador-síncrono-no-estudo-de-estabilidade"></a>Gerador Síncrono no estudo de estabilidade<a class="hash-link" href="#gerador-síncrono-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A relação de valores observados em testes adequados (definidos na <a href="https://doi.org/10.1109/IEEESTD.2020.9050934" target="_blank" rel="noopener noreferrer">IEEE Std. 115-2019</a>), denominados parâmetros padrões, são utilizados para modelar a máquina síncrona no estudo de estabilidade do PSP-UFU.</p><p>Os parâmetros da máquina síncrona que influenciam rapidamente no decaimento de valores são chamados <strong>subtransitórios</strong> (indicados por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">&#x27;&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>), aqueles que influenciam mais lentamente são chamados <strong>transitórios</strong> (indicados por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>) e, finalmente, aqueles que influenciam continuamente são chamados de parâmetros <strong>síncronos</strong> (sem indicação de sobrescrito).</p><p>Um conjunto de equações algébrico-diferenciais determinam o comportamento da máquina síncrona no estudo de estabilidade:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msub><mi>V</mi><mi>q</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>I</mi><mi>d</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msub><mi>V</mi><mi>d</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>I</mi><mi>q</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msub><mi>V</mi><mi>q</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mi>I</mi><mi>d</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msub><mi>V</mi><mi>d</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mi>I</mi><mi>q</mi><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mrow><mi>f</mi><mi>d</mi></mrow></msub><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>d</mi><mn>0</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msub><mi>s</mi><mi>q</mi></msub><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>q</mi><mn>0</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msubsup><mi>x</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>d</mi><mn>0</mn></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msubsup><mi>x</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>q</mi><mn>0</mn></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mfrac><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>ω</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><msub><mi>ω</mi><mi>r</mi></msub><mrow><mn>2</mn><mi>H</mi></mrow></mfrac><mrow><mo fence="true">[</mo><msub><mi>P</mi><mi>m</mi></msub><mo>−</mo><msub><mi>P</mi><mi>e</mi></msub><mo>−</mo><msub><mi>D</mi><mi>a</mi></msub><mrow><mo fence="true">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow><mo fence="true">]</mo></mrow><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>δ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mrow><mo fence="true">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">E_{q}&#x27; - V_q = r_aI_q - x_{ds}&#x27;Id\\ E_{d}&#x27; - V_d = r_aI_d - x_{qs}&#x27;Iq\\ E_{q}&#x27;&#x27; - V_q = r_aI_q - x_{ds}&#x27;&#x27;Id\\ E_{d}&#x27;&#x27; - V_d = r_aI_d - x_{qs}&#x27;&#x27;Iq\\ ~\\ \frac{dE_{q}&#x27;}{dt} = \frac{V_{fd} + \left( x_d - x_{d}&#x27; \right)I_d - s_d E_{q}&#x27;}{T_{d0}&#x27;}\\ \frac{dE_{d}&#x27;}{dt} = \frac{- \left( x_q - x_{q}&#x27; \right)I_q - s_q E_{d}&#x27;}{T_{q0}&#x27;}\\ \frac{dE_{q}&#x27;&#x27;}{dt} = \frac{s_dE_{q}&#x27; + \left( x_{d}&#x27; - x_{d}&#x27;&#x27; \right)I_d - s_d E_{q}&#x27;&#x27;}{T_{d0}&#x27;&#x27;}\\ \frac{dE_{d}&#x27;&#x27;}{dt} = \frac{s_dE_{d}&#x27; + \left( x_{q}&#x27; - x_{q}&#x27;&#x27; \right)I_q - s_d E_{d}&#x27;&#x27;}{T_{q0}&#x27;&#x27;}\\ ~\\ \frac{d\omega}{dt} = \frac{\omega_r}{2H} \left[ P_m - P_e - D_a \left( \omega - \omega_r \right) \right]\\ \frac{d\delta}{dt} = \Omega_b\left( \omega - \omega_r \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault">d</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault" style="margin-right:0.03588em">q</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault">d</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault" style="margin-right:0.03588em">q</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.211em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.512308em;vertical-align:-0.9873080000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.398692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.30130799999999996em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9873080000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.7115240000000003em;vertical-align:-1.088416em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6231080000000002em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.4336920000000006em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4024159999999999em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.088416em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.211em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.512308em;vertical-align:-0.9873080000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.398692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.30130799999999996em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9873080000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.7115240000000003em;vertical-align:-1.088416em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6231080000000002em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.4336920000000006em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4024159999999999em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.088416em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.7935599999999998em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mclose delimcenter" style="top:0em">]</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03785em">δ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></div><p>As duas últimas equações diferenciais são as equações mecânicas da máquina; e as demais são equações elétricas (consulte essa <a href="https://dx.doi.org/10.14393/ufu.te.2019.2444" target="_blank" rel="noopener noreferrer">tese</a> para maiores detalhes acerca dos parâmetros dessas equações).</p><p>Utilizando as equações transitórias e subtransitórias podem-se definir cinco modelos de distintas complexidades.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O modelo da máquina síncrona é <strong>selecionado automaticamente</strong> de acordo com os dados fornecidos ao programa.</p></div></div><p>Na sequência são apresentados tais modelos, incluídos os efeitos da saturação magnética, em conjunto com seus diagramas de blocos:</p><ul><li><strong>Modelo 1</strong>: Corresponde a uma tensão constante atrás de uma reatância transitória de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">x_{d}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.035em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>), não exigindo equações diferenciais;</li><li><strong>Modelo 2</strong>: São representados os efeitos transitórios de eixo direto, sendo necessária a solução de uma equação diferencial (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/pt/images/model2SyncGenerator.svg" alt="Diagrama de blocos do Modelo 2 das máquinas síncronas" title="Diagrama de blocos do Modelo 2 das máquinas síncronas"></center></div><ul><li><strong>Modelo 3</strong>: São representados os efeitos transitórios de eixo direto e em quadratura, exigindo duas equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/pt/images/model3SyncGenerator.svg" alt="Diagrama de blocos do Modelo 3 das máquinas síncronas" title="Diagrama de blocos do Modelo 3 das máquinas síncronas"></center></div><ul><li><strong>Modelo 4</strong>: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de três equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/pt/images/model4SyncGenerator.svg" alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas"></center></div><ul><li><strong>Modelo 5</strong>: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de quatro equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}&#x27;&#x27;}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/pt/images/model5SyncGenerator.svg" alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas"></center></div><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Em todos os modelos as equações diferenciais mecânicas são solucionadas.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="saturação"></a>Saturação<a class="hash-link" href="#saturação" title="Direct link to heading">#</a></h4><p>Para representar matematicamente o efeito da saturação nas equações das máquinas síncronas são introduzidos “fatores de saturação” que modificam as impedâncias do circuito equivalente, os quais dependem de uma reatância de dispersão efetiva, chamada de reatância de Potier (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>).</p><p>Tal reatância pode ser obtida por meio de ensaios (utilizando curvas de saturação de circuito aberto e carga de fator de potência zero) ou estimadas de forma aproximada por outros parâmetros da máquina. A reatância de dispersão (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>l</mi></msub></mrow><annotation encoding="application/x-tex">x_l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), aqui substituída de forma aproximada por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, representa a parcela da reatância da máquina originados do fluxo magnético que percorrem o ar na maioria de seu caminho e, portanto, é independente da saturação.</p><p>O método implementado no programa permite reproduzir a saturação em ambos os eixos (direto e em quadratura), diferindo entre si devido à divergência no tamanho do entreferro. É assumido que a soma vetorial das duas componentes do fluxo magnético saturado está em fase com a f.m.m. e proporcional à Tensão de Potier (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">E_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, a qual é a tensão atrás da reatância de Potier).</p><p>Para isso, são utilizados internamente dois fatores de saturação, sendo um no eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">s_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e outro no eixo em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">s_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>). Esses fatores de saturação são <strong>automaticamente calculados</strong> a cada passo de integração e dependem da curva de saturação da máquina definida pelo <a href="/PSP/pt/docs/syncGenerator#fator-de-satura%C3%A7%C3%A3o">fator de saturação inserido no formulário de edição de dados</a>.</p><p>Portanto as reatâncias saturadas, que devem ser inseridas nas equações algébricas da máquina, são definidas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>d</mi></msub><mo>−</mo><msub><mi>x</mi><mi>p</mi></msub></mrow><mrow><msub><mi>s</mi><mi>d</mi></msub><mo>+</mo><msub><mi>x</mi><mi>p</mi></msub></mrow></mfrac><mspace linebreak="newline"></mspace><msub><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>q</mi></msub><mo>−</mo><msub><mi>x</mi><mi>p</mi></msub></mrow><mrow><msub><mi>s</mi><mi>q</mi></msub><mo>+</mo><msub><mi>x</mi><mi>p</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">x_{ds}=\frac{x_d-x_p}{s_d +x_p}\\ x_{qs}=\frac{x_q-x_p}{s_q +x_p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.232438em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.232438em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Essas equações também são utilizadas para as reatâncias transitórias e subtransitórias, visto que o valor da reatância de Potier (ou de dispersão) não é alterada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="barramento-infinito"></a>Barramento infinito<a class="hash-link" href="#barramento-infinito" title="Direct link to heading">#</a></h4><p>Algumas referências incluem um modelo sem equações diferenciais, em que a máquina é somente representada por uma tensão constante atrás de uma reatância transitória de eixo direto. Tal é utilizado na representação de um <strong>barramento infinito</strong>, o qual é normalmente constituído de um subsistema muito maior àquele simulado.</p><p>No PSP-UFU a <strong>representação de um barramento infinito</strong> pode ser obtido por meio da utilização de uma máquina representada pelo Modelo 1 cujo valor da constante de inércia (H) é infinito ou muito grande (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9999</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">9999~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span>, por exemplo) em relação às demais máquinas do sistema, e o valor de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">x_{d}&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.035em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span> deve ser um valor muito pequeno (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">10^{-3}~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="centro-de-inércia"></a>Centro de inércia<a class="hash-link" href="#centro-de-inércia" title="Direct link to heading">#</a></h4><p>Normalmente utiliza-se a velocidade de referência como sendo a síncrona e portanto, nesse caso, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>r</mi></msub><mo>=</mo><msub><mi>ω</mi><mi>b</mi></msub><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\omega_r = \omega_b = 1,0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> Essa abordagem, adotada por vários livros de estabilidade, considera como referência uma máquina fictícia girando sempre na velocidade síncrona independente das perturbações aplicadas no sistema. No PSP-UFU foi implementado o conceito de centro de inércia (COI, do inglês, <em>Center of Inertia</em>), que constitui uma soma ponderada das velocidades das máquinas presentes no sistema:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>r</mi></msub><mo>=</mo><mfrac><mrow><mo fence="true">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>H</mi><mi>i</mi></msub><msub><mi>ω</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>H</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\omega_r=\frac{\left( \sum_{i=1}^{n} H_i \omega_i \right)}{\left( \sum_{i=1}^{n} H_i \right)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.488004em;vertical-align:-0.994002em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.494002em"><span style="top:-2.305708em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.994002em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:
+<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">n</span></span></span></span></span>	é o número de máquinas síncronas <strong>conectadas</strong> no sistema.</p><p>A aplicação do COI resulta em dados de saída, como o ângulo do rotor, mais fáceis de serem analisados. Na implementação realizada no programa a utilização ou não desse recurso é opcional e pode ser <a href="/PSP/pt/docs/simulationConfig">definida pelo usuário</a>.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-geradores-síncronos"></a>Formulário de edição dos geradores síncronos<a class="hash-link" href="#formulário-de-edição-dos-geradores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos geradores síncronos:</p><div><center><img src="/PSP/pt/images/syncGeneratorForm.png" alt="Formulário dos geradores síncronos no PSP-UFU" title="Formulário dos geradores síncronos no PSP-UFU"></center></div><p>Um segundo pelos dados de estabilidade, como mostra a figura abaixo, acessado ao clicar no botão &quot;Estabilidade&quot; do formulário principal. Nele é possível também acessar aos controles das máquinas síncronas manipulados pelo <a href="/PSP/pt/docs/controlEditor">editor de controle</a>.</p><div><center><img src="/PSP/pt/images/syncGeneratorStabForm.png" alt="Formulário de estabilidade dos geradores síncronos no PSP-UFU" title="Formulário de estabilidade dos geradores síncronos no PSP-UFU"></center></div><p>No formulário de estabilidade pode ser observado o botão &quot;Chaveamento&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do gerador durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/pt/images/syncGeneratorSw.png" alt="Formulário de chaveamento do gerador síncrono" title="Formulário de chaveamento do gerador síncrono"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Chaveamento</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do gerador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do gerador.</p><p>Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso mais de um gerador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-reativas-máxima-e-mínima"></a>Potências reativas máxima e mínima<a class="hash-link" href="#potências-reativas-máxima-e-mínima" title="Direct link to heading">#</a></h4><p>Limites de potência reativa máxima e mínima do gerador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a <a href="/PSP/pt/docs/simulationConfig">potência base do sistema</a>.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedâncias-de-sequência"></a>Impedâncias de sequência<a class="hash-link" href="#impedâncias-de-sequência" title="Direct link to heading">#</a></h4><p>Valores de resistência e reatância para cálculo das correntes de falta. São inseridos dados de sequência positiva, negativa e zero.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedância-de-aterramento"></a>Impedância de aterramento<a class="hash-link" href="#impedância-de-aterramento" title="Direct link to heading">#</a></h4><p>Valores utilizados para o cálculo das correntes de falta do tipo fase-terra e fase-fase-terra. Caso o neutro do gerador não seja aterrado, o valor inserido nesse campo é ignorado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="neutro-aterrado"></a>Neutro aterrado<a class="hash-link" href="#neutro-aterrado" title="Direct link to heading">#</a></h4><p>Indica se o neutro do gerador é aterrado.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="imprimir-dados-da-máquina-síncrona"></a>Imprimir dados da máquina síncrona<a class="hash-link" href="#imprimir-dados-da-máquina-síncrona" title="Direct link to heading">#</a></h4><p>Exibe os dados do gerador síncrono nos gráficos no tempo. Os seguintes dados são exibidos:</p><ul><li>tensão terminal</li><li>potências ativa e reativas</li><li>potência mecânica</li><li>frequência</li><li>tensão de campo</li><li>ângulo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03785em">δ</span></span></span></span></span>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-avr-e-regulador-de-velocidade"></a>Utilizar AVR e regulador de velocidade<a class="hash-link" href="#utilizar-avr-e-regulador-de-velocidade" title="Direct link to heading">#</a></h4><p>Esses campos acionam ou inibem os controles da máquina síncrona, assim como modelos de turbina e excitatrizes.</p><p>Caso selecionadas os controles podem ser inseridos e modificados pelos botões “Editar AVR” e “Editar regulador de velocidade”, os quais acessam o <a href="/PSP/pt/docs/controlEditor">editor de controle do PSP-UFU</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="resistência-de-armadura"></a>Resistência de armadura<a class="hash-link" href="#resistência-de-armadura" title="Direct link to heading">#</a></h4><p>Resistência de armadura da máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatância-de-potier"></a>Reatância de Potier<a class="hash-link" href="#reatância-de-potier" title="Direct link to heading">#</a></h4><p>Reatância de Potier para cálculo da saturação da máquina, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="fator-de-saturação"></a>Fator de saturação<a class="hash-link" href="#fator-de-saturação" title="Direct link to heading">#</a></h4><p>Valor utilizado no cálculo da curva de saturação.</p><p>Representa o valor (em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) de corrente de campo necessária para atingir 1,2 p.u. de tensão terminal.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Esse valor deve ser <strong>maior que 1,2</strong>, ou irá gerar erros na simulação. Caso não seja informado, a saturação da máquina não é considerada nos cálculos.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="frequência-de-circuito-aberto"></a>Frequência de circuito aberto<a class="hash-link" href="#frequência-de-circuito-aberto" title="Direct link to heading">#</a></h4><p>Indica a velocidade da máquina no caso de início da simulação desconectada da rede.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Essa informação é particularmente útil na análise de conexão de geradores dessincronizados na rede.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatâncias-síncronas"></a>Reatâncias síncronas<a class="hash-link" href="#reatâncias-síncronas" title="Direct link to heading">#</a></h4><p>Valores de reatância síncrona (regime permanente) da máquina. Os valores de eixo direto e em quadratura devem ser iguais ou muito próximos para representação de uma máquina de polos lisos, enquanto para polos salientes esses valores são distintos.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatâncias-e-constantes-de-tempo-transitórias"></a>Reatâncias e constantes de tempo transitórias<a class="hash-link" href="#reatâncias-e-constantes-de-tempo-transitórias" title="Direct link to heading">#</a></h4><p>Parâmetros transitórios da máquina síncrona em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> ou segundos.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O valor da reatância transitória de eixo direto deve ser diferente de zero ou levará o programa a erro.</p></div></div><p>De acordo com a quantidade de parâmetros inseridos é definido internamente pelo programa qual o modelo a ser utilizado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="reatâncias-e-constantes-de-tempo-subtransitórias"></a>Reatâncias e constantes de tempo subtransitórias<a class="hash-link" href="#reatâncias-e-constantes-de-tempo-subtransitórias" title="Direct link to heading">#</a></h4><p>Parâmetros subtransitórios da máquina síncrona em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> ou segundos, representando em detalhes a presença de enrolamentos amortecedores. Assim como os dados transitórios, esses parâmetros definem o modelo da máquina.</p></div><div role="tabpanel" hidden=""><p>O botão &quot;Chaveamento&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do gerador durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="acesso-aos-controles-da-máquina-síncrona"></a>Acesso aos controles da máquina síncrona<a class="hash-link" href="#acesso-aos-controles-da-máquina-síncrona" title="Direct link to heading">#</a></h2><p>Como já <a href="/PSP/pt/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">mencionado anteriormente</a>, os reguladores de velocidade e tensão da máquina síncrona podem ser acionados ou inibidos por meio das caixas de seleção &quot;<a href="/PSP/pt/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Utilizar AVR e regulador de velocidade</a>&quot;. Ambas as opções irão acessar o <a href="/PSP/pt/docs/controlEditor">editor de controles</a>.</p><p>O acesso aos controles do <strong>AVR</strong> poderão então ser criados e manipulados ao clicar no botão &quot;Editar AVR&quot;, assim como o <strong>Regulador de Velocidade</strong> é acessado no botão &quot;Editar regulador de velocidade&quot;.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU a opção de editar o <strong>AVR</strong> engloba mais que somente o controle de tensão da máquina. Nele <strong>deve</strong> ser inserida a malha de controle da máquina assim como a <strong>excitatriz da máquina síncrona</strong>. Outras estratégias de controle (opcionais), como PSS (<em>Power System Stabilizer</em>) e/ou controles de sobre e sub excitação, são também implementadas em conjunto.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Assim como no AVR, o <strong>Regulador de Velocidade</strong> engloba mais que a regulação primária da máquina. Nessa opção <strong>deve</strong> ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o <strong>modelo da turbina</strong>. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley &amp; Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994.</li><li>DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: <a href="https://doi.org/10.1109/TPAS.1972.293341" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/TPAS.1972.293341</a></li><li>IEEE Std 1110-2002 IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses. IEEE, New York, nov. 2003. doi: <a href="https://doi.org/10.1109/IEEESTD.2003.94408" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/IEEESTD.2003.94408</a></li><li>KIMBARK, E. W. Power System Stability: Volume III – Synchronous Machine. New York: Wiley-IEEE Press, 1995.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/syncGenerator.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/harmSource"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Fonte de Corrente Harmônica</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/syncMotor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Compensador Síncrono »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#gerador-síncrono-no-psp-ufu" class="table-of-contents__link">Gerador Síncrono no PSP-UFU</a><ul><li><a href="#gerador-síncrono-no-fluxo-de-carga" class="table-of-contents__link">Gerador Síncrono no fluxo de carga</a></li><li><a href="#gerador-síncrono-no-estudo-de-curto-circuito" class="table-of-contents__link">Gerador Síncrono no estudo de curto-circuito</a></li><li><a href="#gerador-síncrono-no-estudo-de-estabilidade" class="table-of-contents__link">Gerador Síncrono no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-dos-geradores-síncronos" class="table-of-contents__link">Formulário de edição dos geradores síncronos</a></li><li><a href="#acesso-aos-controles-da-máquina-síncrona" class="table-of-contents__link">Acesso aos controles da máquina síncrona</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Compensador Síncrono</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><p>Uma máquina síncrona operando sem carga mecânica e fornecendo ou absorvendo energia reativa. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=411-34-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="compensador-síncrono-no-psp-ufu"></a>Compensador Síncrono no PSP-UFU<a class="hash-link" href="#compensador-síncrono-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O compensador síncrono nada mais é que um motor síncrono operando sem carga em seu eixo com o intuito de fornecer ou absorver potência reativa na rede. Em relação ao <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>, no PSP-UFU esse elemento tem comportamento idêntico ao <a href="/PSP/pt/docs/syncGenerator">gerador síncrono</a>, porém a sua potência ativa é inserida nos algoritmos de solução com sinal negativo.</p><p>Sua parametrização para o estudo de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a>, <a href="/PSP/pt/docs/fault">falta</a> e <a href="/PSP/pt/docs/harmonics">harmônicos</a> é idêntica ao <a href="/PSP/pt/docs/syncGenerator">gerador síncrono</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O compensador síncrono <strong>não foi implementado no estudo de <a href="/PSP/pt/docs/stability">estabilidade</a></strong> e sua presença no circuito pode causar erros durante os cálculos.</p><p>Não inclua esse elemento para os estudos de <a href="/PSP/pt/docs/stability">estabilidade</a>. Versões futuras do PSP-UFU irão contemplar esse elemento para esse estudo.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>É possível utilizar um <a href="/PSP/pt/docs/syncGenerator">gerador síncrono</a> com potência ativa <em>nula</em> ou <em>negativa</em> para representar o compensador síncrono nos estudos de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-compensadores-síncronos"></a>Formulário de edição dos compensadores síncronos<a class="hash-link" href="#formulário-de-edição-dos-compensadores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos compensadores síncronos:</p><div><center><img src="/PSP/pt/images/syncMotorForm.png" alt="Formulário dos compensadores síncronos no PSP-UFU" title="Formulário dos compensadores síncronos no PSP-UFU"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do compensador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do compensador.</p><p>Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso mais de um compensador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potências-reativas-máxima-e-mínima"></a>Potências reativas máxima e mínima<a class="hash-link" href="#potências-reativas-máxima-e-mínima" title="Direct link to heading">#</a></h4><p>Limites de potência reativa máxima e mínima do compensador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do compensador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a <a href="/PSP/pt/docs/simulationConfig">potência base do sistema</a>.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedâncias-de-sequência"></a>Impedâncias de sequência<a class="hash-link" href="#impedâncias-de-sequência" title="Direct link to heading">#</a></h4><p>Valores de resistência e reatância para cálculo das correntes de falta. São inseridos dados de sequência positiva, negativa e zero.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedância-de-aterramento"></a>Impedância de aterramento<a class="hash-link" href="#impedância-de-aterramento" title="Direct link to heading">#</a></h4><p>Valores utilizados para o cálculo das correntes de falta do tipo fase-terra e fase-fase-terra. Caso o neutro do gerador não seja aterrado, o valor inserido nesse campo é ignorado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="neutro-aterrado"></a>Neutro aterrado<a class="hash-link" href="#neutro-aterrado" title="Direct link to heading">#</a></h4><p>Indica se o neutro do gerador é aterrado.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/syncMotor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/syncGenerator"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Gerador Síncrono</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/indMotor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Motor de Indução »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#compensador-síncrono-no-psp-ufu" class="table-of-contents__link">Compensador Síncrono no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-compensadores-síncronos" class="table-of-contents__link">Formulário de edição dos compensadores síncronos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/pt/docs/installation">Instalação</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mainScreen">Tela Principal</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/cadTools">Ferramentas CAD</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Editor de Elementos de Potência</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/powerEditor">Editor de Potência</a></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!" tabindex="-1">Eelementos de Potência</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/bus">Barramento</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/line">Linha</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/transformer">Transformador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/load">Carga</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/capacitor">Capacitor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/inductor">Indutor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmSource">Fonte de Corrente Harmônica</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/syncGenerator">Gerador Síncrono</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/syncMotor">Compensador Síncrono</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/indMotor">Motor de Indução</a></li></ul></li></ul></li><li 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href="/PSP/pt/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Relatórios Tabulares</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>Para visualização de todos os resultados dos cálculos de <a href="/PSP/pt/docs/powerFlow">fluxo de carga</a> e <a href="/PSP/pt/docs/fault">curto-circuito</a>, desenvolveu-se um visualizador de relatórios tabulares:</p><div><center><img src="/PSP/pt/images/tabReport.png" alt="Visualizador de dados tabulados no PSP-UFU" title="Visualizador de dados tabulados no PSP-UFU"></center></div><p>Tal visualizador pode ser acessado no <a href="/PSP/pt/docs/mainScreen#menu-ribbon">submenu Ribbon</a> <strong>Ferramentas</strong> e então no botão <strong>Relatório de dados</strong>, sendo exibido o formulário organizado em abas.</p><p>As grandezas de alguns dados desse relatório podem ser modificadas, sendo selecionadas no cabeçalho da tabela, como mostra a figura acima.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Os valores podem ser exportados para editores de texto e tabelas externos, selecionando e copiando (Ctrl+C) para a área de transferência.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/tabularReport.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/text"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Texto Vinculado</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/graphViewer"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Visualizador de Gráficos »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. Construído com Docusaurus.</div></div></div></footer></div>
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href="/PSP/pt/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Texto Vinculado</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>O elemento de texto vinculado (ao elemento de potência) pode ser inserido no editor de <a href="/PSP/pt/docs/powerEditor">elementos de potência</a>. Esse elemento de texto pode ser vinculado a inúmeros resultados e parâmetros do sistema de potência.</p><p>Esse elemento é útil para identificação e análise dos resultados de elementos de potência.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Outra aplicação importante é a atualização de seu valor ao aplicar a “solução contínua” no projeto, auxiliando na visualização das mudanças de estado da rede elétrica.</p><p>Também pode-se utilizar os resultados apresentados por esses elementos juntamente com o diagrama unifilar na criação de relatórios do projeto.</p></div></div><div><center><img src="/PSP/pt/images/textExp.png" alt="Exemplo do elemento de texto no PSP-UFU" title="Exemplo de elemento de texto no PSP-UFU"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="inserindo-elementos-de-texto-no-psp-ufu"></a>Inserindo elementos de texto no PSP-UFU<a class="hash-link" href="#inserindo-elementos-de-texto-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Por meio do submenu <strong>Ferramentas</strong> pode-se inserir os elementos de texto, como é apresentado na imagem abaixo.</p><div><center><img src="/PSP/pt/images/addTextElement.svg" alt="Adicionar elementos de texto" title="Adicionar elementos de texto"></center></div><p>Ao selecionar o botão &quot;Texto&quot; ou utilizar a <strong>tecla de atalho &quot;A&quot;</strong>, o usuário deve clicar na posição desejada da <a href="/PSP/pt/docs/powerEditor">área de trabalho</a>. Com isso, será adicionado um elemento temporário com o valor &quot;Texto&quot;. Para parametrizar esse elemento clique duas vezes sobre ele.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O elemento de texto, assim como os <a href="/PSP/pt/docs/controlEditor">elementos de controle</a> não exibem automaticamente o formulário de edição de dados ao inseri-lo pela primeira vez.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-do-elemento-de-texto"></a>Formulário de edição de dados do elemento de texto<a class="hash-link" href="#formulário-de-edição-de-dados-do-elemento-de-texto" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados desse componente gráfico.</p><div><center><img src="/PSP/pt/images/textForm.png" alt="Formulário do elemento de texto no PSP-UFU" title="Formulário do elemento de texto no PSP-UFU"></center></div><p>Sua parametrização é realizada por campos que associam o componente aos elementos elétricos de potência, além do formato dos resultados numéricos. Os campos são liberados a medida que os dados anteriores são inseridos e uma mensagem de erro é apresentada ao usuário caso algum parâmetro necessário não seja preenchido.</p><p>O processo de associação se inicia na escolha do tipo de elemento de potência será vinculado ao texto. Em seguida é exibida uma lista com o nome de todos os componentes previamente inseridos no editor de elementos de potência do tipo definido no campo anterior, associando, assim, o texto ao elemento.</p><p>Em “Tipo de texto” define-se qual será a saída apresentada. Tal campo depende do tipo de elemento selecionado, como mostra a tabela abaixo. Alguns outros dados são necessários para os elementos de ramo, como barra de origem e destino, com o intuito de representar o fluxo correto de potência e corrente. Finalmente o usuário deve inserir a quantidade de casas decimais para dados numéricos. A pré-visualização pode ser observada no formulário.</p><table><thead><tr><th align="center">Elemento de potência</th><th>Tipo de texto</th><th>Unidades disponíveis</th></tr></thead><tbody><tr><td align="center"><a href="/PSP/pt/docs/bus">Barra</a></td><td>Nome<br>Tensão<br>Ângulo<br>Corrente de falta<br>Tensão de falta<br>Nível de curto-circuito<br>THD de tensão</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, V, kV<br>Graus, Radianos<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, V, kV<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, VA, kVA, MVA<br>%</td></tr><tr><td align="center"><a href="/PSP/pt/docs/syncGenerator">Gerador</a></td><td>Nome<br>Potência ativa<br>Potência reativa<br>Corrente de falta</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA</td></tr><tr><td align="center"><a href="/PSP/pt/docs/line">Linha</a> e <a href="/PSP/pt/docs/transformer">transformador</a></td><td>Nome<br>Fluxo de potência ativo<br>Fluxo de potência reativo<br>Perdas<br>Corrente<br>Corrente de falta</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, A, kA</td></tr><tr><td align="center"><a href="/PSP/pt/docs/load">Carga</a>, <a href="/PSP/pt/docs/indMotor">Motor de indução</a><br>e <a href="/PSP/pt/docs/syncMotor">Compensador síncrono</a></td><td>Nome<br>Potência ativa<br>Potência reativa</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, W, kW, MW<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar</td></tr><tr><td align="center"><a href="/PSP/pt/docs/capacitor">Capacitor</a> e <a href="/PSP/pt/docs/inductor">Indutor</a></td><td>Nome<br>Potência reativa</td><td>-<br><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, var, kvar, Mvar</td></tr></tbody></table><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Uma vez inseridos, esses elementos de texto podem ser manipulados da <a href="/PSP/pt/docs/cadTools">mesma forma que os elementos de potência</a>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/text.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 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tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Função Transferência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><p>A Função Transferência é um importante bloco elementar e essencial em um sistema de controle, visto que esse componente possibilita a representação de sistemas dinâmicos por meio de equações algébricas no domínio da frequência.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Nesses blocos é utilizado a letra “<strong>s</strong>” para representação no domínio da frequência, pois essa é a terminologia mais utilizada em sistemas de controle.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="função-transferência-no-psp-ufu"></a>Função Transferência no PSP-UFU<a class="hash-link" href="#função-transferência-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a inserção da função transferência, o programa a transforma na representação em espaço de estado, visto que sua solução é sistemática tornando-a, assim, mais eficiente computacionalmente.</p><p>Dada uma função transferência genérica abaixo:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>b</mi><mn>0</mn></msub><msup><mi mathvariant="bold">s</mi><mi>k</mi></msup><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><msup><mi mathvariant="bold">s</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>b</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="bold">s</mi><mo>+</mo><msub><mi>b</mi><mi>k</mi></msub></mrow><mrow><msup><mi mathvariant="bold">s</mi><mi>k</mi></msup><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msup><mi mathvariant="bold">s</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="bold">s</mi><mo>+</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{Y(\bold{s})}{U(\bold{s})} = \frac{b_0 \bold{s}^k + b_1 \bold{s}^{k-1} + \cdots + b_{k-1} \bold{s} + b_k}{\bold{s}^k + a_1 \bold{s}^{k-1} + \cdots + a_{k-1} \bold{s} + a_k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em">U</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span></span><span class="mclose">)</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.420439em;vertical-align:-0.894331em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.526108em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751079999999999em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751079999999999em"><span style="top:-2.989em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">s</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">s</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.894331em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A transformação em espaço de estado na <em>forma canônica controlável</em> ficará da seguinte forma:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><msub><mi>u</mi><mi>n</mi></msub><mspace linebreak="newline"></mspace><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>b</mi><mn>0</mn></msub><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \bold{s}x_1\\ \bold{s}x_2\\ \vdots\\ \bold{s}x_{k-1}\\ \bold{s}x_k \end{bmatrix} = \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; \cdots &amp; 0\\ 0 &amp; 0 &amp; 1 &amp; \cdots &amp; 0\\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 1\\ -a_k &amp; -a_{k-1} &amp; -a_{k-2} &amp; \cdots &amp; -a_1 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_{k-1}\\ x_k \end{bmatrix} + \begin{bmatrix} 0\\ 0\\ \vdots\\ 0\\ 1 \end{bmatrix} u_n\\ y_n = \begin{bmatrix} \left( b_n - a_n b_0 \right) &amp; \left( b_{n-1} - a_{n-1} b_0 \right) &amp; \cdots &amp; \left( b_1 - a_1 b_0 \right) \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_k \end{bmatrix} + b_0 u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.240000000000001em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-5.04em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.18em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-1.9800000000000009em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-0.7800000000000009em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953995em"><span style="top:-1.3499850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.4999850000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0959850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6919850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.712975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em"><span style="top:-5.8275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-1.5675000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953995em"><span style="top:-1.3499850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.4999850000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0959850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6919850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.712975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>A equação diferencial matricial é resolvida pelo método <strong>Trapezoidal Implícito</strong>. Um processo iterativo é realizado até que o erro entre os cálculos se torne menor que uma tolerância pré-estipulada e caso as iterações se tornem excessivas, ultrapassando um número máximo definido pelo usuário, o processo é interrompido com erro. Com os valores do vetor de estado (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>) calculados obtém-se o valor da saída do bloco (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>).</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-de-dados-da-função-transferência"></a>Formulário de edição de dados da Função Transferência<a class="hash-link" href="#formulário-de-edição-de-dados-da-função-transferência" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados da função transferência.</p><div><center><img src="/PSP/pt/images/transferFunctionForm.png" alt="Formulário de edição de dados da função transferência no PSP-UFU" title="Formulário de edição de dados da função transferência no PSP-UFU"></center></div><p>A função transferência é definida pelos coeficientes do numerador e denominador, <strong>separados por espaço</strong>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Os elementos não presentes são representados como coeficientes de valor zero.</p><p>Por exemplo, um parâmetro inserido igual a “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} 1 &amp; 0{,}5 &amp; 0 &amp; 2 \end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2000000000000002em;vertical-align:-0.35000000000000003em"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span></span></span></span></span>” irá gerar: “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>s</mi><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">s^3+0{,}5s^2+2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span>”.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="referências"></a>Referências<a class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>OGATA, K. Modern Control Engineering. Prentice Hall Inc., New Jersey, 2004.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/transferFunction.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/io"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Entrada / Saída</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/sum"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Somador »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#função-transferência-no-psp-ufu" class="table-of-contents__link">Função Transferência no PSP-UFU</a></li><li><a href="#formulário-de-edição-de-dados-da-função-transferência" class="table-of-contents__link">Formulário de edição de dados da Função Transferência</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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href="/PSP/pt/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/pt/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_3ufF"><div class="container padding-vert--lg docItemWrapper_3FMP"><div class="row"><div class="col docItemCol_3FnS"><div class="docItemContainer_33ec"><article><header><h1 class="docTitle_3a4h">Transformador</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/pt/katex/katex.min.css"><blockquote><p>Aparelho estático com dois ou mais enrolamentos que, por indução eletromagnética, transforma um sistema de tensão e corrente alternada em outro sistema de tensão e corrente geralmente de valores diferentes e na mesma frequência com o objetivo de transmitir energia elétrica. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&amp;ievref=421-01-01" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="transformador-no-psp-ufu"></a>Transformador no PSP-UFU<a class="hash-link" href="#transformador-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Transformadores com quaisquer conexões e defasagens entre as tensões primárias e secundárias podem ser inseridos no PSP-UFU. Para isso, os transformadores devem ser modelados representando seu tape (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em"></span><span class="mord mathdefault">t</span></span></span></span></span>) e defasagem (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">ϕ</span></span></span></span></span>), utilizando um transformador ideal de relação de transformação <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>:</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\overline{t}:1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.81508em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> em série com sua impedância (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mo>=</mo><msub><mi>r</mi><mi>T</mi></msub><mo>+</mo><mi>j</mi><msub><mi>x</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">\overline{y}_T = r_T + jx_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8747em;vertical-align:-0.24414em"></span><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.81508em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span> é um número complexo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi mathvariant="normal">∠</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">t\angle\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">t</span><span class="mord">∠</span><span class="mord mathdefault">ϕ</span></span></span></span></span>)</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O modelo utilizado no PSP-UFU <strong>não considera</strong> o ramo magnetizante ou impedâncias mútuas entre as fases.</p></div></div><p>A figura abaixo mostra o modelo do transformador implementado no PSP-UFU:</p><div><center><img src="/PSP/pt/images/transformerModel.svg" alt="Modelo PI da linha" title="Modelo PI da linha"></center></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mi mathvariant="bold">T</mi></msub></mrow><annotation encoding="application/x-tex">\bold{r_T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a resistência total do transformador;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">T</mi></msub></mrow><annotation encoding="application/x-tex">\bold{x_T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span>	é a reatância indutiva total do transformador;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">t</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\bold{\overline{t}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8349200000000001em;vertical-align:0em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8349200000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathbf">t</span></span></span><span style="top:-3.7549200000000003em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span></span>	é a o tape complexo do transformador.</li></ul><p>A relação entre as tensões e correntes das barras genéricas <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span></span></span></span></span> é obtida pela equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>j</mi><mi>i</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mi>t</mi><mn>2</mn></msup></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>∗</mo></msup></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mfrac></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>×</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>j</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \dot{I}_{ij}\\ \dot{I}_{ji} \end{bmatrix} = \begin{bmatrix} \displaystyle \frac{\overline{y}_T}{t^2} &amp; -\displaystyle \frac{\overline{y}_T}{\overline{t}^*}\\ -\displaystyle \frac{\overline{y}_T}{\overline{t}} &amp; \overline{y}_T \end{bmatrix} \times \begin{bmatrix} \dot{V}_{i}\\ \dot{V}_{j} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.5603800000000003em;vertical-align:-1.0301900000000004em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.53019em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3298099999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0301900000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.202030000000001em;vertical-align:-1.850025em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000007em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2107020000000004em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.29492em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.70508em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019379999999998em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000007em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.216324em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.893676em"><span style="top:-3.26798em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.783676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2107020000000004em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019379999999998em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.5603800000000003em;vertical-align:-1.0301900000000004em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.53019em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3298099999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0301900000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span></span></span></span></span></div><p>Assim como nas <a href="/PSP/pt/docs/line">linhas</a>, o transformador de dois enrolamentos deve ser inserido entre barras, porém não há a necessidade das tensões nominais desses barramentos serem idênticas.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A primeira barra selecionada será o lado <strong>primário</strong>, o qual é indicado por um círculo. Para alterar o lado primário após a inserção basta desconectar os nós dos transformadores e reconectá-los alternando as barras, utilizando as <a href="/PSP/pt/docs/cadTools#conex%C3%A3o-e-desconex%C3%A3o-de-elemetos-na-barra">ferramentas CAD</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="transformadores-no-estudo-de-curto-circuito"></a>Transformadores no estudo de curto-circuito<a class="hash-link" href="#transformadores-no-estudo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas <a href="/PSP/pt/docs/fault">faltas desbalanceadas</a> é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.</p><p>A matriz admitância de sequência positiva é construída da mesma forma que os estudos de fluxo de carga. Na sequência negativa o ângulo de defasagem entre tensões primária e secundária (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">ϕ</span></span></span></span></span>) deve ser invertido, uma vez que os fasores da sequência negativa <strong>são deslocados na direção oposta</strong>. Com isso, a matriz que representa as admitâncias da equação anterior deve ser substituída pela seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>j</mi><mi>i</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>j</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mi>t</mi><mn>2</mn></msup></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>∗</mo></msup></mfrac></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \overline{Y}_{ii} &amp; \overline{Y}_{ij}\\ \overline{Y}_{ji} &amp; \overline{Y}_{jj} \end{bmatrix} = \begin{bmatrix} \displaystyle \frac{\overline{y}_T}{t^2} &amp; -\displaystyle \frac{\overline{y}_T}{\overline{t}}\\ -\displaystyle \frac{\overline{y}_T}{\overline{t}^*} &amp; \overline{y}_T \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4866600000000005em;vertical-align:-0.9933300000000005em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.49333em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3666699999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9933300000000005em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.49333em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3666699999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9933300000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.202030000000001em;vertical-align:-1.850025em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000003em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2892979999999996em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.216324em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.893676em"><span style="top:-3.26798em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.783676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019380000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000003em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.29492em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.70508em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2892979999999996em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8019380000000003em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>A impedância de sequência negativa nos transformadores deve ser tratada de maneira específica devido aos diferentes tipos de conexão. Na figura abaixo são mostrados os circuitos equivalentes para cada tipo de conexão de transformadores de dois enrolamentos. As setas indicam os caminhos possíveis para circulação da corrente de sequência zero.</p><div><center><img src="/PSP/pt/images/seqZeroTrafo.svg" alt="Circuitos equivalentes de sequência zero dos transformadores" title="Circuitos equivalentes de sequência zero dos transformadores"></center></div><p>Para os transformadores com conexão estrela aterrado, se a ligação do neutro para a terra apresentar uma impedância <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, o circuito equivalente de sequência zero deve ter impedância de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">3\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> em série com a resistência e reatância equivalentes do transformador.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="formulário-de-edição-dos-transformadores"></a>Formulário de edição dos transformadores<a class="hash-link" href="#formulário-de-edição-dos-transformadores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos transformadores:</p><div><center><img src="/PSP/pt/images/trafoForm.png" alt="Formulário dos transformadores no PSP-UFU" title="Formulário dos transformadores no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais dos transformadores, informações do fluxo de carga, sua conexão e defasagem;</li><li><strong>Falta</strong>: local onde as impedâncias de sequência zero e impedâncias de aterramento são inseridas.</li></ul><p>Além desses dois contextos, pode ser observado o botão &quot;Estabilidade&quot; na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/pt/images/trafoSw.png" alt="Formulário de chaveamento do transformador" title="Formulário de chaveamento do transformador"></center></div><div class="tabs-container"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_1uMI tabs__item--active">Geral</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Falta</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_1uMI">Botão Estabilidade</li></ul><div class="margin-vert--md"><div role="tabpanel"><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="nome"></a>Nome<a class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-nominal"></a>Tensão nominal<a class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Campo de informação <em>não editável</em> que apresenta a tensão primária e secundária do transformador. Para alterar esse campo é necessário editar o campo correspondente dos <a href="/PSP/pt/docs/bus#geral">barramento</a> conectados.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tensão-base"></a>Tensão base<a class="hash-link" href="#tensão-base" title="Direct link to heading">#</a></h4><p>Tensão utilizada para converter os parâmetros de impedância inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> As tensões selecionadas por uma caixa de escolha são editadas nos formulários dos <a href="/PSP/pt/docs/bus#geral">barramentos</a> conectados.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A tensão base é ignorada caso os dados inseridos estejam em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="potência-nominal"></a>Potência nominal<a class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do transformador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção &quot;Utilizar a potência nominal como base&quot; esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedância-do-transformador"></a>Impedância do transformador<a class="hash-link" href="#impedância-do-transformador" title="Direct link to heading">#</a></h4><p>Impedância série do transformador (resistência e reatância indutiva), inserida em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span>. Caso as impedâncias sejam inseridas em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> será utilizada a tensão base selecionada para conversão para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="conexão"></a>Conexão<a class="hash-link" href="#conexão" title="Direct link to heading">#</a></h4><p>Conexão do transformador utilizada para cálculo das correntes de sequência zero nas faltas desbalanceadas com presença de terra (fase-terra e fase-fase-terra).</p><p>As seguintes conexões estão disponíveis:</p><ol><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y_{aterrado}-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y_{aterrado}-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\Delta-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">Y_{aterrado}-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">Y-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li></ol><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="tap"></a>TAP<a class="hash-link" href="#tap" title="Direct link to heading">#</a></h4><p>Tape do transformador <strong>em relação ao primário</strong>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O tape no PSP-UFU deve ser inserido pela relação entre a tensão primária nominal e a tensão de tape <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtext>TAP</mtext><mo>=</mo><mfrac><msub><mi>V</mi><mrow><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>l</mi></mrow></msub><msub><mi>V</mi><mrow><mi>t</mi><mi>a</mi><mi>p</mi><mi>e</mi></mrow></msub></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\text{TAP} = \frac{V_{nominal}}{V_{tape}}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord text"><span class="mord">TAP</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.894191em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em"><span style="top:-2.357em;margin-left:-0.22222em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.41586em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3487714285714287em;margin-left:-0.22222em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5423199999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span></span>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="defasagem"></a>Defasagem<a class="hash-link" href="#defasagem" title="Direct link to heading">#</a></h4><p>Defasagem do transformador em relação ao primário, em graus. Esse valor independe do tipo de conexão, portanto para conexões diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span>(aterrado ou não) e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span>, deve-se inserir o ângulo de defasagem correto.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do transformador como base para a conversão das unidades, caso contrário será usada a <a href="/PSP/pt/docs/simulationConfig">potência base do sistema</a>.</p></div><div role="tabpanel" hidden=""><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedância-de-sequência-zero-do-transformador"></a>Impedância de sequência zero do transformador<a class="hash-link" href="#impedância-de-sequência-zero-do-transformador" title="Direct link to heading">#</a></h4><p>Esses parâmetros são necessários para o correto cálculo das correntes de <a href="/PSP/pt/docs/fault">falta desbalanceadas</a> (fase-fase, fase-fase-terra e fase-terra) e devem ser inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A não inserção desses dados acarretarão em resultados incorretos para <a href="/PSP/pt/docs/fault">faltas desbalanceadas</a>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_2LWZ" id="impedâncias-de-aterramento"></a>Impedâncias de aterramento<a class="hash-link" href="#impedâncias-de-aterramento" title="Direct link to heading">#</a></h4><p>As impedâncias de aterramento são somente utilizadas para conexão <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e devem ser inseridas em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p></div><div role="tabpanel" hidden=""><p>O botão &quot;Estabilidade&quot; irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/pt/docs/stability">estabilidade</a>.</p><p>Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões &quot;Adicionar&quot; e &quot;Remover&quot;, respectivamente.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/transformer.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" role="img" viewBox="0 0 40 40" class="iconEdit_2_ui" aria-label="Edit page"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Docs pages navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/pt/docs/line"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Linha</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/pt/docs/load"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Carga »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_35-E thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#transformador-no-psp-ufu" class="table-of-contents__link">Transformador no PSP-UFU</a><ul><li><a href="#transformadores-no-estudo-de-curto-circuito" class="table-of-contents__link">Transformadores no estudo de curto-circuito</a></li></ul></li><li><a href="#formulário-de-edição-dos-transformadores" class="table-of-contents__link">Formulário de edição dos transformadores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. 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diff --git a/docs/pt/images/mainScreen.svg b/docs/pt/images/mainScreen.svg
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diff --git a/docs/pt/images/stabInit.svg b/docs/pt/images/stabInit.svg
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new file mode 100644
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hero--primary heroBanner_3P7f"><div class="container"><h1 class="hero__title">PSP-UFU</h1><p class="hero__subtitle">Plataforma de Sistemas de Potência da Universidade Federal de Uberlândia</p><p><center><iframe src="https://ghbtns.com/github-btn.html?user=thales1330&amp;repo=psp&amp;type=star&amp;count=true&amp;size=large" frameborder="0" scrolling="0" width="130" height="30" title="GitHub"></iframe></center></p><div class="buttons_1r9m"><a href="https://github.com/Thales1330/PSP/releases/latest" target="_blank" rel="noopener noreferrer" class="button button--outline button--secondary button--lg">Download</a></div><div class="buttons_1r9m"><a class="button button--outline button--secondary button--lg" href="/PSP/pt/docs/">Manual do Usuário</a></div><div class="buttons_1r9m"><a href="https://minhaskamal.github.io/DownGit/#/home?url=https://github.com/Thales1330/PSP/tree/master/Project/data/samples" target="_blank" rel="noopener noreferrer" class="button button--outline button--secondary button--lg">Amostras</a></div></div></header><main><section class="features_3azU"><div class="container"><div class="row"><div class="col col--4"><div class="text--center"><img class="featureImage_ZtzX" src="/PSP/pt/img/complete-platform.svg" alt="[object Object]"></div><h3>Plataforma Completa</h3><p>O PSP-UFU é um <b>Software Livre e de Código Aberto (FOSS), multiplataforma e multi-idiomas</b> com <b>recursos de GUI (Interface Gráfica do Usuário) avançados</b> e <b>ferramentas CAD (Desenho Assistido por Computador)</b> para estudos de sistemas elétricos de potência..</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_ZtzX" src="/PSP/pt/img/advanced-cad.svg" alt="[object Object]"></div><h3>Recursos CAD avançados</h3><p>O software permite a construção de qualquer <b>rede de transmissão elétrica</b> e <b>sistemas de controle</b> por meio da inserção de elementos visuais.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_ZtzX" src="/PSP/pt/img/visualization.svg" alt="[object Object]"></div><h3>Fácil Vizualização</h3><p>Para a visualização dos resultados, o programa oferece elementos de texto vinculados na tela principal, mapa de calor de tensão e também editores de tabelas e gráficos.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_ZtzX" src="/PSP/pt/img/application.svg" alt="[object Object]"></div><h3>Aplicação</h3><p>O PSP-UFU tem como objetivo fornecer ferramentas eficientes de simulação computacional para <b>fins de pesquisa e educação</b>, além de <b>aplicações industriais</b> no sistema elétrico de potência.</p></div></div></div></section><div class="container"><h2>Estudos Realizados</h2></div><section class="features_3azU"><div class="container"><div class="row"><div class="col col--4"><h3>Fluxo de carga</h3><ul><li>Newton-Raphson</li><li>Gauss-Seidel</li><li>Newton-Gauss Híbrido</li><li>Motores de indução trifásicos inclusos</li></ul></div><div class="col col--4"><h3>Cálculod e curto-circuito</h3><ul><li>Balanceado</li><li>Desbalanceado</li><li>Potência de curto-circuito em todas as barras</li></ul></div><div class="col col--4"><h3>Harmônicos</h3><ul><li>Cálculo das tensões harmônicas e DHT (Distorção Harmônica de Tensão)</li><li>Varredura de frequências</li></ul></div><div class="col col--4"><h3>Estabilidade transitória e dinâmica</h3><ul><li>Vários modelos de máquinas síncronas selecionados automaticamente</li><li>Motores de indução trifásicos</li><li>Controles de máquina definidos pelo usuário, excitatrizes e reguladores de velocidade criados usando diagramas de blocos (Excitatrizes, AVR, PSS, turbinas hidrelétricas e térmicas, regulador de velocidade, etc.)</li></ul></div></div></div></section><div class="container"><h2>Artigos Publicados</h2>Maiores detalhes podem ser encontrados nos artigos publicados:<blockquote cite="https://doi.org/10.1002/2050-7038.12185">Oliveira, T. L., Guimarães, G. C., &amp; Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. <i>International Transactions on Electrical Energy Systems</i>, e12185. doi: <a href="https://doi.org/10.1002/2050-7038.12185">10.1002/2050-7038.12185</a></blockquote><blockquote cite="https://doi.org/10.1177/0020720919879058">Oliveira, T. L., Guimarães, G. C., Silva, L. R., &amp; Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. <i>The International Journal of Electrical Engineering &amp; Education</i>, 0020720919879058. doi: <a href="https://doi.org/10.1177/0020720919879058">10.1177/0020720919879058</a></blockquote></div><div class="container"><h2>Documentação do Código-Fonte</h2><p>Todas as descrições detalhadas do código-fonte podem ser encontradas na <b><a href="https://thales1330.github.io/PSP/doxygen/html/index.html">Documentação Online</a></b>, gerada pelo <a href="http://www.doxygen.org/">Doxygen</a>.</p></div><div class="container"><h2>Screenshots</h2><center><iframe src="gallery.html" style="position:relative;width:100%;height:56.25vw;display:block;overflow:hidden;border:none">Seu navegador não oferece suporte para esse recurso.</iframe></center></div></main></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Manual</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/pt/docs/">Manual do PSP-UFU</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. Construído com Docusaurus.</div></div></div></footer></div>
+<script src="/PSP/pt/assets/js/styles.cc4b2cfe.js"></script>
+<script src="/PSP/pt/assets/js/runtime~main.b7ffaf86.js"></script>
+<script src="/PSP/pt/assets/js/main.168dd2d0.js"></script>
+<script src="/PSP/pt/assets/js/1.c1697df3.js"></script>
+<script src="/PSP/pt/assets/js/2.6b1d437b.js"></script>
+<script src="/PSP/pt/assets/js/c4f5d8e4.4dc43757.js"></script>
+</body>
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diff --git a/docs/pt/js/blueimp-gallery-fullscreen.js b/docs/pt/js/blueimp-gallery-fullscreen.js
new file mode 100644
index 0000000..7a1139d
--- /dev/null
+++ b/docs/pt/js/blueimp-gallery-fullscreen.js
@@ -0,0 +1,86 @@
+/*
+ * blueimp Gallery Fullscreen JS
+ * https://github.com/blueimp/Gallery
+ *
+ * Copyright 2013, Sebastian Tschan
+ * https://blueimp.net
+ *
+ * Licensed under the MIT license:
+ * https://opensource.org/licenses/MIT
+ */
+
+/* global define */
+
+;(function (factory) {
+  'use strict'
+  if (typeof define === 'function' && define.amd) {
+    // Register as an anonymous AMD module:
+    define(['./blueimp-helper', './blueimp-gallery'], factory)
+  } else {
+    // Browser globals:
+    factory(window.blueimp.helper || window.jQuery, window.blueimp.Gallery)
+  }
+})(function ($, Gallery) {
+  'use strict'
+
+  var galleryPrototype = Gallery.prototype
+
+  $.extend(galleryPrototype.options, {
+    // Defines if the gallery should open in fullscreen mode:
+    fullscreen: false
+  })
+
+  var initialize = galleryPrototype.initialize
+  var close = galleryPrototype.close
+
+  $.extend(galleryPrototype, {
+    getFullScreenElement: function () {
+      return (
+        document.fullscreenElement ||
+        document.webkitFullscreenElement ||
+        document.mozFullScreenElement ||
+        document.msFullscreenElement
+      )
+    },
+
+    requestFullScreen: function (element) {
+      if (element.requestFullscreen) {
+        element.requestFullscreen()
+      } else if (element.webkitRequestFullscreen) {
+        element.webkitRequestFullscreen()
+      } else if (element.mozRequestFullScreen) {
+        element.mozRequestFullScreen()
+      } else if (element.msRequestFullscreen) {
+        element.msRequestFullscreen()
+      }
+    },
+
+    exitFullScreen: function () {
+      if (document.exitFullscreen) {
+        document.exitFullscreen()
+      } else if (document.webkitCancelFullScreen) {
+        document.webkitCancelFullScreen()
+      } else if (document.mozCancelFullScreen) {
+        document.mozCancelFullScreen()
+      } else if (document.msExitFullscreen) {
+        document.msExitFullscreen()
+      }
+    },
+
+    initialize: function () {
+      initialize.call(this)
+      if (this.options.fullscreen && !this.getFullScreenElement()) {
+        this.requestFullScreen(this.container[0])
+      }
+    },
+
+    close: function () {
+      if (this.getFullScreenElement() === this.container[0]) {
+        this.exitFullScreen()
+      }
+      close.call(this)
+    }
+  })
+
+  return Gallery
+})
diff --git a/docs/pt/js/blueimp-gallery-indicator.js b/docs/pt/js/blueimp-gallery-indicator.js
new file mode 100644
index 0000000..ee1b373
--- /dev/null
+++ b/docs/pt/js/blueimp-gallery-indicator.js
@@ -0,0 +1,148 @@
+/*
+ * blueimp Gallery Indicator JS
+ * https://github.com/blueimp/Gallery
+ *
+ * Copyright 2013, Sebastian Tschan
+ * https://blueimp.net
+ *
+ * Licensed under the MIT license:
+ * https://opensource.org/licenses/MIT
+ */
+
+/* global define */
+
+;(function (factory) {
+  'use strict'
+  if (typeof define === 'function' && define.amd) {
+    // Register as an anonymous AMD module:
+    define(['./blueimp-helper', './blueimp-gallery'], factory)
+  } else {
+    // Browser globals:
+    factory(window.blueimp.helper || window.jQuery, window.blueimp.Gallery)
+  }
+})(function ($, Gallery) {
+  'use strict'
+
+  var galleryPrototype = Gallery.prototype
+
+  $.extend(galleryPrototype.options, {
+    // The tag name, Id, element or querySelector of the indicator container:
+    indicatorContainer: 'ol',
+    // The class for the active indicator:
+    activeIndicatorClass: 'active',
+    // The list object property (or data attribute) with the thumbnail URL,
+    // used as alternative to a thumbnail child element:
+    thumbnailProperty: 'thumbnail',
+    // Defines if the gallery indicators should display a thumbnail:
+    thumbnailIndicators: true
+  })
+
+  var initSlides = galleryPrototype.initSlides
+  var addSlide = galleryPrototype.addSlide
+  var resetSlides = galleryPrototype.resetSlides
+  var handleClick = galleryPrototype.handleClick
+  var handleSlide = galleryPrototype.handleSlide
+  var handleClose = galleryPrototype.handleClose
+
+  $.extend(galleryPrototype, {
+    createIndicator: function (obj) {
+      var indicator = this.indicatorPrototype.cloneNode(false)
+      var title = this.getItemProperty(obj, this.options.titleProperty)
+      var thumbnailProperty = this.options.thumbnailProperty
+      var thumbnailUrl
+      var thumbnail
+      if (this.options.thumbnailIndicators) {
+        if (thumbnailProperty) {
+          thumbnailUrl = this.getItemProperty(obj, thumbnailProperty)
+        }
+        if (thumbnailUrl === undefined) {
+          thumbnail = obj.getElementsByTagName && $(obj).find('img')[0]
+          if (thumbnail) {
+            thumbnailUrl = thumbnail.src
+          }
+        }
+        if (thumbnailUrl) {
+          indicator.style.backgroundImage = 'url("' + thumbnailUrl + '")'
+        }
+      }
+      if (title) {
+        indicator.title = title
+      }
+      indicator.setAttribute('role', 'link')
+      return indicator
+    },
+
+    addIndicator: function (index) {
+      if (this.indicatorContainer.length) {
+        var indicator = this.createIndicator(this.list[index])
+        indicator.setAttribute('data-index', index)
+        this.indicatorContainer[0].appendChild(indicator)
+        this.indicators.push(indicator)
+      }
+    },
+
+    setActiveIndicator: function (index) {
+      if (this.indicators) {
+        if (this.activeIndicator) {
+          this.activeIndicator.removeClass(this.options.activeIndicatorClass)
+        }
+        this.activeIndicator = $(this.indicators[index])
+        this.activeIndicator.addClass(this.options.activeIndicatorClass)
+      }
+    },
+
+    initSlides: function (reload) {
+      if (!reload) {
+        this.indicatorContainer = this.container.find(
+          this.options.indicatorContainer
+        )
+        if (this.indicatorContainer.length) {
+          this.indicatorPrototype = document.createElement('li')
+          this.indicators = this.indicatorContainer[0].children
+        }
+      }
+      initSlides.call(this, reload)
+    },
+
+    addSlide: function (index) {
+      addSlide.call(this, index)
+      this.addIndicator(index)
+    },
+
+    resetSlides: function () {
+      resetSlides.call(this)
+      this.indicatorContainer.empty()
+      this.indicators = []
+    },
+
+    handleClick: function (event) {
+      var target = event.target || event.srcElement
+      var parent = target.parentNode
+      if (parent === this.indicatorContainer[0]) {
+        // Click on indicator element
+        this.preventDefault(event)
+        this.slide(this.getNodeIndex(target))
+      } else if (parent.parentNode === this.indicatorContainer[0]) {
+        // Click on indicator child element
+        this.preventDefault(event)
+        this.slide(this.getNodeIndex(parent))
+      } else {
+        return handleClick.call(this, event)
+      }
+    },
+
+    handleSlide: function (oldIndex, newIndex) {
+      handleSlide.call(this, oldIndex, newIndex)
+      this.setActiveIndicator(newIndex)
+    },
+
+    handleClose: function () {
+      if (this.activeIndicator) {
+        this.activeIndicator.removeClass(this.options.activeIndicatorClass)
+      }
+      handleClose.call(this)
+    }
+  })
+
+  return Gallery
+})
diff --git a/docs/pt/js/blueimp-gallery-video.js b/docs/pt/js/blueimp-gallery-video.js
new file mode 100644
index 0000000..d8b287c
--- /dev/null
+++ b/docs/pt/js/blueimp-gallery-video.js
@@ -0,0 +1,188 @@
+/*
+ * blueimp Gallery Video Factory JS
+ * https://github.com/blueimp/Gallery
+ *
+ * Copyright 2013, Sebastian Tschan
+ * https://blueimp.net
+ *
+ * Licensed under the MIT license:
+ * https://opensource.org/licenses/MIT
+ */
+
+/* global define */
+
+;(function (factory) {
+  'use strict'
+  if (typeof define === 'function' && define.amd) {
+    // Register as an anonymous AMD module:
+    define(['./blueimp-helper', './blueimp-gallery'], factory)
+  } else {
+    // Browser globals:
+    factory(window.blueimp.helper || window.jQuery, window.blueimp.Gallery)
+  }
+})(function ($, Gallery) {
+  'use strict'
+
+  var galleryPrototype = Gallery.prototype
+
+  $.extend(galleryPrototype.options, {
+    // The class for video content elements:
+    videoContentClass: 'video-content',
+    // The class for video when it is loading:
+    videoLoadingClass: 'video-loading',
+    // The class for video when it is playing:
+    videoPlayingClass: 'video-playing',
+    // The class for video content displayed in an iframe:
+    videoIframeClass: 'video-iframe',
+    // The class for the video cover element:
+    videoCoverClass: 'video-cover',
+    // The class for the video play control:
+    videoPlayClass: 'video-play',
+    // Play videos inline by default:
+    videoPlaysInline: true,
+    // The list object property (or data attribute) for video preload:
+    videoPreloadProperty: 'preload',
+    // The list object property (or data attribute) for the video poster URL:
+    videoPosterProperty: 'poster'
+  })
+
+  var handleSlide = galleryPrototype.handleSlide
+
+  $.extend(galleryPrototype, {
+    handleSlide: function (oldIndex, newIndex) {
+      handleSlide.call(this, oldIndex, newIndex)
+      this.setTimeout(function () {
+        if (this.activeVideo) {
+          this.activeVideo.pause()
+        }
+      })
+    },
+
+    videoFactory: function (obj, callback, videoInterface) {
+      var that = this
+      var options = this.options
+      var videoContainerNode = this.elementPrototype.cloneNode(false)
+      var videoContainer = $(videoContainerNode)
+      var errorArgs = [
+        {
+          type: 'error',
+          target: videoContainerNode
+        }
+      ]
+      var video = videoInterface || document.createElement('video')
+      var coverElement = this.elementPrototype.cloneNode(false)
+      var playElement = document.createElement('a')
+      var url = this.getItemProperty(obj, options.urlProperty)
+      var sources = this.getItemProperty(obj, options.sourcesProperty)
+      var title = this.getItemProperty(obj, options.titleProperty)
+      var posterUrl = this.getItemProperty(obj, options.videoPosterProperty)
+      var playControls = [playElement]
+      var hasGalleryControls
+      var isLoading
+      var i
+      videoContainer.addClass(options.videoContentClass)
+      $(playElement).addClass(options.videoPlayClass)
+      if (
+        !$(coverElement)
+          .addClass(options.videoCoverClass)
+          .hasClass(options.toggleClass)
+      ) {
+        playControls.push(coverElement)
+      }
+      coverElement.draggable = false
+      if (title) {
+        videoContainerNode.title = title
+        playElement.setAttribute('aria-label', title)
+      }
+      if (posterUrl) {
+        // Set as background image instead of as poster video element property:
+        // - Is accessible for browsers that do not support the video element
+        // - Is accessible for both video element and iframe video players
+        // - Avoids visual artifacts in IE with the poster property set
+        coverElement.style.backgroundImage = 'url("' + posterUrl + '")'
+      }
+      if (video.setAttribute) {
+        if (options.videoPlaysInline) video.setAttribute('playsinline', '')
+      } else {
+        videoContainer.addClass(options.videoIframeClass)
+      }
+      video.preload =
+        this.getItemProperty(obj, options.videoPreloadProperty) || 'none'
+      if (this.support.source && sources) {
+        for (i = 0; i < sources.length; i += 1) {
+          video.appendChild(
+            $.extend(this.sourcePrototype.cloneNode(false), sources[i])
+          )
+        }
+      }
+      if (url) video.src = url
+      playElement.href = url || (sources && sources.length && sources[0].src)
+      if (video.play && video.pause) {
+        ;(videoInterface || $(video))
+          .on('error', function () {
+            that.setTimeout(callback, errorArgs)
+          })
+          .on('pause', function () {
+            if (video.seeking) return
+            isLoading = false
+            videoContainer
+              .removeClass(that.options.videoLoadingClass)
+              .removeClass(that.options.videoPlayingClass)
+            if (hasGalleryControls) {
+              that.container.addClass(that.options.controlsClass)
+            }
+            video.controls = false
+            if (video === that.activeVideo) delete that.activeVideo
+            if (that.interval) {
+              // Continue slideshow interval
+              that.play()
+            }
+          })
+          .on('playing', function () {
+            isLoading = false
+            coverElement.removeAttribute('style')
+            videoContainer
+              .removeClass(that.options.videoLoadingClass)
+              .addClass(that.options.videoPlayingClass)
+          })
+          .on('play', function () {
+            // Clear slideshow timeout:
+            window.clearTimeout(that.timeout)
+            isLoading = true
+            videoContainer.addClass(that.options.videoLoadingClass)
+            if (that.container.hasClass(that.options.controlsClass)) {
+              hasGalleryControls = true
+              that.container.removeClass(that.options.controlsClass)
+            } else {
+              hasGalleryControls = false
+            }
+            video.controls = true
+            that.activeVideo = video
+          })
+        $(playControls).on('click', function (event) {
+          that.preventDefault(event)
+          that.activeVideo = video
+          if (isLoading) {
+            video.pause()
+          } else {
+            video.play()
+          }
+        })
+        videoContainerNode.appendChild(
+          (videoInterface && videoInterface.element) || video
+        )
+      }
+      videoContainerNode.appendChild(coverElement)
+      videoContainerNode.appendChild(playElement)
+      this.setTimeout(callback, [
+        {
+          type: 'load',
+          target: videoContainerNode
+        }
+      ])
+      return videoContainerNode
+    }
+  })
+
+  return Gallery
+})
diff --git a/docs/pt/js/blueimp-gallery-vimeo.js b/docs/pt/js/blueimp-gallery-vimeo.js
new file mode 100644
index 0000000..c274f54
--- /dev/null
+++ b/docs/pt/js/blueimp-gallery-vimeo.js
@@ -0,0 +1,211 @@
+/*
+ * blueimp Gallery Vimeo Video Factory JS
+ * https://github.com/blueimp/Gallery
+ *
+ * Copyright 2013, Sebastian Tschan
+ * https://blueimp.net
+ *
+ * Licensed under the MIT license:
+ * https://opensource.org/licenses/MIT
+ */
+
+/* global define, $f */
+
+;(function (factory) {
+  'use strict'
+  if (typeof define === 'function' && define.amd) {
+    // Register as an anonymous AMD module:
+    define(['./blueimp-helper', './blueimp-gallery-video'], factory)
+  } else {
+    // Browser globals:
+    factory(window.blueimp.helper || window.jQuery, window.blueimp.Gallery)
+  }
+})(function ($, Gallery) {
+  'use strict'
+
+  if (!window.postMessage) {
+    return Gallery
+  }
+
+  var galleryPrototype = Gallery.prototype
+
+  $.extend(galleryPrototype.options, {
+    // The list object property (or data attribute) with the Vimeo video id:
+    vimeoVideoIdProperty: 'vimeo',
+    // The URL for the Vimeo video player, can be extended with custom parameters:
+    // https://developer.vimeo.com/player/embedding
+    vimeoPlayerUrl:
+      '//player.vimeo.com/video/VIDEO_ID?api=1&player_id=PLAYER_ID',
+    // The prefix for the Vimeo video player ID:
+    vimeoPlayerIdPrefix: 'vimeo-player-',
+    // Require a click on the native Vimeo player for the initial playback:
+    vimeoClickToPlay: false
+  })
+
+  var textFactory =
+    galleryPrototype.textFactory || galleryPrototype.imageFactory
+  var VimeoPlayer = function (url, videoId, playerId, clickToPlay) {
+    this.url = url
+    this.videoId = videoId
+    this.playerId = playerId
+    this.clickToPlay = clickToPlay
+    this.element = document.createElement('div')
+    this.listeners = {}
+  }
+  var counter = 0
+
+  $.extend(VimeoPlayer.prototype, {
+    on: function (type, func) {
+      this.listeners[type] = func
+      return this
+    },
+
+    loadAPI: function () {
+      var that = this
+      var apiUrl = '//f.vimeocdn.com/js/froogaloop2.min.js'
+      var scriptTags = document.getElementsByTagName('script')
+      var i = scriptTags.length
+      var scriptTag
+      var called
+      /**
+       * Callback function
+       */
+      function callback() {
+        if (!called && that.playOnReady) {
+          that.play()
+        }
+        called = true
+      }
+      while (i) {
+        i -= 1
+        if (scriptTags[i].src === apiUrl) {
+          scriptTag = scriptTags[i]
+          break
+        }
+      }
+      if (!scriptTag) {
+        scriptTag = document.createElement('script')
+        scriptTag.src = apiUrl
+      }
+      $(scriptTag).on('load', callback)
+      scriptTags[0].parentNode.insertBefore(scriptTag, scriptTags[0])
+      // Fix for cached scripts on IE 8:
+      if (/loaded|complete/.test(scriptTag.readyState)) {
+        callback()
+      }
+    },
+
+    onReady: function () {
+      var that = this
+      this.ready = true
+      this.player.addEvent('play', function () {
+        that.hasPlayed = true
+        that.onPlaying()
+      })
+      this.player.addEvent('pause', function () {
+        that.onPause()
+      })
+      this.player.addEvent('finish', function () {
+        that.onPause()
+      })
+      if (this.playOnReady) {
+        this.play()
+      }
+    },
+
+    onPlaying: function () {
+      if (this.playStatus < 2) {
+        this.listeners.playing()
+        this.playStatus = 2
+      }
+    },
+
+    onPause: function () {
+      this.listeners.pause()
+      delete this.playStatus
+    },
+
+    insertIframe: function () {
+      var iframe = document.createElement('iframe')
+      iframe.src = this.url
+        .replace('VIDEO_ID', this.videoId)
+        .replace('PLAYER_ID', this.playerId)
+      iframe.id = this.playerId
+      this.element.parentNode.replaceChild(iframe, this.element)
+      this.element = iframe
+    },
+
+    play: function () {
+      var that = this
+      if (!this.playStatus) {
+        this.listeners.play()
+        this.playStatus = 1
+      }
+      if (this.ready) {
+        if (
+          !this.hasPlayed &&
+          (this.clickToPlay ||
+            (window.navigator &&
+              /iP(hone|od|ad)/.test(window.navigator.platform)))
+        ) {
+          // Manually trigger the playing callback if clickToPlay
+          // is enabled and to workaround a limitation in iOS,
+          // which requires synchronous user interaction to start
+          // the video playback:
+          this.onPlaying()
+        } else {
+          this.player.api('play')
+        }
+      } else {
+        this.playOnReady = true
+        if (!window.$f) {
+          this.loadAPI()
+        } else if (!this.player) {
+          this.insertIframe()
+          this.player = $f(this.element)
+          this.player.addEvent('ready', function () {
+            that.onReady()
+          })
+        }
+      }
+    },
+
+    pause: function () {
+      if (this.ready) {
+        this.player.api('pause')
+      } else if (this.playStatus) {
+        delete this.playOnReady
+        this.listeners.pause()
+        delete this.playStatus
+      }
+    }
+  })
+
+  $.extend(galleryPrototype, {
+    VimeoPlayer: VimeoPlayer,
+
+    textFactory: function (obj, callback) {
+      var options = this.options
+      var videoId = this.getItemProperty(obj, options.vimeoVideoIdProperty)
+      if (videoId) {
+        if (this.getItemProperty(obj, options.urlProperty) === undefined) {
+          obj[options.urlProperty] = '//vimeo.com/' + videoId
+        }
+        counter += 1
+        return this.videoFactory(
+          obj,
+          callback,
+          new VimeoPlayer(
+            options.vimeoPlayerUrl,
+            videoId,
+            options.vimeoPlayerIdPrefix + counter,
+            options.vimeoClickToPlay
+          )
+        )
+      }
+      return textFactory.call(this, obj, callback)
+    }
+  })
+
+  return Gallery
+})
diff --git a/docs/pt/js/blueimp-gallery-youtube.js b/docs/pt/js/blueimp-gallery-youtube.js
new file mode 100644
index 0000000..e81258c
--- /dev/null
+++ b/docs/pt/js/blueimp-gallery-youtube.js
@@ -0,0 +1,223 @@
+/*
+ * blueimp Gallery YouTube Video Factory JS
+ * https://github.com/blueimp/Gallery
+ *
+ * Copyright 2013, Sebastian Tschan
+ * https://blueimp.net
+ *
+ * Licensed under the MIT license:
+ * https://opensource.org/licenses/MIT
+ */
+
+/* global define, YT */
+
+;(function (factory) {
+  'use strict'
+  if (typeof define === 'function' && define.amd) {
+    // Register as an anonymous AMD module:
+    define(['./blueimp-helper', './blueimp-gallery-video'], factory)
+  } else {
+    // Browser globals:
+    factory(window.blueimp.helper || window.jQuery, window.blueimp.Gallery)
+  }
+})(function ($, Gallery) {
+  'use strict'
+
+  if (!window.postMessage) {
+    return Gallery
+  }
+
+  var galleryPrototype = Gallery.prototype
+
+  $.extend(galleryPrototype.options, {
+    // The list object property (or data attribute) with the YouTube video id:
+    youTubeVideoIdProperty: 'youtube',
+    // Optional object with parameters passed to the YouTube video player:
+    // https://developers.google.com/youtube/player_parameters
+    youTubePlayerVars: {
+      wmode: 'transparent'
+    },
+    // Require a click on the native YouTube player for the initial playback:
+    youTubeClickToPlay: false
+  })
+
+  var textFactory =
+    galleryPrototype.textFactory || galleryPrototype.imageFactory
+  var YouTubePlayer = function (videoId, playerVars, clickToPlay) {
+    this.videoId = videoId
+    this.playerVars = playerVars
+    this.clickToPlay = clickToPlay
+    this.element = document.createElement('div')
+    this.listeners = {}
+  }
+
+  $.extend(YouTubePlayer.prototype, {
+    on: function (type, func) {
+      this.listeners[type] = func
+      return this
+    },
+
+    loadAPI: function () {
+      var that = this
+      var onYouTubeIframeAPIReady = window.onYouTubeIframeAPIReady
+      var apiUrl = '//www.youtube.com/iframe_api'
+      var scriptTags = document.getElementsByTagName('script')
+      var i = scriptTags.length
+      var scriptTag
+      window.onYouTubeIframeAPIReady = function () {
+        if (onYouTubeIframeAPIReady) {
+          onYouTubeIframeAPIReady.apply(this)
+        }
+        if (that.playOnReady) {
+          that.play()
+        }
+      }
+      while (i) {
+        i -= 1
+        if (scriptTags[i].src === apiUrl) {
+          return
+        }
+      }
+      scriptTag = document.createElement('script')
+      scriptTag.src = apiUrl
+      scriptTags[0].parentNode.insertBefore(scriptTag, scriptTags[0])
+    },
+
+    onReady: function () {
+      this.ready = true
+      if (this.playOnReady) {
+        this.play()
+      }
+    },
+
+    onPlaying: function () {
+      if (this.playStatus < 2) {
+        this.listeners.playing()
+        this.playStatus = 2
+      }
+    },
+
+    onPause: function () {
+      this.listeners.pause()
+      delete this.playStatus
+    },
+
+    onStateChange: function (event) {
+      window.clearTimeout(this.pauseTimeout)
+      switch (event.data) {
+        case YT.PlayerState.PLAYING:
+          this.hasPlayed = true
+          this.onPlaying()
+          break
+        case YT.PlayerState.UNSTARTED:
+        case YT.PlayerState.PAUSED:
+          // YouTube sends an unstarted event if pause is triggered before the
+          // video has started.
+          // YouTube sends a pause event when seeking.
+          // In both cases, we initiate a pause in a timeout that gets cleared
+          // if followed by another event within the timeout window.
+          this.pauseTimeout = galleryPrototype.setTimeout.call(
+            this,
+            this.onPause,
+            null,
+            500
+          )
+          break
+        case YT.PlayerState.ENDED:
+          this.onPause()
+          break
+      }
+    },
+
+    onError: function (event) {
+      this.listeners.error(event)
+    },
+
+    play: function () {
+      var that = this
+      if (!this.playStatus) {
+        this.listeners.play()
+        this.playStatus = 1
+      }
+      if (this.ready) {
+        if (
+          !this.hasPlayed &&
+          (this.clickToPlay ||
+            (window.navigator &&
+              /iP(hone|od|ad)/.test(window.navigator.platform)))
+        ) {
+          // Manually trigger the playing callback if clickToPlay
+          // is enabled and to workaround a limitation in iOS,
+          // which requires synchronous user interaction to start
+          // the video playback:
+          this.onPlaying()
+        } else {
+          this.player.playVideo()
+        }
+      } else {
+        this.playOnReady = true
+        if (!(window.YT && YT.Player)) {
+          this.loadAPI()
+        } else if (!this.player) {
+          this.player = new YT.Player(this.element, {
+            videoId: this.videoId,
+            playerVars: this.playerVars,
+            events: {
+              onReady: function () {
+                that.onReady()
+              },
+              onStateChange: function (event) {
+                that.onStateChange(event)
+              },
+              onError: function (event) {
+                that.onError(event)
+              }
+            }
+          })
+        }
+      }
+    },
+
+    pause: function () {
+      if (this.ready) {
+        this.player.pauseVideo()
+      } else if (this.playStatus) {
+        delete this.playOnReady
+        this.listeners.pause()
+        delete this.playStatus
+      }
+    }
+  })
+
+  $.extend(galleryPrototype, {
+    YouTubePlayer: YouTubePlayer,
+
+    textFactory: function (obj, callback) {
+      var options = this.options
+      var videoId = this.getItemProperty(obj, options.youTubeVideoIdProperty)
+      if (videoId) {
+        if (this.getItemProperty(obj, options.urlProperty) === undefined) {
+          obj[options.urlProperty] = '//www.youtube.com/watch?v=' + videoId
+        }
+        if (
+          this.getItemProperty(obj, options.videoPosterProperty) === undefined
+        ) {
+          obj[options.videoPosterProperty] =
+            '//img.youtube.com/vi/' + videoId + '/maxresdefault.jpg'
+        }
+        return this.videoFactory(
+          obj,
+          callback,
+          new YouTubePlayer(
+            videoId,
+            options.youTubePlayerVars,
+            options.youTubeClickToPlay
+          )
+        )
+      }
+      return textFactory.call(this, obj, callback)
+    }
+  })
+
+  return Gallery
+})
diff --git a/docs/pt/js/blueimp-gallery.js b/docs/pt/js/blueimp-gallery.js
new file mode 100644
index 0000000..5ddea4c
--- /dev/null
+++ b/docs/pt/js/blueimp-gallery.js
@@ -0,0 +1,1551 @@
+/*
+ * blueimp Gallery JS
+ * https://github.com/blueimp/Gallery
+ *
+ * Copyright 2013, Sebastian Tschan
+ * https://blueimp.net
+ *
+ * Swipe implementation based on
+ * https://github.com/bradbirdsall/Swipe
+ *
+ * Licensed under the MIT license:
+ * https://opensource.org/licenses/MIT
+ */
+
+/* global define, DocumentTouch */
+
+/* eslint-disable no-param-reassign */
+
+;(function (factory) {
+  'use strict'
+  if (typeof define === 'function' && define.amd) {
+    // Register as an anonymous AMD module:
+    define(['./blueimp-helper'], factory)
+  } else {
+    // Browser globals:
+    window.blueimp = window.blueimp || {}
+    window.blueimp.Gallery = factory(window.blueimp.helper || window.jQuery)
+  }
+})(function ($) {
+  'use strict'
+
+  /**
+   * Gallery constructor
+   *
+   * @class
+   * @param {Array|NodeList} list Gallery content
+   * @param {object} [options] Gallery options
+   * @returns {object} Gallery object
+   */
+  function Gallery(list, options) {
+    if (document.body.style.maxHeight === undefined) {
+      // document.body.style.maxHeight is undefined on IE6 and lower
+      return null
+    }
+    if (!this || this.options !== Gallery.prototype.options) {
+      // Called as function instead of as constructor,
+      // so we simply return a new instance:
+      return new Gallery(list, options)
+    }
+    if (!list || !list.length) {
+      this.console.log(
+        'blueimp Gallery: No or empty list provided as first argument.',
+        list
+      )
+      return
+    }
+    this.list = list
+    this.num = list.length
+    this.initOptions(options)
+    this.initialize()
+  }
+
+  $.extend(Gallery.prototype, {
+    options: {
+      // The Id, element or querySelector of the gallery widget:
+      container: '#blueimp-gallery',
+      // The tag name, Id, element or querySelector of the slides container:
+      slidesContainer: 'div',
+      // The tag name, Id, element or querySelector of the title element:
+      titleElement: 'h3',
+      // The class to add when the gallery is visible:
+      displayClass: 'blueimp-gallery-display',
+      // The class to add when the gallery controls are visible:
+      controlsClass: 'blueimp-gallery-controls',
+      // The class to add when the gallery only displays one element:
+      singleClass: 'blueimp-gallery-single',
+      // The class to add when the left edge has been reached:
+      leftEdgeClass: 'blueimp-gallery-left',
+      // The class to add when the right edge has been reached:
+      rightEdgeClass: 'blueimp-gallery-right',
+      // The class to add when the automatic slideshow is active:
+      playingClass: 'blueimp-gallery-playing',
+      // The class to add when the browser supports SVG as img (or background):
+      svgasimgClass: 'blueimp-gallery-svgasimg',
+      // The class to add when the browser supports SMIL (animated SVGs):
+      smilClass: 'blueimp-gallery-smil',
+      // The class for all slides:
+      slideClass: 'slide',
+      // The slide class for the active (current index) slide:
+      slideActiveClass: 'slide-active',
+      // The slide class for the previous (before current index) slide:
+      slidePrevClass: 'slide-prev',
+      // The slide class for the next (after current index) slide:
+      slideNextClass: 'slide-next',
+      // The slide class for loading elements:
+      slideLoadingClass: 'slide-loading',
+      // The slide class for elements that failed to load:
+      slideErrorClass: 'slide-error',
+      // The class for the content element loaded into each slide:
+      slideContentClass: 'slide-content',
+      // The class for the "toggle" control:
+      toggleClass: 'toggle',
+      // The class for the "prev" control:
+      prevClass: 'prev',
+      // The class for the "next" control:
+      nextClass: 'next',
+      // The class for the "close" control:
+      closeClass: 'close',
+      // The class for the "play-pause" toggle control:
+      playPauseClass: 'play-pause',
+      // The list object property (or data attribute) with the object type:
+      typeProperty: 'type',
+      // The list object property (or data attribute) with the object title:
+      titleProperty: 'title',
+      // The list object property (or data attribute) with the object alt text:
+      altTextProperty: 'alt',
+      // The list object property (or data attribute) with the object URL:
+      urlProperty: 'href',
+      // The list object property (or data attribute) with the object srcset:
+      srcsetProperty: 'srcset',
+      // The list object property (or data attribute) with the object sizes:
+      sizesProperty: 'sizes',
+      // The list object property (or data attribute) with the object sources:
+      sourcesProperty: 'sources',
+      // The gallery listens for transitionend events before triggering the
+      // opened and closed events, unless the following option is set to false:
+      displayTransition: true,
+      // Defines if the gallery slides are cleared from the gallery modal,
+      // or reused for the next gallery initialization:
+      clearSlides: true,
+      // Toggle the controls on pressing the Enter key:
+      toggleControlsOnEnter: true,
+      // Toggle the controls on slide click:
+      toggleControlsOnSlideClick: true,
+      // Toggle the automatic slideshow interval on pressing the Space key:
+      toggleSlideshowOnSpace: true,
+      // Navigate the gallery by pressing the ArrowLeft and ArrowRight keys:
+      enableKeyboardNavigation: true,
+      // Close the gallery on pressing the Escape key:
+      closeOnEscape: true,
+      // Close the gallery when clicking on an empty slide area:
+      closeOnSlideClick: true,
+      // Close the gallery by swiping up or down:
+      closeOnSwipeUpOrDown: true,
+      // Close the gallery when the URL hash changes:
+      closeOnHashChange: true,
+      // Emulate touch events on mouse-pointer devices such as desktop browsers:
+      emulateTouchEvents: true,
+      // Stop touch events from bubbling up to ancestor elements of the Gallery:
+      stopTouchEventsPropagation: false,
+      // Hide the page scrollbars:
+      hidePageScrollbars: true,
+      // Stops any touches on the container from scrolling the page:
+      disableScroll: true,
+      // Carousel mode (shortcut for carousel specific options):
+      carousel: false,
+      // Allow continuous navigation, moving from last to first
+      // and from first to last slide:
+      continuous: true,
+      // Remove elements outside of the preload range from the DOM:
+      unloadElements: true,
+      // Start with the automatic slideshow:
+      startSlideshow: false,
+      // Delay in milliseconds between slides for the automatic slideshow:
+      slideshowInterval: 5000,
+      // The direction the slides are moving: ltr=LeftToRight or rtl=RightToLeft
+      slideshowDirection: 'ltr',
+      // The starting index as integer.
+      // Can also be an object of the given list,
+      // or an equal object with the same url property:
+      index: 0,
+      // The number of elements to load around the current index:
+      preloadRange: 2,
+      // The transition duration between slide changes in milliseconds:
+      transitionDuration: 300,
+      // The transition duration for automatic slide changes, set to an integer
+      // greater 0 to override the default transition duration:
+      slideshowTransitionDuration: 500,
+      // The event object for which the default action will be canceled
+      // on Gallery initialization (e.g. the click event to open the Gallery):
+      event: undefined,
+      // Callback function executed when the Gallery is initialized.
+      // Is called with the gallery instance as "this" object:
+      onopen: undefined,
+      // Callback function executed when the Gallery has been initialized
+      // and the initialization transition has been completed.
+      // Is called with the gallery instance as "this" object:
+      onopened: undefined,
+      // Callback function executed on slide change.
+      // Is called with the gallery instance as "this" object and the
+      // current index and slide as arguments:
+      onslide: undefined,
+      // Callback function executed after the slide change transition.
+      // Is called with the gallery instance as "this" object and the
+      // current index and slide as arguments:
+      onslideend: undefined,
+      // Callback function executed on slide content load.
+      // Is called with the gallery instance as "this" object and the
+      // slide index and slide element as arguments:
+      onslidecomplete: undefined,
+      // Callback function executed when the Gallery is about to be closed.
+      // Is called with the gallery instance as "this" object:
+      onclose: undefined,
+      // Callback function executed when the Gallery has been closed
+      // and the closing transition has been completed.
+      // Is called with the gallery instance as "this" object:
+      onclosed: undefined
+    },
+
+    carouselOptions: {
+      hidePageScrollbars: false,
+      toggleControlsOnEnter: false,
+      toggleSlideshowOnSpace: false,
+      enableKeyboardNavigation: false,
+      closeOnEscape: false,
+      closeOnSlideClick: false,
+      closeOnSwipeUpOrDown: false,
+      closeOnHashChange: false,
+      disableScroll: false,
+      startSlideshow: true
+    },
+
+    console:
+      window.console && typeof window.console.log === 'function'
+        ? window.console
+        : { log: function () {} },
+
+    // Detect touch, transition, transform and background-size support:
+    support: (function (element) {
+      var support = {
+        source: !!window.HTMLSourceElement,
+        picture: !!window.HTMLPictureElement,
+        svgasimg: document.implementation.hasFeature(
+          'http://www.w3.org/TR/SVG11/feature#Image',
+          '1.1'
+        ),
+        smil:
+          !!document.createElementNS &&
+          /SVGAnimate/.test(
+            document
+              .createElementNS('http://www.w3.org/2000/svg', 'animate')
+              .toString()
+          ),
+        touch:
+          window.ontouchstart !== undefined ||
+          (window.DocumentTouch && document instanceof DocumentTouch)
+      }
+      var transitions = {
+        webkitTransition: {
+          end: 'webkitTransitionEnd',
+          prefix: '-webkit-'
+        },
+        MozTransition: {
+          end: 'transitionend',
+          prefix: '-moz-'
+        },
+        OTransition: {
+          end: 'otransitionend',
+          prefix: '-o-'
+        },
+        transition: {
+          end: 'transitionend',
+          prefix: ''
+        }
+      }
+      var prop
+      for (prop in transitions) {
+        if (
+          Object.prototype.hasOwnProperty.call(transitions, prop) &&
+          element.style[prop] !== undefined
+        ) {
+          support.transition = transitions[prop]
+          support.transition.name = prop
+          break
+        }
+      }
+      /**
+       * Tests browser support
+       */
+      function elementTests() {
+        var transition = support.transition
+        var prop
+        var translateZ
+        document.body.appendChild(element)
+        if (transition) {
+          prop = transition.name.slice(0, -9) + 'ransform'
+          if (element.style[prop] !== undefined) {
+            element.style[prop] = 'translateZ(0)'
+            translateZ = window
+              .getComputedStyle(element)
+              .getPropertyValue(transition.prefix + 'transform')
+            support.transform = {
+              prefix: transition.prefix,
+              name: prop,
+              translate: true,
+              translateZ: !!translateZ && translateZ !== 'none'
+            }
+          }
+        }
+        document.body.removeChild(element)
+      }
+      if (document.body) {
+        elementTests()
+      } else {
+        $(document).on('DOMContentLoaded', elementTests)
+      }
+      return support
+      // Test element, has to be standard HTML and must not be hidden
+      // for the CSS3 tests using window.getComputedStyle to be applicable:
+    })(document.createElement('div')),
+
+    requestAnimationFrame:
+      window.requestAnimationFrame ||
+      window.webkitRequestAnimationFrame ||
+      window.mozRequestAnimationFrame,
+
+    cancelAnimationFrame:
+      window.cancelAnimationFrame ||
+      window.webkitCancelRequestAnimationFrame ||
+      window.webkitCancelAnimationFrame ||
+      window.mozCancelAnimationFrame,
+
+    initialize: function () {
+      this.initStartIndex()
+      if (this.initWidget() === false) {
+        return false
+      }
+      this.initEventListeners()
+      // Load the slide at the given index:
+      this.onslide(this.index)
+      // Manually trigger the slideend event for the initial slide:
+      this.ontransitionend()
+      // Start the automatic slideshow if applicable:
+      if (this.options.startSlideshow) {
+        this.play()
+      }
+    },
+
+    slide: function (to, duration) {
+      window.clearTimeout(this.timeout)
+      var index = this.index
+      var direction
+      var naturalDirection
+      var diff
+      if (index === to || this.num === 1) {
+        return
+      }
+      if (!duration) {
+        duration = this.options.transitionDuration
+      }
+      if (this.support.transform) {
+        if (!this.options.continuous) {
+          to = this.circle(to)
+        }
+        // 1: backward, -1: forward:
+        direction = Math.abs(index - to) / (index - to)
+        // Get the actual position of the slide:
+        if (this.options.continuous) {
+          naturalDirection = direction
+          direction = -this.positions[this.circle(to)] / this.slideWidth
+          // If going forward but to < index, use to = slides.length + to
+          // If going backward but to > index, use to = -slides.length + to
+          if (direction !== naturalDirection) {
+            to = -direction * this.num + to
+          }
+        }
+        diff = Math.abs(index - to) - 1
+        // Move all the slides between index and to in the right direction:
+        while (diff) {
+          diff -= 1
+          this.move(
+            this.circle((to > index ? to : index) - diff - 1),
+            this.slideWidth * direction,
+            0
+          )
+        }
+        to = this.circle(to)
+        this.move(index, this.slideWidth * direction, duration)
+        this.move(to, 0, duration)
+        if (this.options.continuous) {
+          this.move(
+            this.circle(to - direction),
+            -(this.slideWidth * direction),
+            0
+          )
+        }
+      } else {
+        to = this.circle(to)
+        this.animate(index * -this.slideWidth, to * -this.slideWidth, duration)
+      }
+      this.onslide(to)
+    },
+
+    getIndex: function () {
+      return this.index
+    },
+
+    getNumber: function () {
+      return this.num
+    },
+
+    prev: function () {
+      if (this.options.continuous || this.index) {
+        this.slide(this.index - 1)
+      }
+    },
+
+    next: function () {
+      if (this.options.continuous || this.index < this.num - 1) {
+        this.slide(this.index + 1)
+      }
+    },
+
+    play: function (time) {
+      var that = this
+      var nextIndex =
+        this.index + (this.options.slideshowDirection === 'rtl' ? -1 : 1)
+      window.clearTimeout(this.timeout)
+      this.interval = time || this.options.slideshowInterval
+      if (this.elements[this.index] > 1) {
+        this.timeout = this.setTimeout(
+          (!this.requestAnimationFrame && this.slide) ||
+            function (to, duration) {
+              that.animationFrameId = that.requestAnimationFrame.call(
+                window,
+                function () {
+                  that.slide(to, duration)
+                }
+              )
+            },
+          [nextIndex, this.options.slideshowTransitionDuration],
+          this.interval
+        )
+      }
+      this.container.addClass(this.options.playingClass)
+      this.slidesContainer[0].setAttribute('aria-live', 'off')
+      if (this.playPauseElement.length) {
+        this.playPauseElement[0].setAttribute('aria-pressed', 'true')
+      }
+    },
+
+    pause: function () {
+      window.clearTimeout(this.timeout)
+      this.interval = null
+      if (this.cancelAnimationFrame) {
+        this.cancelAnimationFrame.call(window, this.animationFrameId)
+        this.animationFrameId = null
+      }
+      this.container.removeClass(this.options.playingClass)
+      this.slidesContainer[0].setAttribute('aria-live', 'polite')
+      if (this.playPauseElement.length) {
+        this.playPauseElement[0].setAttribute('aria-pressed', 'false')
+      }
+    },
+
+    add: function (list) {
+      var i
+      if (!list.concat) {
+        // Make a real array out of the list to add:
+        list = Array.prototype.slice.call(list)
+      }
+      if (!this.list.concat) {
+        // Make a real array out of the Gallery list:
+        this.list = Array.prototype.slice.call(this.list)
+      }
+      this.list = this.list.concat(list)
+      this.num = this.list.length
+      if (this.num > 2 && this.options.continuous === null) {
+        this.options.continuous = true
+        this.container.removeClass(this.options.leftEdgeClass)
+      }
+      this.container
+        .removeClass(this.options.rightEdgeClass)
+        .removeClass(this.options.singleClass)
+      for (i = this.num - list.length; i < this.num; i += 1) {
+        this.addSlide(i)
+        this.positionSlide(i)
+      }
+      this.positions.length = this.num
+      this.initSlides(true)
+    },
+
+    resetSlides: function () {
+      this.slidesContainer.empty()
+      this.unloadAllSlides()
+      this.slides = []
+    },
+
+    handleClose: function () {
+      var options = this.options
+      this.destroyEventListeners()
+      // Cancel the slideshow:
+      this.pause()
+      this.container[0].style.display = 'none'
+      this.container
+        .removeClass(options.displayClass)
+        .removeClass(options.singleClass)
+        .removeClass(options.leftEdgeClass)
+        .removeClass(options.rightEdgeClass)
+      if (options.hidePageScrollbars) {
+        document.body.style.overflow = this.bodyOverflowStyle
+      }
+      if (this.options.clearSlides) {
+        this.resetSlides()
+      }
+      if (this.options.onclosed) {
+        this.options.onclosed.call(this)
+      }
+    },
+
+    close: function () {
+      var that = this
+      /**
+       * Close handler
+       *
+       * @param {event} event Close event
+       */
+      function closeHandler(event) {
+        if (event.target === that.container[0]) {
+          that.container.off(that.support.transition.end, closeHandler)
+          that.handleClose()
+        }
+      }
+      if (this.options.onclose) {
+        this.options.onclose.call(this)
+      }
+      if (this.support.transition && this.options.displayTransition) {
+        this.container.on(this.support.transition.end, closeHandler)
+        this.container.removeClass(this.options.displayClass)
+      } else {
+        this.handleClose()
+      }
+    },
+
+    circle: function (index) {
+      // Always return a number inside of the slides index range:
+      return (this.num + (index % this.num)) % this.num
+    },
+
+    move: function (index, dist, duration) {
+      this.translateX(index, dist, duration)
+      this.positions[index] = dist
+    },
+
+    translate: function (index, x, y, duration) {
+      if (!this.slides[index]) return
+      var style = this.slides[index].style
+      var transition = this.support.transition
+      var transform = this.support.transform
+      style[transition.name + 'Duration'] = duration + 'ms'
+      style[transform.name] =
+        'translate(' +
+        x +
+        'px, ' +
+        y +
+        'px)' +
+        (transform.translateZ ? ' translateZ(0)' : '')
+    },
+
+    translateX: function (index, x, duration) {
+      this.translate(index, x, 0, duration)
+    },
+
+    translateY: function (index, y, duration) {
+      this.translate(index, 0, y, duration)
+    },
+
+    animate: function (from, to, duration) {
+      if (!duration) {
+        this.slidesContainer[0].style.left = to + 'px'
+        return
+      }
+      var that = this
+      var start = new Date().getTime()
+      var timer = window.setInterval(function () {
+        var timeElap = new Date().getTime() - start
+        if (timeElap > duration) {
+          that.slidesContainer[0].style.left = to + 'px'
+          that.ontransitionend()
+          window.clearInterval(timer)
+          return
+        }
+        that.slidesContainer[0].style.left =
+          (to - from) * (Math.floor((timeElap / duration) * 100) / 100) +
+          from +
+          'px'
+      }, 4)
+    },
+
+    preventDefault: function (event) {
+      if (event.preventDefault) {
+        event.preventDefault()
+      } else {
+        event.returnValue = false
+      }
+    },
+
+    stopPropagation: function (event) {
+      if (event.stopPropagation) {
+        event.stopPropagation()
+      } else {
+        event.cancelBubble = true
+      }
+    },
+
+    onresize: function () {
+      this.initSlides(true)
+    },
+
+    onhashchange: function () {
+      if (this.options.closeOnHashChange) {
+        this.close()
+      }
+    },
+
+    onmousedown: function (event) {
+      // Trigger on clicks of the left mouse button only
+      // and exclude video & audio elements:
+      if (
+        event.which &&
+        event.which === 1 &&
+        event.target.nodeName !== 'VIDEO' &&
+        event.target.nodeName !== 'AUDIO'
+      ) {
+        // Preventing the default mousedown action is required
+        // to make touch emulation work with Firefox:
+        event.preventDefault()
+        ;(event.originalEvent || event).touches = [
+          {
+            pageX: event.pageX,
+            pageY: event.pageY
+          }
+        ]
+        this.ontouchstart(event)
+      }
+    },
+
+    onmousemove: function (event) {
+      if (this.touchStart) {
+        ;(event.originalEvent || event).touches = [
+          {
+            pageX: event.pageX,
+            pageY: event.pageY
+          }
+        ]
+        this.ontouchmove(event)
+      }
+    },
+
+    onmouseup: function (event) {
+      if (this.touchStart) {
+        this.ontouchend(event)
+        delete this.touchStart
+      }
+    },
+
+    onmouseout: function (event) {
+      if (this.touchStart) {
+        var target = event.target
+        var related = event.relatedTarget
+        if (!related || (related !== target && !$.contains(target, related))) {
+          this.onmouseup(event)
+        }
+      }
+    },
+
+    ontouchstart: function (event) {
+      if (this.options.stopTouchEventsPropagation) {
+        this.stopPropagation(event)
+      }
+      // jQuery doesn't copy touch event properties by default,
+      // so we have to access the originalEvent object:
+      var touch = (event.originalEvent || event).touches[0]
+      this.touchStart = {
+        // Remember the initial touch coordinates:
+        x: touch.pageX,
+        y: touch.pageY,
+        // Store the time to determine touch duration:
+        time: Date.now()
+      }
+      // Helper variable to detect scroll movement:
+      this.isScrolling = undefined
+      // Reset delta values:
+      this.touchDelta = {}
+    },
+
+    ontouchmove: function (event) {
+      if (this.options.stopTouchEventsPropagation) {
+        this.stopPropagation(event)
+      }
+      // jQuery doesn't copy touch event properties by default,
+      // so we have to access the originalEvent object:
+      var touches = (event.originalEvent || event).touches
+      var touch = touches[0]
+      var scale = (event.originalEvent || event).scale
+      var index = this.index
+      var touchDeltaX
+      var indices
+      // Ensure this is a one touch swipe and not, e.g. a pinch:
+      if (touches.length > 1 || (scale && scale !== 1)) {
+        return
+      }
+      if (this.options.disableScroll) {
+        event.preventDefault()
+      }
+      // Measure change in x and y coordinates:
+      this.touchDelta = {
+        x: touch.pageX - this.touchStart.x,
+        y: touch.pageY - this.touchStart.y
+      }
+      touchDeltaX = this.touchDelta.x
+      // Detect if this is a vertical scroll movement (run only once per touch):
+      if (this.isScrolling === undefined) {
+        this.isScrolling =
+          this.isScrolling ||
+          Math.abs(touchDeltaX) < Math.abs(this.touchDelta.y)
+      }
+      if (!this.isScrolling) {
+        // Always prevent horizontal scroll:
+        event.preventDefault()
+        // Stop the slideshow:
+        window.clearTimeout(this.timeout)
+        if (this.options.continuous) {
+          indices = [this.circle(index + 1), index, this.circle(index - 1)]
+        } else {
+          // Increase resistance if first slide and sliding left
+          // or last slide and sliding right:
+          this.touchDelta.x = touchDeltaX =
+            touchDeltaX /
+            ((!index && touchDeltaX > 0) ||
+            (index === this.num - 1 && touchDeltaX < 0)
+              ? Math.abs(touchDeltaX) / this.slideWidth + 1
+              : 1)
+          indices = [index]
+          if (index) {
+            indices.push(index - 1)
+          }
+          if (index < this.num - 1) {
+            indices.unshift(index + 1)
+          }
+        }
+        while (indices.length) {
+          index = indices.pop()
+          this.translateX(index, touchDeltaX + this.positions[index], 0)
+        }
+      } else if (!this.options.carousel) {
+        this.translateY(index, this.touchDelta.y + this.positions[index], 0)
+      }
+    },
+
+    ontouchend: function (event) {
+      if (this.options.stopTouchEventsPropagation) {
+        this.stopPropagation(event)
+      }
+      var index = this.index
+      var absTouchDeltaX = Math.abs(this.touchDelta.x)
+      var slideWidth = this.slideWidth
+      var duration = Math.ceil(
+        (this.options.transitionDuration * (1 - absTouchDeltaX / slideWidth)) /
+          2
+      )
+      // Determine if slide attempt triggers next/prev slide:
+      var isValidSlide = absTouchDeltaX > 20
+      // Determine if slide attempt is past start or end:
+      var isPastBounds =
+        (!index && this.touchDelta.x > 0) ||
+        (index === this.num - 1 && this.touchDelta.x < 0)
+      var isValidClose =
+        !isValidSlide &&
+        this.options.closeOnSwipeUpOrDown &&
+        Math.abs(this.touchDelta.y) > 20
+      var direction
+      var indexForward
+      var indexBackward
+      var distanceForward
+      var distanceBackward
+      if (this.options.continuous) {
+        isPastBounds = false
+      }
+      // Determine direction of swipe (true: right, false: left):
+      direction = this.touchDelta.x < 0 ? -1 : 1
+      if (!this.isScrolling) {
+        if (isValidSlide && !isPastBounds) {
+          indexForward = index + direction
+          indexBackward = index - direction
+          distanceForward = slideWidth * direction
+          distanceBackward = -slideWidth * direction
+          if (this.options.continuous) {
+            this.move(this.circle(indexForward), distanceForward, 0)
+            this.move(this.circle(index - 2 * direction), distanceBackward, 0)
+          } else if (indexForward >= 0 && indexForward < this.num) {
+            this.move(indexForward, distanceForward, 0)
+          }
+          this.move(index, this.positions[index] + distanceForward, duration)
+          this.move(
+            this.circle(indexBackward),
+            this.positions[this.circle(indexBackward)] + distanceForward,
+            duration
+          )
+          index = this.circle(indexBackward)
+          this.onslide(index)
+        } else {
+          // Move back into position
+          if (this.options.continuous) {
+            this.move(this.circle(index - 1), -slideWidth, duration)
+            this.move(index, 0, duration)
+            this.move(this.circle(index + 1), slideWidth, duration)
+          } else {
+            if (index) {
+              this.move(index - 1, -slideWidth, duration)
+            }
+            this.move(index, 0, duration)
+            if (index < this.num - 1) {
+              this.move(index + 1, slideWidth, duration)
+            }
+          }
+        }
+      } else {
+        if (isValidClose) {
+          this.close()
+        } else {
+          // Move back into position
+          this.translateY(index, 0, duration)
+        }
+      }
+    },
+
+    ontouchcancel: function (event) {
+      if (this.touchStart) {
+        this.ontouchend(event)
+        delete this.touchStart
+      }
+    },
+
+    ontransitionend: function (event) {
+      var slide = this.slides[this.index]
+      if (!event || slide === event.target) {
+        if (this.interval) {
+          this.play()
+        }
+        this.setTimeout(this.options.onslideend, [this.index, slide])
+      }
+    },
+
+    oncomplete: function (event) {
+      var target = event.target || event.srcElement
+      var parent = target && target.parentNode
+      var index
+      if (!target || !parent) {
+        return
+      }
+      index = this.getNodeIndex(parent)
+      $(parent).removeClass(this.options.slideLoadingClass)
+      if (event.type === 'error') {
+        $(parent).addClass(this.options.slideErrorClass)
+        this.elements[index] = 3 // Fail
+      } else {
+        this.elements[index] = 2 // Done
+      }
+      // Fix for IE7's lack of support for percentage max-height:
+      if (target.clientHeight > this.container[0].clientHeight) {
+        target.style.maxHeight = this.container[0].clientHeight
+      }
+      if (this.interval && this.slides[this.index] === parent) {
+        this.play()
+      }
+      this.setTimeout(this.options.onslidecomplete, [index, parent])
+    },
+
+    onload: function (event) {
+      this.oncomplete(event)
+    },
+
+    onerror: function (event) {
+      this.oncomplete(event)
+    },
+
+    onkeydown: function (event) {
+      switch (event.which || event.keyCode) {
+        case 13: // Enter
+          if (this.options.toggleControlsOnEnter) {
+            this.preventDefault(event)
+            this.toggleControls()
+          }
+          break
+        case 27: // Escape
+          if (this.options.closeOnEscape) {
+            this.close()
+            // prevent Escape from closing other things
+            event.stopImmediatePropagation()
+          }
+          break
+        case 32: // Space
+          if (this.options.toggleSlideshowOnSpace) {
+            this.preventDefault(event)
+            this.toggleSlideshow()
+          }
+          break
+        case 37: // ArrowLeft
+          if (this.options.enableKeyboardNavigation) {
+            this.preventDefault(event)
+            this.prev()
+          }
+          break
+        case 39: // ArrowRight
+          if (this.options.enableKeyboardNavigation) {
+            this.preventDefault(event)
+            this.next()
+          }
+          break
+      }
+    },
+
+    handleClick: function (event) {
+      var options = this.options
+      var target = event.target || event.srcElement
+      var parent = target.parentNode
+      /**
+       * Checks if the target from the close has the given class
+       *
+       * @param {string} className Class name
+       * @returns {boolean} Returns true if the target has the class name
+       */
+      function isTarget(className) {
+        return $(target).hasClass(className) || $(parent).hasClass(className)
+      }
+      if (isTarget(options.toggleClass)) {
+        // Click on "toggle" control
+        this.preventDefault(event)
+        this.toggleControls()
+      } else if (isTarget(options.prevClass)) {
+        // Click on "prev" control
+        this.preventDefault(event)
+        this.prev()
+      } else if (isTarget(options.nextClass)) {
+        // Click on "next" control
+        this.preventDefault(event)
+        this.next()
+      } else if (isTarget(options.closeClass)) {
+        // Click on "close" control
+        this.preventDefault(event)
+        this.close()
+      } else if (isTarget(options.playPauseClass)) {
+        // Click on "play-pause" control
+        this.preventDefault(event)
+        this.toggleSlideshow()
+      } else if (parent === this.slidesContainer[0]) {
+        // Click on slide background
+        if (options.closeOnSlideClick) {
+          this.preventDefault(event)
+          this.close()
+        } else if (options.toggleControlsOnSlideClick) {
+          this.preventDefault(event)
+          this.toggleControls()
+        }
+      } else if (
+        parent.parentNode &&
+        parent.parentNode === this.slidesContainer[0]
+      ) {
+        // Click on displayed element
+        if (options.toggleControlsOnSlideClick) {
+          this.preventDefault(event)
+          this.toggleControls()
+        }
+      }
+    },
+
+    onclick: function (event) {
+      if (
+        this.options.emulateTouchEvents &&
+        this.touchDelta &&
+        (Math.abs(this.touchDelta.x) > 20 || Math.abs(this.touchDelta.y) > 20)
+      ) {
+        delete this.touchDelta
+        return
+      }
+      return this.handleClick(event)
+    },
+
+    updateEdgeClasses: function (index) {
+      if (!index) {
+        this.container.addClass(this.options.leftEdgeClass)
+      } else {
+        this.container.removeClass(this.options.leftEdgeClass)
+      }
+      if (index === this.num - 1) {
+        this.container.addClass(this.options.rightEdgeClass)
+      } else {
+        this.container.removeClass(this.options.rightEdgeClass)
+      }
+    },
+
+    updateActiveSlide: function (oldIndex, newIndex) {
+      var slides = this.slides
+      var options = this.options
+      var list = [
+        {
+          index: newIndex,
+          method: 'addClass',
+          hidden: false
+        },
+        {
+          index: oldIndex,
+          method: 'removeClass',
+          hidden: true
+        }
+      ]
+      var item, index
+      while (list.length) {
+        item = list.pop()
+        $(slides[item.index])[item.method](options.slideActiveClass)
+        index = this.circle(item.index - 1)
+        if (options.continuous || index < item.index) {
+          $(slides[index])[item.method](options.slidePrevClass)
+        }
+        index = this.circle(item.index + 1)
+        if (options.continuous || index > item.index) {
+          $(slides[index])[item.method](options.slideNextClass)
+        }
+      }
+      this.slides[oldIndex].setAttribute('aria-hidden', 'true')
+      this.slides[newIndex].removeAttribute('aria-hidden')
+    },
+
+    handleSlide: function (oldIndex, newIndex) {
+      if (!this.options.continuous) {
+        this.updateEdgeClasses(newIndex)
+      }
+      this.updateActiveSlide(oldIndex, newIndex)
+      this.loadElements(newIndex)
+      if (this.options.unloadElements) {
+        this.unloadElements(oldIndex, newIndex)
+      }
+      this.setTitle(newIndex)
+    },
+
+    onslide: function (index) {
+      this.handleSlide(this.index, index)
+      this.index = index
+      this.setTimeout(this.options.onslide, [index, this.slides[index]])
+    },
+
+    setTitle: function (index) {
+      var firstChild = this.slides[index].firstChild
+      var text = firstChild.title || firstChild.alt
+      var titleElement = this.titleElement
+      if (titleElement.length) {
+        this.titleElement.empty()
+        if (text) {
+          titleElement[0].appendChild(document.createTextNode(text))
+        }
+      }
+    },
+
+    setTimeout: function (func, args, wait) {
+      var that = this
+      return (
+        func &&
+        window.setTimeout(function () {
+          func.apply(that, args || [])
+        }, wait || 0)
+      )
+    },
+
+    imageFactory: function (obj, callback) {
+      var options = this.options
+      var that = this
+      var url = obj
+      var img = this.imagePrototype.cloneNode(false)
+      var picture
+      var called
+      var sources
+      var srcset
+      var sizes
+      var title
+      var altText
+      var i
+      /**
+       * Wraps the callback function for the load/error event
+       *
+       * @param {event} event load/error event
+       * @returns {number} timeout ID
+       */
+      function callbackWrapper(event) {
+        if (!called) {
+          event = {
+            type: event.type,
+            target: picture || img
+          }
+          if (!event.target.parentNode) {
+            // Fix for browsers (e.g. IE7) firing the load event for
+            // cached images before the element could
+            // be added to the DOM:
+            return that.setTimeout(callbackWrapper, [event])
+          }
+          called = true
+          $(img).off('load error', callbackWrapper)
+          callback(event)
+        }
+      }
+      if (typeof url !== 'string') {
+        url = this.getItemProperty(obj, options.urlProperty)
+        sources =
+          this.support.picture &&
+          this.support.source &&
+          this.getItemProperty(obj, options.sourcesProperty)
+        srcset = this.getItemProperty(obj, options.srcsetProperty)
+        sizes = this.getItemProperty(obj, options.sizesProperty)
+        title = this.getItemProperty(obj, options.titleProperty)
+        altText = this.getItemProperty(obj, options.altTextProperty) || title
+      }
+      img.draggable = false
+      if (title) {
+        img.title = title
+      }
+      if (altText) {
+        img.alt = altText
+      }
+      $(img).on('load error', callbackWrapper)
+      if (sources && sources.length) {
+        picture = this.picturePrototype.cloneNode(false)
+        for (i = 0; i < sources.length; i += 1) {
+          picture.appendChild(
+            $.extend(this.sourcePrototype.cloneNode(false), sources[i])
+          )
+        }
+        picture.appendChild(img)
+        $(picture).addClass(options.toggleClass)
+      }
+      if (srcset) {
+        if (sizes) {
+          img.sizes = sizes
+        }
+        img.srcset = srcset
+      }
+      img.src = url
+      if (picture) return picture
+      return img
+    },
+
+    createElement: function (obj, callback) {
+      var type = obj && this.getItemProperty(obj, this.options.typeProperty)
+      var factory =
+        (type && this[type.split('/')[0] + 'Factory']) || this.imageFactory
+      var element = obj && factory.call(this, obj, callback)
+      if (!element) {
+        element = this.elementPrototype.cloneNode(false)
+        this.setTimeout(callback, [
+          {
+            type: 'error',
+            target: element
+          }
+        ])
+      }
+      $(element).addClass(this.options.slideContentClass)
+      return element
+    },
+
+    iteratePreloadRange: function (index, func) {
+      var num = this.num
+      var options = this.options
+      var limit = Math.min(num, options.preloadRange * 2 + 1)
+      var j = index
+      var i
+      for (i = 0; i < limit; i += 1) {
+        // First iterate to the current index (0),
+        // then the next one (+1),
+        // then the previous one (-1),
+        // then the next after next (+2),
+        // then the one before the previous one (-2), etc.:
+        j += i * (i % 2 === 0 ? -1 : 1)
+        if (j < 0 || j >= num) {
+          if (!options.continuous) continue
+          // Connect the ends of the list to load slide elements for
+          // continuous iteration:
+          j = this.circle(j)
+        }
+        func.call(this, j)
+      }
+    },
+
+    loadElement: function (index) {
+      if (!this.elements[index]) {
+        if (this.slides[index].firstChild) {
+          this.elements[index] = $(this.slides[index]).hasClass(
+            this.options.slideErrorClass
+          )
+            ? 3
+            : 2
+        } else {
+          this.elements[index] = 1 // Loading
+          $(this.slides[index]).addClass(this.options.slideLoadingClass)
+          this.slides[index].appendChild(
+            this.createElement(this.list[index], this.proxyListener)
+          )
+        }
+      }
+    },
+
+    loadElements: function (index) {
+      this.iteratePreloadRange(index, this.loadElement)
+    },
+
+    unloadElements: function (oldIndex, newIndex) {
+      var preloadRange = this.options.preloadRange
+      this.iteratePreloadRange(oldIndex, function (i) {
+        var diff = Math.abs(i - newIndex)
+        if (diff > preloadRange && diff + preloadRange < this.num) {
+          this.unloadSlide(i)
+          delete this.elements[i]
+        }
+      })
+    },
+
+    addSlide: function (index) {
+      var slide = this.slidePrototype.cloneNode(false)
+      slide.setAttribute('data-index', index)
+      slide.setAttribute('aria-hidden', 'true')
+      this.slidesContainer[0].appendChild(slide)
+      this.slides.push(slide)
+    },
+
+    positionSlide: function (index) {
+      var slide = this.slides[index]
+      slide.style.width = this.slideWidth + 'px'
+      if (this.support.transform) {
+        slide.style.left = index * -this.slideWidth + 'px'
+        this.move(
+          index,
+          this.index > index
+            ? -this.slideWidth
+            : this.index < index
+            ? this.slideWidth
+            : 0,
+          0
+        )
+      }
+    },
+
+    initSlides: function (reload) {
+      var clearSlides, i
+      if (!reload) {
+        this.positions = []
+        this.positions.length = this.num
+        this.elements = {}
+        this.picturePrototype =
+          this.support.picture && document.createElement('picture')
+        this.sourcePrototype =
+          this.support.source && document.createElement('source')
+        this.imagePrototype = document.createElement('img')
+        this.elementPrototype = document.createElement('div')
+        this.slidePrototype = this.elementPrototype.cloneNode(false)
+        $(this.slidePrototype).addClass(this.options.slideClass)
+        this.slides = this.slidesContainer[0].children
+        clearSlides =
+          this.options.clearSlides || this.slides.length !== this.num
+      }
+      this.slideWidth = this.container[0].clientWidth
+      this.slideHeight = this.container[0].clientHeight
+      this.slidesContainer[0].style.width = this.num * this.slideWidth + 'px'
+      if (clearSlides) {
+        this.resetSlides()
+      }
+      for (i = 0; i < this.num; i += 1) {
+        if (clearSlides) {
+          this.addSlide(i)
+        }
+        this.positionSlide(i)
+      }
+      // Reposition the slides before and after the given index:
+      if (this.options.continuous && this.support.transform) {
+        this.move(this.circle(this.index - 1), -this.slideWidth, 0)
+        this.move(this.circle(this.index + 1), this.slideWidth, 0)
+      }
+      if (!this.support.transform) {
+        this.slidesContainer[0].style.left =
+          this.index * -this.slideWidth + 'px'
+      }
+    },
+
+    unloadSlide: function (index) {
+      var slide, firstChild
+      slide = this.slides[index]
+      firstChild = slide.firstChild
+      if (firstChild !== null) {
+        slide.removeChild(firstChild)
+      }
+    },
+
+    unloadAllSlides: function () {
+      var i, len
+      for (i = 0, len = this.slides.length; i < len; i++) {
+        this.unloadSlide(i)
+      }
+    },
+
+    toggleControls: function () {
+      var controlsClass = this.options.controlsClass
+      if (this.container.hasClass(controlsClass)) {
+        this.container.removeClass(controlsClass)
+      } else {
+        this.container.addClass(controlsClass)
+      }
+    },
+
+    toggleSlideshow: function () {
+      if (!this.interval) {
+        this.play()
+      } else {
+        this.pause()
+      }
+    },
+
+    getNodeIndex: function (element) {
+      return parseInt(element.getAttribute('data-index'), 10)
+    },
+
+    getNestedProperty: function (obj, property) {
+      property.replace(
+        // Matches native JavaScript notation in a String,
+        // e.g. '["doubleQuoteProp"].dotProp[2]'
+        // eslint-disable-next-line no-useless-escape
+        /\[(?:'([^']+)'|"([^"]+)"|(\d+))\]|(?:(?:^|\.)([^\.\[]+))/g,
+        function (str, singleQuoteProp, doubleQuoteProp, arrayIndex, dotProp) {
+          var prop =
+            dotProp ||
+            singleQuoteProp ||
+            doubleQuoteProp ||
+            (arrayIndex && parseInt(arrayIndex, 10))
+          if (str && obj) {
+            obj = obj[prop]
+          }
+        }
+      )
+      return obj
+    },
+
+    getDataProperty: function (obj, property) {
+      var key
+      var prop
+      if (obj.dataset) {
+        key = property.replace(/-([a-z])/g, function (_, b) {
+          return b.toUpperCase()
+        })
+        prop = obj.dataset[key]
+      } else if (obj.getAttribute) {
+        prop = obj.getAttribute(
+          'data-' + property.replace(/([A-Z])/g, '-$1').toLowerCase()
+        )
+      }
+      if (typeof prop === 'string') {
+        // eslint-disable-next-line no-useless-escape
+        if (
+          /^(true|false|null|-?\d+(\.\d+)?|\{[\s\S]*\}|\[[\s\S]*\])$/.test(prop)
+        ) {
+          try {
+            return $.parseJSON(prop)
+          } catch (ignore) {
+            // ignore JSON parsing errors
+          }
+        }
+        return prop
+      }
+    },
+
+    getItemProperty: function (obj, property) {
+      var prop = this.getDataProperty(obj, property)
+      if (prop === undefined) {
+        prop = obj[property]
+      }
+      if (prop === undefined) {
+        prop = this.getNestedProperty(obj, property)
+      }
+      return prop
+    },
+
+    initStartIndex: function () {
+      var index = this.options.index
+      var urlProperty = this.options.urlProperty
+      var i
+      // Check if the index is given as a list object:
+      if (index && typeof index !== 'number') {
+        for (i = 0; i < this.num; i += 1) {
+          if (
+            this.list[i] === index ||
+            this.getItemProperty(this.list[i], urlProperty) ===
+              this.getItemProperty(index, urlProperty)
+          ) {
+            index = i
+            break
+          }
+        }
+      }
+      // Make sure the index is in the list range:
+      this.index = this.circle(parseInt(index, 10) || 0)
+    },
+
+    initEventListeners: function () {
+      var that = this
+      var slidesContainer = this.slidesContainer
+      /**
+       * Proxy listener
+       *
+       * @param {event} event original event
+       */
+      function proxyListener(event) {
+        var type =
+          that.support.transition && that.support.transition.end === event.type
+            ? 'transitionend'
+            : event.type
+        that['on' + type](event)
+      }
+      $(window).on('resize', proxyListener)
+      $(window).on('hashchange', proxyListener)
+      $(document.body).on('keydown', proxyListener)
+      this.container.on('click', proxyListener)
+      if (this.support.touch) {
+        slidesContainer.on(
+          'touchstart touchmove touchend touchcancel',
+          proxyListener
+        )
+      } else if (this.options.emulateTouchEvents && this.support.transition) {
+        slidesContainer.on(
+          'mousedown mousemove mouseup mouseout',
+          proxyListener
+        )
+      }
+      if (this.support.transition) {
+        slidesContainer.on(this.support.transition.end, proxyListener)
+      }
+      this.proxyListener = proxyListener
+    },
+
+    destroyEventListeners: function () {
+      var slidesContainer = this.slidesContainer
+      var proxyListener = this.proxyListener
+      $(window).off('resize', proxyListener)
+      $(document.body).off('keydown', proxyListener)
+      this.container.off('click', proxyListener)
+      if (this.support.touch) {
+        slidesContainer.off(
+          'touchstart touchmove touchend touchcancel',
+          proxyListener
+        )
+      } else if (this.options.emulateTouchEvents && this.support.transition) {
+        slidesContainer.off(
+          'mousedown mousemove mouseup mouseout',
+          proxyListener
+        )
+      }
+      if (this.support.transition) {
+        slidesContainer.off(this.support.transition.end, proxyListener)
+      }
+    },
+
+    handleOpen: function () {
+      if (this.options.onopened) {
+        this.options.onopened.call(this)
+      }
+    },
+
+    initWidget: function () {
+      var that = this
+      /**
+       * Open handler
+       *
+       * @param {event} event Gallery open event
+       */
+      function openHandler(event) {
+        if (event.target === that.container[0]) {
+          that.container.off(that.support.transition.end, openHandler)
+          that.handleOpen()
+        }
+      }
+      this.container = $(this.options.container)
+      if (!this.container.length) {
+        this.console.log(
+          'blueimp Gallery: Widget container not found.',
+          this.options.container
+        )
+        return false
+      }
+      this.slidesContainer = this.container
+        .find(this.options.slidesContainer)
+        .first()
+      if (!this.slidesContainer.length) {
+        this.console.log(
+          'blueimp Gallery: Slides container not found.',
+          this.options.slidesContainer
+        )
+        return false
+      }
+      this.titleElement = this.container.find(this.options.titleElement).first()
+      this.playPauseElement = this.container
+        .find('.' + this.options.playPauseClass)
+        .first()
+      if (this.num === 1) {
+        this.container.addClass(this.options.singleClass)
+      }
+      if (this.support.svgasimg) {
+        this.container.addClass(this.options.svgasimgClass)
+      }
+      if (this.support.smil) {
+        this.container.addClass(this.options.smilClass)
+      }
+      if (this.options.onopen) {
+        this.options.onopen.call(this)
+      }
+      if (this.support.transition && this.options.displayTransition) {
+        this.container.on(this.support.transition.end, openHandler)
+      } else {
+        this.handleOpen()
+      }
+      if (this.options.hidePageScrollbars) {
+        // Hide the page scrollbars:
+        this.bodyOverflowStyle = document.body.style.overflow
+        document.body.style.overflow = 'hidden'
+      }
+      this.container[0].style.display = 'block'
+      this.initSlides()
+      this.container.addClass(this.options.displayClass)
+    },
+
+    initOptions: function (options) {
+      // Create a copy of the prototype options:
+      this.options = $.extend({}, this.options)
+      // Check if carousel mode is enabled:
+      if (
+        (options && options.carousel) ||
+        (this.options.carousel && (!options || options.carousel !== false))
+      ) {
+        $.extend(this.options, this.carouselOptions)
+      }
+      // Override any given options:
+      $.extend(this.options, options)
+      if (this.num < 3) {
+        // 1 or 2 slides cannot be displayed continuous,
+        // remember the original option by setting to null instead of false:
+        this.options.continuous = this.options.continuous ? null : false
+      }
+      if (!this.support.transition) {
+        this.options.emulateTouchEvents = false
+      }
+      if (this.options.event) {
+        this.preventDefault(this.options.event)
+      }
+    }
+  })
+
+  return Gallery
+})
diff --git a/docs/pt/js/blueimp-gallery.min.js b/docs/pt/js/blueimp-gallery.min.js
new file mode 100644
index 0000000..c3b72a9
--- /dev/null
+++ b/docs/pt/js/blueimp-gallery.min.js
@@ -0,0 +1,2 @@
+!function(){"use strict";function t(t,e){var i;for(i in e)Object.prototype.hasOwnProperty.call(e,i)&&(t[i]=e[i]);return t}function i(t){if(!this||this.find!==i.prototype.find)return new i(t);if(this.length=0,t)if("string"==typeof t&&(t=this.find(t)),t.nodeType||t===t.window)this.length=1,this[0]=t;else{var e=t.length;for(this.length=e;e;)this[--e]=t[e]}}i.extend=t,i.contains=function(t,e){do{if((e=e.parentNode)===t)return!0}while(e);return!1},i.parseJSON=function(t){return JSON.parse(t)},t(i.prototype,{find:function(t){var e=this[0]||document;return"string"==typeof t&&(t=e.querySelectorAll?e.querySelectorAll(t):"#"===t.charAt(0)?e.getElementById(t.slice(1)):e.getElementsByTagName(t)),new i(t)},hasClass:function(t){return!!this[0]&&new RegExp("(?:^|\\s+)"+t+"(?:\\s+|$)").test(this[0].className)},addClass:function(t){for(var e,i,s,n=this.length;n;)if((i=this[--n]).className)for(e=e||t.split(/\s+/),s=0;s<e.length;s+=1)this.hasClass(e[s])||(i.className+=" "+e[s]);else i.className=t;return this},removeClass:function(t){for(var e,i=new RegExp("^(?:"+t.split(/\s+/).join("|")+")$"),s=/(\S+)(?:\s+|$)/g,n=function(t,e){return i.test(e)?"":t},o=/\s+$/,a=this.length;a;)(e=this[--a]).className=e.className.replace(s,n).replace(o,"");return this},on:function(t,e){for(var i,s,n=t.split(/\s+/);n.length;)for(t=n.shift(),i=this.length;i;)(s=this[--i]).addEventListener?s.addEventListener(t,e,!1):s.attachEvent&&s.attachEvent("on"+t,e);return this},off:function(t,e){for(var i,s,n=t.split(/\s+/);n.length;)for(t=n.shift(),i=this.length;i;)(s=this[--i]).removeEventListener?s.removeEventListener(t,e,!1):s.detachEvent&&s.detachEvent("on"+t,e);return this},empty:function(){for(var t,e=this.length;e;)for(t=this[--e];t.hasChildNodes();)t.removeChild(t.lastChild);return this},first:function(){return new i(this[0])}}),"function"==typeof define&&define.amd?define(function(){return i}):(window.blueimp=window.blueimp||{},window.blueimp.helper=i)}(),function(t){"use strict";"function"==typeof define&&define.amd?define(["./blueimp-helper"],t):(window.blueimp=window.blueimp||{},window.blueimp.Gallery=t(window.blueimp.helper||window.jQuery))}(function(f){"use strict";function i(t,e){return document.body.style.maxHeight===undefined?null:this&&this.options===i.prototype.options?void(t&&t.length?(this.list=t,this.num=t.length,this.initOptions(e),this.initialize()):this.console.log("blueimp Gallery: No or empty list provided as first argument.",t)):new i(t,e)}return f.extend(i.prototype,{options:{container:"#blueimp-gallery",slidesContainer:"div",titleElement:"h3",displayClass:"blueimp-gallery-display",controlsClass:"blueimp-gallery-controls",singleClass:"blueimp-gallery-single",leftEdgeClass:"blueimp-gallery-left",rightEdgeClass:"blueimp-gallery-right",playingClass:"blueimp-gallery-playing",svgasimgClass:"blueimp-gallery-svgasimg",smilClass:"blueimp-gallery-smil",slideClass:"slide",slideActiveClass:"slide-active",slidePrevClass:"slide-prev",slideNextClass:"slide-next",slideLoadingClass:"slide-loading",slideErrorClass:"slide-error",slideContentClass:"slide-content",toggleClass:"toggle",prevClass:"prev",nextClass:"next",closeClass:"close",playPauseClass:"play-pause",typeProperty:"type",titleProperty:"title",altTextProperty:"alt",urlProperty:"href",srcsetProperty:"srcset",sizesProperty:"sizes",sourcesProperty:"sources",displayTransition:!0,clearSlides:!0,toggleControlsOnEnter:!0,toggleControlsOnSlideClick:!0,toggleSlideshowOnSpace:!0,enableKeyboardNavigation:!0,closeOnEscape:!0,closeOnSlideClick:!0,closeOnSwipeUpOrDown:!0,closeOnHashChange:!0,emulateTouchEvents:!0,stopTouchEventsPropagation:!1,hidePageScrollbars:!0,disableScroll:!0,carousel:!1,continuous:!0,unloadElements:!0,startSlideshow:!1,slideshowInterval:5e3,slideshowDirection:"ltr",index:0,preloadRange:2,transitionDuration:300,slideshowTransitionDuration:500,event:undefined,onopen:undefined,onopened:undefined,onslide:undefined,onslideend:undefined,onslidecomplete:undefined,onclose:undefined,onclosed:undefined},carouselOptions:{hidePageScrollbars:!1,toggleControlsOnEnter:!1,toggleSlideshowOnSpace:!1,enableKeyboardNavigation:!1,closeOnEscape:!1,closeOnSlideClick:!1,closeOnSwipeUpOrDown:!1,closeOnHashChange:!1,disableScroll:!1,startSlideshow:!0},console:window.console&&"function"==typeof window.console.log?window.console:{log:function(){}},support:function(s){var t,n={source:!!window.HTMLSourceElement,picture:!!window.HTMLPictureElement,svgasimg:document.implementation.hasFeature("http://www.w3.org/TR/SVG11/feature#Image","1.1"),smil:!!document.createElementNS&&/SVGAnimate/.test(document.createElementNS("http://www.w3.org/2000/svg","animate").toString()),touch:window.ontouchstart!==undefined||window.DocumentTouch&&document instanceof DocumentTouch},e={webkitTransition:{end:"webkitTransitionEnd",prefix:"-webkit-"},MozTransition:{end:"transitionend",prefix:"-moz-"},OTransition:{end:"otransitionend",prefix:"-o-"},transition:{end:"transitionend",prefix:""}};for(t in e)if(Object.prototype.hasOwnProperty.call(e,t)&&s.style[t]!==undefined){n.transition=e[t],n.transition.name=t;break}function i(){var t,e,i=n.transition;document.body.appendChild(s),i&&(t=i.name.slice(0,-9)+"ransform",s.style[t]!==undefined&&(s.style[t]="translateZ(0)",e=window.getComputedStyle(s).getPropertyValue(i.prefix+"transform"),n.transform={prefix:i.prefix,name:t,translate:!0,translateZ:!!e&&"none"!==e})),document.body.removeChild(s)}return document.body?i():f(document).on("DOMContentLoaded",i),n}(document.createElement("div")),requestAnimationFrame:window.requestAnimationFrame||window.webkitRequestAnimationFrame||window.mozRequestAnimationFrame,cancelAnimationFrame:window.cancelAnimationFrame||window.webkitCancelRequestAnimationFrame||window.webkitCancelAnimationFrame||window.mozCancelAnimationFrame,initialize:function(){if(this.initStartIndex(),!1===this.initWidget())return!1;this.initEventListeners(),this.onslide(this.index),this.ontransitionend(),this.options.startSlideshow&&this.play()},slide:function(t,e){window.clearTimeout(this.timeout);var i,s,n,o=this.index;if(o!==t&&1!==this.num){if(e=e||this.options.transitionDuration,this.support.transform){for(this.options.continuous||(t=this.circle(t)),i=Math.abs(o-t)/(o-t),this.options.continuous&&(s=i,(i=-this.positions[this.circle(t)]/this.slideWidth)!==s&&(t=-i*this.num+t)),n=Math.abs(o-t)-1;n;)--n,this.move(this.circle((o<t?t:o)-n-1),this.slideWidth*i,0);t=this.circle(t),this.move(o,this.slideWidth*i,e),this.move(t,0,e),this.options.continuous&&this.move(this.circle(t-i),-this.slideWidth*i,0)}else t=this.circle(t),this.animate(o*-this.slideWidth,t*-this.slideWidth,e);this.onslide(t)}},getIndex:function(){return this.index},getNumber:function(){return this.num},prev:function(){(this.options.continuous||this.index)&&this.slide(this.index-1)},next:function(){(this.options.continuous||this.index<this.num-1)&&this.slide(this.index+1)},play:function(t){var i=this,e=this.index+("rtl"===this.options.slideshowDirection?-1:1);window.clearTimeout(this.timeout),this.interval=t||this.options.slideshowInterval,1<this.elements[this.index]&&(this.timeout=this.setTimeout(!this.requestAnimationFrame&&this.slide||function(t,e){i.animationFrameId=i.requestAnimationFrame.call(window,function(){i.slide(t,e)})},[e,this.options.slideshowTransitionDuration],this.interval)),this.container.addClass(this.options.playingClass),this.slidesContainer[0].setAttribute("aria-live","off"),this.playPauseElement.length&&this.playPauseElement[0].setAttribute("aria-pressed","true")},pause:function(){window.clearTimeout(this.timeout),this.interval=null,this.cancelAnimationFrame&&(this.cancelAnimationFrame.call(window,this.animationFrameId),this.animationFrameId=null),this.container.removeClass(this.options.playingClass),this.slidesContainer[0].setAttribute("aria-live","polite"),this.playPauseElement.length&&this.playPauseElement[0].setAttribute("aria-pressed","false")},add:function(t){var e;for(t.concat||(t=Array.prototype.slice.call(t)),this.list.concat||(this.list=Array.prototype.slice.call(this.list)),this.list=this.list.concat(t),this.num=this.list.length,2<this.num&&null===this.options.continuous&&(this.options.continuous=!0,this.container.removeClass(this.options.leftEdgeClass)),this.container.removeClass(this.options.rightEdgeClass).removeClass(this.options.singleClass),e=this.num-t.length;e<this.num;e+=1)this.addSlide(e),this.positionSlide(e);this.positions.length=this.num,this.initSlides(!0)},resetSlides:function(){this.slidesContainer.empty(),this.unloadAllSlides(),this.slides=[]},handleClose:function(){var t=this.options;this.destroyEventListeners(),this.pause(),this.container[0].style.display="none",this.container.removeClass(t.displayClass).removeClass(t.singleClass).removeClass(t.leftEdgeClass).removeClass(t.rightEdgeClass),t.hidePageScrollbars&&(document.body.style.overflow=this.bodyOverflowStyle),this.options.clearSlides&&this.resetSlides(),this.options.onclosed&&this.options.onclosed.call(this)},close:function(){var e=this;this.options.onclose&&this.options.onclose.call(this),this.support.transition&&this.options.displayTransition?(this.container.on(this.support.transition.end,function i(t){t.target===e.container[0]&&(e.container.off(e.support.transition.end,i),e.handleClose())}),this.container.removeClass(this.options.displayClass)):this.handleClose()},circle:function(t){return(this.num+t%this.num)%this.num},move:function(t,e,i){this.translateX(t,e,i),this.positions[t]=e},translate:function(t,e,i,s){var n,o,a;this.slides[t]&&(n=this.slides[t].style,o=this.support.transition,a=this.support.transform,n[o.name+"Duration"]=s+"ms",n[a.name]="translate("+e+"px, "+i+"px)"+(a.translateZ?" translateZ(0)":""))},translateX:function(t,e,i){this.translate(t,e,0,i)},translateY:function(t,e,i){this.translate(t,0,e,i)},animate:function(e,i,s){var n,o,a;s?(n=this,o=(new Date).getTime(),a=window.setInterval(function(){var t=(new Date).getTime()-o;if(s<t)return n.slidesContainer[0].style.left=i+"px",n.ontransitionend(),void window.clearInterval(a);n.slidesContainer[0].style.left=(i-e)*(Math.floor(t/s*100)/100)+e+"px"},4)):this.slidesContainer[0].style.left=i+"px"},preventDefault:function(t){t.preventDefault?t.preventDefault():t.returnValue=!1},stopPropagation:function(t){t.stopPropagation?t.stopPropagation():t.cancelBubble=!0},onresize:function(){this.initSlides(!0)},onhashchange:function(){this.options.closeOnHashChange&&this.close()},onmousedown:function(t){t.which&&1===t.which&&"VIDEO"!==t.target.nodeName&&"AUDIO"!==t.target.nodeName&&(t.preventDefault(),(t.originalEvent||t).touches=[{pageX:t.pageX,pageY:t.pageY}],this.ontouchstart(t))},onmousemove:function(t){this.touchStart&&((t.originalEvent||t).touches=[{pageX:t.pageX,pageY:t.pageY}],this.ontouchmove(t))},onmouseup:function(t){this.touchStart&&(this.ontouchend(t),delete this.touchStart)},onmouseout:function(t){var e,i;this.touchStart&&(e=t.target,(i=t.relatedTarget)&&(i===e||f.contains(e,i))||this.onmouseup(t))},ontouchstart:function(t){this.options.stopTouchEventsPropagation&&this.stopPropagation(t);var e=(t.originalEvent||t).touches[0];this.touchStart={x:e.pageX,y:e.pageY,time:Date.now()},this.isScrolling=undefined,this.touchDelta={}},ontouchmove:function(t){this.options.stopTouchEventsPropagation&&this.stopPropagation(t);var e,i,s=(t.originalEvent||t).touches,n=s[0],o=(t.originalEvent||t).scale,a=this.index;if(!(1<s.length||o&&1!==o))if(this.options.disableScroll&&t.preventDefault(),this.touchDelta={x:n.pageX-this.touchStart.x,y:n.pageY-this.touchStart.y},e=this.touchDelta.x,this.isScrolling===undefined&&(this.isScrolling=this.isScrolling||Math.abs(e)<Math.abs(this.touchDelta.y)),this.isScrolling)this.options.carousel||this.translateY(a,this.touchDelta.y+this.positions[a],0);else for(t.preventDefault(),window.clearTimeout(this.timeout),this.options.continuous?i=[this.circle(a+1),a,this.circle(a-1)]:(this.touchDelta.x=e/=!a&&0<e||a===this.num-1&&e<0?Math.abs(e)/this.slideWidth+1:1,i=[a],a&&i.push(a-1),a<this.num-1&&i.unshift(a+1));i.length;)a=i.pop(),this.translateX(a,e+this.positions[a],0)},ontouchend:function(t){this.options.stopTouchEventsPropagation&&this.stopPropagation(t);var e,i,s,n,o,a=this.index,l=Math.abs(this.touchDelta.x),r=this.slideWidth,h=Math.ceil(this.options.transitionDuration*(1-l/r)/2),d=20<l,c=!a&&0<this.touchDelta.x||a===this.num-1&&this.touchDelta.x<0,u=!d&&this.options.closeOnSwipeUpOrDown&&20<Math.abs(this.touchDelta.y);this.options.continuous&&(c=!1),e=this.touchDelta.x<0?-1:1,this.isScrolling?u?this.close():this.translateY(a,0,h):d&&!c?(i=a+e,s=a-e,n=r*e,o=-r*e,this.options.continuous?(this.move(this.circle(i),n,0),this.move(this.circle(a-2*e),o,0)):0<=i&&i<this.num&&this.move(i,n,0),this.move(a,this.positions[a]+n,h),this.move(this.circle(s),this.positions[this.circle(s)]+n,h),a=this.circle(s),this.onslide(a)):this.options.continuous?(this.move(this.circle(a-1),-r,h),this.move(a,0,h),this.move(this.circle(a+1),r,h)):(a&&this.move(a-1,-r,h),this.move(a,0,h),a<this.num-1&&this.move(a+1,r,h))},ontouchcancel:function(t){this.touchStart&&(this.ontouchend(t),delete this.touchStart)},ontransitionend:function(t){var e=this.slides[this.index];t&&e!==t.target||(this.interval&&this.play(),this.setTimeout(this.options.onslideend,[this.index,e]))},oncomplete:function(t){var e,i=t.target||t.srcElement,s=i&&i.parentNode;i&&s&&(e=this.getNodeIndex(s),f(s).removeClass(this.options.slideLoadingClass),"error"===t.type?(f(s).addClass(this.options.slideErrorClass),this.elements[e]=3):this.elements[e]=2,i.clientHeight>this.container[0].clientHeight&&(i.style.maxHeight=this.container[0].clientHeight),this.interval&&this.slides[this.index]===s&&this.play(),this.setTimeout(this.options.onslidecomplete,[e,s]))},onload:function(t){this.oncomplete(t)},onerror:function(t){this.oncomplete(t)},onkeydown:function(t){switch(t.which||t.keyCode){case 13:this.options.toggleControlsOnEnter&&(this.preventDefault(t),this.toggleControls());break;case 27:this.options.closeOnEscape&&(this.close(),t.stopImmediatePropagation());break;case 32:this.options.toggleSlideshowOnSpace&&(this.preventDefault(t),this.toggleSlideshow());break;case 37:this.options.enableKeyboardNavigation&&(this.preventDefault(t),this.prev());break;case 39:this.options.enableKeyboardNavigation&&(this.preventDefault(t),this.next())}},handleClick:function(t){var e=this.options,i=t.target||t.srcElement,s=i.parentNode;function n(t){return f(i).hasClass(t)||f(s).hasClass(t)}n(e.toggleClass)?(this.preventDefault(t),this.toggleControls()):n(e.prevClass)?(this.preventDefault(t),this.prev()):n(e.nextClass)?(this.preventDefault(t),this.next()):n(e.closeClass)?(this.preventDefault(t),this.close()):n(e.playPauseClass)?(this.preventDefault(t),this.toggleSlideshow()):s===this.slidesContainer[0]?e.closeOnSlideClick?(this.preventDefault(t),this.close()):e.toggleControlsOnSlideClick&&(this.preventDefault(t),this.toggleControls()):s.parentNode&&s.parentNode===this.slidesContainer[0]&&e.toggleControlsOnSlideClick&&(this.preventDefault(t),this.toggleControls())},onclick:function(t){if(!(this.options.emulateTouchEvents&&this.touchDelta&&(20<Math.abs(this.touchDelta.x)||20<Math.abs(this.touchDelta.y))))return this.handleClick(t);delete this.touchDelta},updateEdgeClasses:function(t){t?this.container.removeClass(this.options.leftEdgeClass):this.container.addClass(this.options.leftEdgeClass),t===this.num-1?this.container.addClass(this.options.rightEdgeClass):this.container.removeClass(this.options.rightEdgeClass)},updateActiveSlide:function(t,e){for(var i,s,n=this.slides,o=this.options,a=[{index:e,method:"addClass",hidden:!1},{index:t,method:"removeClass",hidden:!0}];a.length;)i=a.pop(),f(n[i.index])[i.method](o.slideActiveClass),s=this.circle(i.index-1),(o.continuous||s<i.index)&&f(n[s])[i.method](o.slidePrevClass),s=this.circle(i.index+1),(o.continuous||s>i.index)&&f(n[s])[i.method](o.slideNextClass);this.slides[t].setAttribute("aria-hidden","true"),this.slides[e].removeAttribute("aria-hidden")},handleSlide:function(t,e){this.options.continuous||this.updateEdgeClasses(e),this.updateActiveSlide(t,e),this.loadElements(e),this.options.unloadElements&&this.unloadElements(t,e),this.setTitle(e)},onslide:function(t){this.handleSlide(this.index,t),this.index=t,this.setTimeout(this.options.onslide,[t,this.slides[t]])},setTitle:function(t){var e=this.slides[t].firstChild,i=e.title||e.alt,s=this.titleElement;s.length&&(this.titleElement.empty(),i&&s[0].appendChild(document.createTextNode(i)))},setTimeout:function(t,e,i){var s=this;return t&&window.setTimeout(function(){t.apply(s,e||[])},i||0)},imageFactory:function(t,e){var i,s,n,o,a,l,r,h,d=this.options,c=this,u=t,p=this.imagePrototype.cloneNode(!1);if("string"!=typeof u&&(u=this.getItemProperty(t,d.urlProperty),n=this.support.picture&&this.support.source&&this.getItemProperty(t,d.sourcesProperty),o=this.getItemProperty(t,d.srcsetProperty),a=this.getItemProperty(t,d.sizesProperty),l=this.getItemProperty(t,d.titleProperty),r=this.getItemProperty(t,d.altTextProperty)||l),p.draggable=!1,l&&(p.title=l),r&&(p.alt=r),f(p).on("load error",function m(t){if(!s){if(!(t={type:t.type,target:i||p}).target.parentNode)return c.setTimeout(m,[t]);s=!0,f(p).off("load error",m),e(t)}}),n&&n.length){for(i=this.picturePrototype.cloneNode(!1),h=0;h<n.length;h+=1)i.appendChild(f.extend(this.sourcePrototype.cloneNode(!1),n[h]));i.appendChild(p),f(i).addClass(d.toggleClass)}return o&&(a&&(p.sizes=a),p.srcset=o),p.src=u,i||p},createElement:function(t,e){var i=t&&this.getItemProperty(t,this.options.typeProperty),s=i&&this[i.split("/")[0]+"Factory"]||this.imageFactory,n=t&&s.call(this,t,e);return n||(n=this.elementPrototype.cloneNode(!1),this.setTimeout(e,[{type:"error",target:n}])),f(n).addClass(this.options.slideContentClass),n},iteratePreloadRange:function(t,e){for(var i=this.num,s=this.options,n=Math.min(i,2*s.preloadRange+1),o=t,a=0;a<n;a+=1){if((o+=a*(a%2==0?-1:1))<0||i<=o){if(!s.continuous)continue;o=this.circle(o)}e.call(this,o)}},loadElement:function(t){this.elements[t]||(this.slides[t].firstChild?this.elements[t]=f(this.slides[t]).hasClass(this.options.slideErrorClass)?3:2:(this.elements[t]=1,f(this.slides[t]).addClass(this.options.slideLoadingClass),this.slides[t].appendChild(this.createElement(this.list[t],this.proxyListener))))},loadElements:function(t){this.iteratePreloadRange(t,this.loadElement)},unloadElements:function(t,i){var s=this.options.preloadRange;this.iteratePreloadRange(t,function(t){var e=Math.abs(t-i);s<e&&e+s<this.num&&(this.unloadSlide(t),delete this.elements[t])})},addSlide:function(t){var e=this.slidePrototype.cloneNode(!1);e.setAttribute("data-index",t),e.setAttribute("aria-hidden","true"),this.slidesContainer[0].appendChild(e),this.slides.push(e)},positionSlide:function(t){var e=this.slides[t];e.style.width=this.slideWidth+"px",this.support.transform&&(e.style.left=t*-this.slideWidth+"px",this.move(t,this.index>t?-this.slideWidth:this.index<t?this.slideWidth:0,0))},initSlides:function(t){var e,i;for(t||(this.positions=[],this.positions.length=this.num,this.elements={},this.picturePrototype=this.support.picture&&document.createElement("picture"),this.sourcePrototype=this.support.source&&document.createElement("source"),this.imagePrototype=document.createElement("img"),this.elementPrototype=document.createElement("div"),this.slidePrototype=this.elementPrototype.cloneNode(!1),f(this.slidePrototype).addClass(this.options.slideClass),this.slides=this.slidesContainer[0].children,e=this.options.clearSlides||this.slides.length!==this.num),this.slideWidth=this.container[0].clientWidth,this.slideHeight=this.container[0].clientHeight,this.slidesContainer[0].style.width=this.num*this.slideWidth+"px",e&&this.resetSlides(),i=0;i<this.num;i+=1)e&&this.addSlide(i),this.positionSlide(i);this.options.continuous&&this.support.transform&&(this.move(this.circle(this.index-1),-this.slideWidth,0),this.move(this.circle(this.index+1),this.slideWidth,0)),this.support.transform||(this.slidesContainer[0].style.left=this.index*-this.slideWidth+"px")},unloadSlide:function(t){var e=this.slides[t],i=e.firstChild;null!==i&&e.removeChild(i)},unloadAllSlides:function(){for(var t=0,e=this.slides.length;t<e;t++)this.unloadSlide(t)},toggleControls:function(){var t=this.options.controlsClass;this.container.hasClass(t)?this.container.removeClass(t):this.container.addClass(t)},toggleSlideshow:function(){this.interval?this.pause():this.play()},getNodeIndex:function(t){return parseInt(t.getAttribute("data-index"),10)},getNestedProperty:function(a,t){return t.replace(/\[(?:'([^']+)'|"([^"]+)"|(\d+))\]|(?:(?:^|\.)([^\.\[]+))/g,function(t,e,i,s,n){var o=n||e||i||s&&parseInt(s,10);t&&a&&(a=a[o])}),a},getDataProperty:function(t,e){var i,s;if(t.dataset?(i=e.replace(/-([a-z])/g,function(t,e){return e.toUpperCase()}),s=t.dataset[i]):t.getAttribute&&(s=t.getAttribute("data-"+e.replace(/([A-Z])/g,"-$1").toLowerCase())),"string"==typeof s){if(/^(true|false|null|-?\d+(\.\d+)?|\{[\s\S]*\}|\[[\s\S]*\])$/.test(s))try{return f.parseJSON(s)}catch(n){}return s}},getItemProperty:function(t,e){var i=this.getDataProperty(t,e);return i===undefined&&(i=t[e]),i===undefined&&(i=this.getNestedProperty(t,e)),i},initStartIndex:function(){var t,e=this.options.index,i=this.options.urlProperty;if(e&&"number"!=typeof e)for(t=0;t<this.num;t+=1)if(this.list[t]===e||this.getItemProperty(this.list[t],i)===this.getItemProperty(e,i)){e=t;break}this.index=this.circle(parseInt(e,10)||0)},initEventListeners:function(){var i=this,t=this.slidesContainer;function e(t){var e=i.support.transition&&i.support.transition.end===t.type?"transitionend":t.type;i["on"+e](t)}f(window).on("resize",e),f(window).on("hashchange",e),f(document.body).on("keydown",e),this.container.on("click",e),this.support.touch?t.on("touchstart touchmove touchend touchcancel",e):this.options.emulateTouchEvents&&this.support.transition&&t.on("mousedown mousemove mouseup mouseout",e),this.support.transition&&t.on(this.support.transition.end,e),this.proxyListener=e},destroyEventListeners:function(){var t=this.slidesContainer,e=this.proxyListener;f(window).off("resize",e),f(document.body).off("keydown",e),this.container.off("click",e),this.support.touch?t.off("touchstart touchmove touchend touchcancel",e):this.options.emulateTouchEvents&&this.support.transition&&t.off("mousedown mousemove mouseup mouseout",e),this.support.transition&&t.off(this.support.transition.end,e)},handleOpen:function(){this.options.onopened&&this.options.onopened.call(this)},initWidget:function(){var e=this;return this.container=f(this.options.container),this.container.length?(this.slidesContainer=this.container.find(this.options.slidesContainer).first(),this.slidesContainer.length?(this.titleElement=this.container.find(this.options.titleElement).first(),this.playPauseElement=this.container.find("."+this.options.playPauseClass).first(),1===this.num&&this.container.addClass(this.options.singleClass),this.support.svgasimg&&this.container.addClass(this.options.svgasimgClass),this.support.smil&&this.container.addClass(this.options.smilClass),this.options.onopen&&this.options.onopen.call(this),this.support.transition&&this.options.displayTransition?this.container.on(this.support.transition.end,function i(t){t.target===e.container[0]&&(e.container.off(e.support.transition.end,i),e.handleOpen())}):this.handleOpen(),this.options.hidePageScrollbars&&(this.bodyOverflowStyle=document.body.style.overflow,document.body.style.overflow="hidden"),this.container[0].style.display="block",this.initSlides(),void this.container.addClass(this.options.displayClass)):(this.console.log("blueimp Gallery: Slides container not found.",this.options.slidesContainer),!1)):(this.console.log("blueimp Gallery: Widget container not found.",this.options.container),!1)},initOptions:function(t){this.options=f.extend({},this.options),(t&&t.carousel||this.options.carousel&&(!t||!1!==t.carousel))&&f.extend(this.options,this.carouselOptions),f.extend(this.options,t),this.num<3&&(this.options.continuous=!!this.options.continuous&&null),this.support.transition||(this.options.emulateTouchEvents=!1),this.options.event&&this.preventDefault(this.options.event)}}),i}),function(t){"use strict";"function"==typeof define&&define.amd?define(["./blueimp-helper","./blueimp-gallery"],t):t(window.blueimp.helper||window.jQuery,window.blueimp.Gallery)}(function(t,e){"use strict";var i=e.prototype;t.extend(i.options,{fullscreen:!1});var s=i.initialize,n=i.close;return t.extend(i,{getFullScreenElement:function(){return document.fullscreenElement||document.webkitFullscreenElement||document.mozFullScreenElement||document.msFullscreenElement},requestFullScreen:function(t){t.requestFullscreen?t.requestFullscreen():t.webkitRequestFullscreen?t.webkitRequestFullscreen():t.mozRequestFullScreen?t.mozRequestFullScreen():t.msRequestFullscreen&&t.msRequestFullscreen()},exitFullScreen:function(){document.exitFullscreen?document.exitFullscreen():document.webkitCancelFullScreen?document.webkitCancelFullScreen():document.mozCancelFullScreen?document.mozCancelFullScreen():document.msExitFullscreen&&document.msExitFullscreen()},initialize:function(){s.call(this),this.options.fullscreen&&!this.getFullScreenElement()&&this.requestFullScreen(this.container[0])},close:function(){this.getFullScreenElement()===this.container[0]&&this.exitFullScreen(),n.call(this)}}),e}),function(t){"use strict";"function"==typeof define&&define.amd?define(["./blueimp-helper","./blueimp-gallery"],t):t(window.blueimp.helper||window.jQuery,window.blueimp.Gallery)}(function(a,t){"use strict";var e=t.prototype;a.extend(e.options,{indicatorContainer:"ol",activeIndicatorClass:"active",thumbnailProperty:"thumbnail",thumbnailIndicators:!0});var i=e.initSlides,s=e.addSlide,n=e.resetSlides,o=e.handleClick,l=e.handleSlide,r=e.handleClose;return a.extend(e,{createIndicator:function(t){var e,i,s=this.indicatorPrototype.cloneNode(!1),n=this.getItemProperty(t,this.options.titleProperty),o=this.options.thumbnailProperty;return this.options.thumbnailIndicators&&(o&&(e=this.getItemProperty(t,o)),e===undefined&&(i=t.getElementsByTagName&&a(t).find("img")[0])&&(e=i.src),e&&(s.style.backgroundImage='url("'+e+'")')),n&&(s.title=n),s.setAttribute("role","link"),s},addIndicator:function(t){var e;this.indicatorContainer.length&&((e=this.createIndicator(this.list[t])).setAttribute("data-index",t),this.indicatorContainer[0].appendChild(e),this.indicators.push(e))},setActiveIndicator:function(t){this.indicators&&(this.activeIndicator&&this.activeIndicator.removeClass(this.options.activeIndicatorClass),this.activeIndicator=a(this.indicators[t]),this.activeIndicator.addClass(this.options.activeIndicatorClass))},initSlides:function(t){t||(this.indicatorContainer=this.container.find(this.options.indicatorContainer),this.indicatorContainer.length&&(this.indicatorPrototype=document.createElement("li"),this.indicators=this.indicatorContainer[0].children)),i.call(this,t)},addSlide:function(t){s.call(this,t),this.addIndicator(t)},resetSlides:function(){n.call(this),this.indicatorContainer.empty(),this.indicators=[]},handleClick:function(t){var e=t.target||t.srcElement,i=e.parentNode;if(i===this.indicatorContainer[0])this.preventDefault(t),this.slide(this.getNodeIndex(e));else{if(i.parentNode!==this.indicatorContainer[0])return o.call(this,t);this.preventDefault(t),this.slide(this.getNodeIndex(i))}},handleSlide:function(t,e){l.call(this,t,e),this.setActiveIndicator(e)},handleClose:function(){this.activeIndicator&&this.activeIndicator.removeClass(this.options.activeIndicatorClass),r.call(this)}}),t}),function(t){"use strict";"function"==typeof define&&define.amd?define(["./blueimp-helper","./blueimp-gallery"],t):t(window.blueimp.helper||window.jQuery,window.blueimp.Gallery)}(function(C,t){"use strict";var e=t.prototype;C.extend(e.options,{videoContentClass:"video-content",videoLoadingClass:"video-loading",videoPlayingClass:"video-playing",videoIframeClass:"video-iframe",videoCoverClass:"video-cover",videoPlayClass:"video-play",videoPlaysInline:!0,videoPreloadProperty:"preload",videoPosterProperty:"poster"});var i=e.handleSlide;return C.extend(e,{handleSlide:function(t,e){i.call(this,t,e),this.setTimeout(function(){this.activeVideo&&this.activeVideo.pause()})},videoFactory:function(t,e,i){var s,n,o,a=this,l=this.options,r=this.elementPrototype.cloneNode(!1),h=C(r),d=[{type:"error",target:r}],c=i||document.createElement("video"),u=this.elementPrototype.cloneNode(!1),p=document.createElement("a"),m=this.getItemProperty(t,l.urlProperty),f=this.getItemProperty(t,l.sourcesProperty),y=this.getItemProperty(t,l.titleProperty),g=this.getItemProperty(t,l.videoPosterProperty),v=[p];if(h.addClass(l.videoContentClass),C(p).addClass(l.videoPlayClass),C(u).addClass(l.videoCoverClass).hasClass(l.toggleClass)||v.push(u),u.draggable=!1,y&&(r.title=y,p.setAttribute("aria-label",y)),g&&(u.style.backgroundImage='url("'+g+'")'),c.setAttribute?l.videoPlaysInline&&c.setAttribute("playsinline",""):h.addClass(l.videoIframeClass),c.preload=this.getItemProperty(t,l.videoPreloadProperty)||"none",this.support.source&&f)for(o=0;o<f.length;o+=1)c.appendChild(C.extend(this.sourcePrototype.cloneNode(!1),f[o]));return m&&(c.src=m),p.href=m||f&&f.length&&f[0].src,c.play&&c.pause&&((i||C(c)).on("error",function(){a.setTimeout(e,d)}).on("pause",function(){c.seeking||(n=!1,h.removeClass(a.options.videoLoadingClass).removeClass(a.options.videoPlayingClass),s&&a.container.addClass(a.options.controlsClass),c.controls=!1,c===a.activeVideo&&delete a.activeVideo,a.interval&&a.play())}).on("playing",function(){n=!1,u.removeAttribute("style"),h.removeClass(a.options.videoLoadingClass).addClass(a.options.videoPlayingClass)}).on("play",function(){window.clearTimeout(a.timeout),n=!0,h.addClass(a.options.videoLoadingClass),a.container.hasClass(a.options.controlsClass)?(s=!0,a.container.removeClass(a.options.controlsClass)):s=!1,c.controls=!0,a.activeVideo=c}),C(v).on("click",function(t){a.preventDefault(t),a.activeVideo=c,n?c.pause():c.play()}),r.appendChild(i&&i.element||c)),r.appendChild(u),r.appendChild(p),this.setTimeout(e,[{type:"load",target:r}]),r}}),t}),function(t){"use strict";"function"==typeof define&&define.amd?define(["./blueimp-helper","./blueimp-gallery-video"],t):t(window.blueimp.helper||window.jQuery,window.blueimp.Gallery)}(function(l,t){"use strict";if(!window.postMessage)return t;var e=t.prototype;l.extend(e.options,{vimeoVideoIdProperty:"vimeo",vimeoPlayerUrl:"//player.vimeo.com/video/VIDEO_ID?api=1&player_id=PLAYER_ID",vimeoPlayerIdPrefix:"vimeo-player-",vimeoClickToPlay:!1});var n=e.textFactory||e.imageFactory,o=function(t,e,i,s){this.url=t,this.videoId=e,this.playerId=i,this.clickToPlay=s,this.element=document.createElement("div"),this.listeners={}},a=0;return l.extend(o.prototype,{on:function(t,e){return this.listeners[t]=e,this},loadAPI:function(){var t,e,i=this,s="//f.vimeocdn.com/js/froogaloop2.min.js",n=document.getElementsByTagName("script"),o=n.length;function a(){!e&&i.playOnReady&&i.play(),e=!0}for(;o;)if(n[--o].src===s){t=n[o];break}t||((t=document.createElement("script")).src=s),l(t).on("load",a),n[0].parentNode.insertBefore(t,n[0]),/loaded|complete/.test(t.readyState)&&a()},onReady:function(){var t=this;this.ready=!0,this.player.addEvent("play",function(){t.hasPlayed=!0,t.onPlaying()}),this.player.addEvent("pause",function(){t.onPause()}),this.player.addEvent("finish",function(){t.onPause()}),this.playOnReady&&this.play()},onPlaying:function(){this.playStatus<2&&(this.listeners.playing(),this.playStatus=2)},onPause:function(){this.listeners.pause(),delete this.playStatus},insertIframe:function(){var t=document.createElement("iframe");t.src=this.url.replace("VIDEO_ID",this.videoId).replace("PLAYER_ID",this.playerId),t.id=this.playerId,this.element.parentNode.replaceChild(t,this.element),this.element=t},play:function(){var t=this;this.playStatus||(this.listeners.play(),this.playStatus=1),this.ready?!this.hasPlayed&&(this.clickToPlay||window.navigator&&/iP(hone|od|ad)/.test(window.navigator.platform))?this.onPlaying():this.player.api("play"):(this.playOnReady=!0,window.$f?this.player||(this.insertIframe(),this.player=$f(this.element),this.player.addEvent("ready",function(){t.onReady()})):this.loadAPI())},pause:function(){this.ready?this.player.api("pause"):this.playStatus&&(delete this.playOnReady,this.listeners.pause(),delete this.playStatus)}}),l.extend(e,{VimeoPlayer:o,textFactory:function(t,e){var i=this.options,s=this.getItemProperty(t,i.vimeoVideoIdProperty);return s?(this.getItemProperty(t,i.urlProperty)===undefined&&(t[i.urlProperty]="//vimeo.com/"+s),a+=1,this.videoFactory(t,e,new o(i.vimeoPlayerUrl,s,i.vimeoPlayerIdPrefix+a,i.vimeoClickToPlay))):n.call(this,t,e)}}),t}),function(t){"use strict";"function"==typeof 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\ No newline at end of file
diff --git a/docs/pt/js/blueimp-helper.js b/docs/pt/js/blueimp-helper.js
new file mode 100644
index 0000000..cdc37e5
--- /dev/null
+++ b/docs/pt/js/blueimp-helper.js
@@ -0,0 +1,217 @@
+/*
+ * blueimp helper JS
+ * https://github.com/blueimp/Gallery
+ *
+ * Copyright 2013, Sebastian Tschan
+ * https://blueimp.net
+ *
+ * Licensed under the MIT license:
+ * https://opensource.org/licenses/MIT
+ */
+
+/* global define */
+
+/* eslint-disable no-param-reassign */
+
+;(function () {
+  'use strict'
+
+  /**
+   * Object.assign polyfill
+   *
+   * @param {object} obj1 First object
+   * @param {object} obj2 Second object
+   * @returns {object} Merged object
+   */
+  function extend(obj1, obj2) {
+    var prop
+    for (prop in obj2) {
+      if (Object.prototype.hasOwnProperty.call(obj2, prop)) {
+        obj1[prop] = obj2[prop]
+      }
+    }
+    return obj1
+  }
+  /**
+   * Helper constructor
+   *
+   * @class
+   * @param {*} query jQuery type query argument
+   */
+  function Helper(query) {
+    if (!this || this.find !== Helper.prototype.find) {
+      // Called as function instead of as constructor,
+      // so we simply return a new instance:
+      return new Helper(query)
+    }
+    this.length = 0
+    if (query) {
+      if (typeof query === 'string') {
+        query = this.find(query)
+      }
+      if (query.nodeType || query === query.window) {
+        // Single HTML element
+        this.length = 1
+        this[0] = query
+      } else {
+        // HTML element collection
+        var i = query.length
+        this.length = i
+        while (i) {
+          i -= 1
+          this[i] = query[i]
+        }
+      }
+    }
+  }
+
+  Helper.extend = extend
+
+  Helper.contains = function (container, element) {
+    do {
+      element = element.parentNode
+      if (element === container) {
+        return true
+      }
+    } while (element)
+    return false
+  }
+
+  Helper.parseJSON = function (string) {
+    return JSON.parse(string)
+  }
+
+  extend(Helper.prototype, {
+    find: function (query) {
+      var container = this[0] || document
+      if (typeof query === 'string') {
+        if (container.querySelectorAll) {
+          query = container.querySelectorAll(query)
+        } else if (query.charAt(0) === '#') {
+          query = container.getElementById(query.slice(1))
+        } else {
+          query = container.getElementsByTagName(query)
+        }
+      }
+      return new Helper(query)
+    },
+
+    hasClass: function (className) {
+      if (!this[0]) return false
+      return new RegExp('(?:^|\\s+)' + className + '(?:\\s+|$)').test(
+        this[0].className
+      )
+    },
+
+    addClass: function (className) {
+      var i = this.length
+      var classNames
+      var element
+      var j
+      while (i) {
+        i -= 1
+        element = this[i]
+        if (!element.className) {
+          element.className = className
+          continue
+        }
+        if (!classNames) classNames = className.split(/\s+/)
+        for (j = 0; j < classNames.length; j += 1) {
+          if (this.hasClass(classNames[j])) {
+            continue
+          }
+          element.className += ' ' + classNames[j]
+        }
+      }
+      return this
+    },
+
+    removeClass: function (className) {
+      // Match any of the given class names
+      var regexp = new RegExp('^(?:' + className.split(/\s+/).join('|') + ')$')
+      // Match any class names and their trailing whitespace
+      var matcher = /(\S+)(?:\s+|$)/g
+      var replacer = function (match, className) {
+        // Replace class names that match the given ones
+        return regexp.test(className) ? '' : match
+      }
+      var trimEnd = /\s+$/
+      var i = this.length
+      var element
+      while (i) {
+        i -= 1
+        element = this[i]
+        element.className = element.className
+          .replace(matcher, replacer)
+          .replace(trimEnd, '')
+      }
+      return this
+    },
+
+    on: function (eventName, handler) {
+      var eventNames = eventName.split(/\s+/)
+      var i
+      var element
+      while (eventNames.length) {
+        eventName = eventNames.shift()
+        i = this.length
+        while (i) {
+          i -= 1
+          element = this[i]
+          if (element.addEventListener) {
+            element.addEventListener(eventName, handler, false)
+          } else if (element.attachEvent) {
+            element.attachEvent('on' + eventName, handler)
+          }
+        }
+      }
+      return this
+    },
+
+    off: function (eventName, handler) {
+      var eventNames = eventName.split(/\s+/)
+      var i
+      var element
+      while (eventNames.length) {
+        eventName = eventNames.shift()
+        i = this.length
+        while (i) {
+          i -= 1
+          element = this[i]
+          if (element.removeEventListener) {
+            element.removeEventListener(eventName, handler, false)
+          } else if (element.detachEvent) {
+            element.detachEvent('on' + eventName, handler)
+          }
+        }
+      }
+      return this
+    },
+
+    empty: function () {
+      var i = this.length
+      var element
+      while (i) {
+        i -= 1
+        element = this[i]
+        while (element.hasChildNodes()) {
+          element.removeChild(element.lastChild)
+        }
+      }
+      return this
+    },
+
+    first: function () {
+      return new Helper(this[0])
+    }
+  })
+
+  if (typeof define === 'function' && define.amd) {
+    define(function () {
+      return Helper
+    })
+  } else {
+    window.blueimp = window.blueimp || {}
+    window.blueimp.helper = Helper
+  }
+})()
diff --git a/docs/pt/js/initGallery.js b/docs/pt/js/initGallery.js
new file mode 100644
index 0000000..735bfcf
--- /dev/null
+++ b/docs/pt/js/initGallery.js
@@ -0,0 +1,23 @@
+/*
+ * blueimp Gallery Demo JS
+ * https://github.com/blueimp/Gallery
+ *
+ * Copyright 2013, Sebastian Tschan
+ * https://blueimp.net
+ *
+ * Licensed under the MIT license:
+ * https://opensource.org/licenses/MIT
+ */
+
+/* global blueimp, $ */
+
+$(function () {
+  'use strict'
+  // Initialize the Gallery as video carousel:
+  // eslint-disable-next-line new-cap
+  blueimp.Gallery(document.getElementById('links').getElementsByTagName('a'), {
+    //container: '#__docusaurus div div #blueimp-gallery-carousel',
+    container: document.getElementById('blueimp-gallery-carousel'),
+    carousel: true
+  })
+})
diff --git a/docs/pt/js/jquery.blueimp-gallery.js b/docs/pt/js/jquery.blueimp-gallery.js
new file mode 100644
index 0000000..92ff4f6
--- /dev/null
+++ b/docs/pt/js/jquery.blueimp-gallery.js
@@ -0,0 +1,75 @@
+/*
+ * blueimp Gallery jQuery plugin
+ * https://github.com/blueimp/Gallery
+ *
+ * Copyright 2013, Sebastian Tschan
+ * https://blueimp.net
+ *
+ * Licensed under the MIT license:
+ * https://opensource.org/licenses/MIT
+ */
+
+/* global define */
+
+;(function (factory) {
+  'use strict'
+  if (typeof define === 'function' && define.amd) {
+    define(['jquery', './blueimp-gallery'], factory)
+  } else {
+    factory(window.jQuery, window.blueimp.Gallery)
+  }
+})(function ($, Gallery) {
+  'use strict'
+
+  // Global click handler to open links with data-gallery attribute
+  // in the Gallery lightbox:
+  $(document).on('click', '[data-gallery]', function (event) {
+    // Get the container id from the data-gallery attribute:
+    var id = $(this).data('gallery')
+    var widget = $(id)
+    var container =
+      (widget.length && widget) || $(Gallery.prototype.options.container)
+    var callbacks = {
+      onopen: function () {
+        container.data('gallery', this).trigger('open')
+      },
+      onopened: function () {
+        container.trigger('opened')
+      },
+      onslide: function () {
+        container.trigger('slide', arguments)
+      },
+      onslideend: function () {
+        container.trigger('slideend', arguments)
+      },
+      onslidecomplete: function () {
+        container.trigger('slidecomplete', arguments)
+      },
+      onclose: function () {
+        container.trigger('close')
+      },
+      onclosed: function () {
+        container.trigger('closed').removeData('gallery')
+      }
+    }
+    var options = $.extend(
+      // Retrieve custom options from data-attributes
+      // on the Gallery widget:
+      container.data(),
+      {
+        container: container[0],
+        index: this,
+        event: event
+      },
+      callbacks
+    )
+    // Select all links with the same data-gallery attribute:
+    var links = $(this)
+      .closest('[data-gallery-group], body')
+      .find('[data-gallery="' + id + '"]')
+    if (options.filter) {
+      links = links.filter(options.filter)
+    }
+    return new Gallery(links, options)
+  })
+})
diff --git a/docs/pt/js/jquery.blueimp-gallery.min.js b/docs/pt/js/jquery.blueimp-gallery.min.js
new file mode 100644
index 0000000..6031332
--- /dev/null
+++ b/docs/pt/js/jquery.blueimp-gallery.min.js
@@ -0,0 +1,2 @@
+!function(t){"use strict";"function"==typeof define&&define.amd?define(["./blueimp-helper"],t):(window.blueimp=window.blueimp||{},window.blueimp.Gallery=t(window.blueimp.helper||window.jQuery))}(function(y){"use strict";function i(t,e){return document.body.style.maxHeight===undefined?null:this&&this.options===i.prototype.options?void(t&&t.length?(this.list=t,this.num=t.length,this.initOptions(e),this.initialize()):this.console.log("blueimp Gallery: No or empty list provided as first argument.",t)):new i(t,e)}return y.extend(i.prototype,{options:{container:"#blueimp-gallery",slidesContainer:"div",titleElement:"h3",displayClass:"blueimp-gallery-display",controlsClass:"blueimp-gallery-controls",singleClass:"blueimp-gallery-single",leftEdgeClass:"blueimp-gallery-left",rightEdgeClass:"blueimp-gallery-right",playingClass:"blueimp-gallery-playing",svgasimgClass:"blueimp-gallery-svgasimg",smilClass:"blueimp-gallery-smil",slideClass:"slide",slideActiveClass:"slide-active",slidePrevClass:"slide-prev",slideNextClass:"slide-next",slideLoadingClass:"slide-loading",slideErrorClass:"slide-error",slideContentClass:"slide-content",toggleClass:"toggle",prevClass:"prev",nextClass:"next",closeClass:"close",playPauseClass:"play-pause",typeProperty:"type",titleProperty:"title",altTextProperty:"alt",urlProperty:"href",srcsetProperty:"srcset",sizesProperty:"sizes",sourcesProperty:"sources",displayTransition:!0,clearSlides:!0,toggleControlsOnEnter:!0,toggleControlsOnSlideClick:!0,toggleSlideshowOnSpace:!0,enableKeyboardNavigation:!0,closeOnEscape:!0,closeOnSlideClick:!0,closeOnSwipeUpOrDown:!0,closeOnHashChange:!0,emulateTouchEvents:!0,stopTouchEventsPropagation:!1,hidePageScrollbars:!0,disableScroll:!0,carousel:!1,continuous:!0,unloadElements:!0,startSlideshow:!1,slideshowInterval:5e3,slideshowDirection:"ltr",index:0,preloadRange:2,transitionDuration:300,slideshowTransitionDuration:500,event:undefined,onopen:undefined,onopened:undefined,onslide:undefined,onslideend:undefined,onslidecomplete:undefined,onclose:undefined,onclosed:undefined},carouselOptions:{hidePageScrollbars:!1,toggleControlsOnEnter:!1,toggleSlideshowOnSpace:!1,enableKeyboardNavigation:!1,closeOnEscape:!1,closeOnSlideClick:!1,closeOnSwipeUpOrDown:!1,closeOnHashChange:!1,disableScroll:!1,startSlideshow:!0},console:window.console&&"function"==typeof window.console.log?window.console:{log:function(){}},support:function(s){var t,n={source:!!window.HTMLSourceElement,picture:!!window.HTMLPictureElement,svgasimg:document.implementation.hasFeature("http://www.w3.org/TR/SVG11/feature#Image","1.1"),smil:!!document.createElementNS&&/SVGAnimate/.test(document.createElementNS("http://www.w3.org/2000/svg","animate").toString()),touch:window.ontouchstart!==undefined||window.DocumentTouch&&document instanceof DocumentTouch},e={webkitTransition:{end:"webkitTransitionEnd",prefix:"-webkit-"},MozTransition:{end:"transitionend",prefix:"-moz-"},OTransition:{end:"otransitionend",prefix:"-o-"},transition:{end:"transitionend",prefix:""}};for(t in e)if(Object.prototype.hasOwnProperty.call(e,t)&&s.style[t]!==undefined){n.transition=e[t],n.transition.name=t;break}function i(){var t,e,i=n.transition;document.body.appendChild(s),i&&(t=i.name.slice(0,-9)+"ransform",s.style[t]!==undefined&&(s.style[t]="translateZ(0)",e=window.getComputedStyle(s).getPropertyValue(i.prefix+"transform"),n.transform={prefix:i.prefix,name:t,translate:!0,translateZ:!!e&&"none"!==e})),document.body.removeChild(s)}return document.body?i():y(document).on("DOMContentLoaded",i),n}(document.createElement("div")),requestAnimationFrame:window.requestAnimationFrame||window.webkitRequestAnimationFrame||window.mozRequestAnimationFrame,cancelAnimationFrame:window.cancelAnimationFrame||window.webkitCancelRequestAnimationFrame||window.webkitCancelAnimationFrame||window.mozCancelAnimationFrame,initialize:function(){if(this.initStartIndex(),!1===this.initWidget())return!1;this.initEventListeners(),this.onslide(this.index),this.ontransitionend(),this.options.startSlideshow&&this.play()},slide:function(t,e){window.clearTimeout(this.timeout);var i,s,n,o=this.index;if(o!==t&&1!==this.num){if(e=e||this.options.transitionDuration,this.support.transform){for(this.options.continuous||(t=this.circle(t)),i=Math.abs(o-t)/(o-t),this.options.continuous&&(s=i,(i=-this.positions[this.circle(t)]/this.slideWidth)!==s&&(t=-i*this.num+t)),n=Math.abs(o-t)-1;n;)--n,this.move(this.circle((o<t?t:o)-n-1),this.slideWidth*i,0);t=this.circle(t),this.move(o,this.slideWidth*i,e),this.move(t,0,e),this.options.continuous&&this.move(this.circle(t-i),-this.slideWidth*i,0)}else t=this.circle(t),this.animate(o*-this.slideWidth,t*-this.slideWidth,e);this.onslide(t)}},getIndex:function(){return this.index},getNumber:function(){return this.num},prev:function(){(this.options.continuous||this.index)&&this.slide(this.index-1)},next:function(){(this.options.continuous||this.index<this.num-1)&&this.slide(this.index+1)},play:function(t){var i=this,e=this.index+("rtl"===this.options.slideshowDirection?-1:1);window.clearTimeout(this.timeout),this.interval=t||this.options.slideshowInterval,1<this.elements[this.index]&&(this.timeout=this.setTimeout(!this.requestAnimationFrame&&this.slide||function(t,e){i.animationFrameId=i.requestAnimationFrame.call(window,function(){i.slide(t,e)})},[e,this.options.slideshowTransitionDuration],this.interval)),this.container.addClass(this.options.playingClass),this.slidesContainer[0].setAttribute("aria-live","off"),this.playPauseElement.length&&this.playPauseElement[0].setAttribute("aria-pressed","true")},pause:function(){window.clearTimeout(this.timeout),this.interval=null,this.cancelAnimationFrame&&(this.cancelAnimationFrame.call(window,this.animationFrameId),this.animationFrameId=null),this.container.removeClass(this.options.playingClass),this.slidesContainer[0].setAttribute("aria-live","polite"),this.playPauseElement.length&&this.playPauseElement[0].setAttribute("aria-pressed","false")},add:function(t){var e;for(t.concat||(t=Array.prototype.slice.call(t)),this.list.concat||(this.list=Array.prototype.slice.call(this.list)),this.list=this.list.concat(t),this.num=this.list.length,2<this.num&&null===this.options.continuous&&(this.options.continuous=!0,this.container.removeClass(this.options.leftEdgeClass)),this.container.removeClass(this.options.rightEdgeClass).removeClass(this.options.singleClass),e=this.num-t.length;e<this.num;e+=1)this.addSlide(e),this.positionSlide(e);this.positions.length=this.num,this.initSlides(!0)},resetSlides:function(){this.slidesContainer.empty(),this.unloadAllSlides(),this.slides=[]},handleClose:function(){var t=this.options;this.destroyEventListeners(),this.pause(),this.container[0].style.display="none",this.container.removeClass(t.displayClass).removeClass(t.singleClass).removeClass(t.leftEdgeClass).removeClass(t.rightEdgeClass),t.hidePageScrollbars&&(document.body.style.overflow=this.bodyOverflowStyle),this.options.clearSlides&&this.resetSlides(),this.options.onclosed&&this.options.onclosed.call(this)},close:function(){var e=this;this.options.onclose&&this.options.onclose.call(this),this.support.transition&&this.options.displayTransition?(this.container.on(this.support.transition.end,function i(t){t.target===e.container[0]&&(e.container.off(e.support.transition.end,i),e.handleClose())}),this.container.removeClass(this.options.displayClass)):this.handleClose()},circle:function(t){return(this.num+t%this.num)%this.num},move:function(t,e,i){this.translateX(t,e,i),this.positions[t]=e},translate:function(t,e,i,s){var n,o,a;this.slides[t]&&(n=this.slides[t].style,o=this.support.transition,a=this.support.transform,n[o.name+"Duration"]=s+"ms",n[a.name]="translate("+e+"px, "+i+"px)"+(a.translateZ?" translateZ(0)":""))},translateX:function(t,e,i){this.translate(t,e,0,i)},translateY:function(t,e,i){this.translate(t,0,e,i)},animate:function(e,i,s){var n,o,a;s?(n=this,o=(new Date).getTime(),a=window.setInterval(function(){var t=(new Date).getTime()-o;if(s<t)return n.slidesContainer[0].style.left=i+"px",n.ontransitionend(),void 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e,i;this.touchStart&&(e=t.target,(i=t.relatedTarget)&&(i===e||y.contains(e,i))||this.onmouseup(t))},ontouchstart:function(t){this.options.stopTouchEventsPropagation&&this.stopPropagation(t);var e=(t.originalEvent||t).touches[0];this.touchStart={x:e.pageX,y:e.pageY,time:Date.now()},this.isScrolling=undefined,this.touchDelta={}},ontouchmove:function(t){this.options.stopTouchEventsPropagation&&this.stopPropagation(t);var e,i,s=(t.originalEvent||t).touches,n=s[0],o=(t.originalEvent||t).scale,a=this.index;if(!(1<s.length||o&&1!==o))if(this.options.disableScroll&&t.preventDefault(),this.touchDelta={x:n.pageX-this.touchStart.x,y:n.pageY-this.touchStart.y},e=this.touchDelta.x,this.isScrolling===undefined&&(this.isScrolling=this.isScrolling||Math.abs(e)<Math.abs(this.touchDelta.y)),this.isScrolling)this.options.carousel||this.translateY(a,this.touchDelta.y+this.positions[a],0);else 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32:this.options.toggleSlideshowOnSpace&&(this.preventDefault(t),this.toggleSlideshow());break;case 37:this.options.enableKeyboardNavigation&&(this.preventDefault(t),this.prev());break;case 39:this.options.enableKeyboardNavigation&&(this.preventDefault(t),this.next())}},handleClick:function(t){var e=this.options,i=t.target||t.srcElement,s=i.parentNode;function n(t){return y(i).hasClass(t)||y(s).hasClass(t)}n(e.toggleClass)?(this.preventDefault(t),this.toggleControls()):n(e.prevClass)?(this.preventDefault(t),this.prev()):n(e.nextClass)?(this.preventDefault(t),this.next()):n(e.closeClass)?(this.preventDefault(t),this.close()):n(e.playPauseClass)?(this.preventDefault(t),this.toggleSlideshow()):s===this.slidesContainer[0]?e.closeOnSlideClick?(this.preventDefault(t),this.close()):e.toggleControlsOnSlideClick&&(this.preventDefault(t),this.toggleControls()):s.parentNode&&s.parentNode===this.slidesContainer[0]&&e.toggleControlsOnSlideClick&&(this.preventDefault(t),this.toggleControls())},onclick:function(t){if(!(this.options.emulateTouchEvents&&this.touchDelta&&(20<Math.abs(this.touchDelta.x)||20<Math.abs(this.touchDelta.y))))return this.handleClick(t);delete 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i=t&&this.getItemProperty(t,this.options.typeProperty),s=i&&this[i.split("/")[0]+"Factory"]||this.imageFactory,n=t&&s.call(this,t,e);return n||(n=this.elementPrototype.cloneNode(!1),this.setTimeout(e,[{type:"error",target:n}])),y(n).addClass(this.options.slideContentClass),n},iteratePreloadRange:function(t,e){for(var i=this.num,s=this.options,n=Math.min(i,2*s.preloadRange+1),o=t,a=0;a<n;a+=1){if((o+=a*(a%2==0?-1:1))<0||i<=o){if(!s.continuous)continue;o=this.circle(o)}e.call(this,o)}},loadElement:function(t){this.elements[t]||(this.slides[t].firstChild?this.elements[t]=y(this.slides[t]).hasClass(this.options.slideErrorClass)?3:2:(this.elements[t]=1,y(this.slides[t]).addClass(this.options.slideLoadingClass),this.slides[t].appendChild(this.createElement(this.list[t],this.proxyListener))))},loadElements:function(t){this.iteratePreloadRange(t,this.loadElement)},unloadElements:function(t,i){var s=this.options.preloadRange;this.iteratePreloadRange(t,function(t){var e=Math.abs(t-i);s<e&&e+s<this.num&&(this.unloadSlide(t),delete this.elements[t])})},addSlide:function(t){var e=this.slidePrototype.cloneNode(!1);e.setAttribute("data-index",t),e.setAttribute("aria-hidden","true"),this.slidesContainer[0].appendChild(e),this.slides.push(e)},positionSlide:function(t){var e=this.slides[t];e.style.width=this.slideWidth+"px",this.support.transform&&(e.style.left=t*-this.slideWidth+"px",this.move(t,this.index>t?-this.slideWidth:this.index<t?this.slideWidth:0,0))},initSlides:function(t){var e,i;for(t||(this.positions=[],this.positions.length=this.num,this.elements={},this.picturePrototype=this.support.picture&&document.createElement("picture"),this.sourcePrototype=this.support.source&&document.createElement("source"),this.imagePrototype=document.createElement("img"),this.elementPrototype=document.createElement("div"),this.slidePrototype=this.elementPrototype.cloneNode(!1),y(this.slidePrototype).addClass(this.options.slideClass),this.slides=this.slidesContainer[0].children,e=this.options.clearSlides||this.slides.length!==this.num),this.slideWidth=this.container[0].clientWidth,this.slideHeight=this.container[0].clientHeight,this.slidesContainer[0].style.width=this.num*this.slideWidth+"px",e&&this.resetSlides(),i=0;i<this.num;i+=1)e&&this.addSlide(i),this.positionSlide(i);this.options.continuous&&this.support.transform&&(this.move(this.circle(this.index-1),-this.slideWidth,0),this.move(this.circle(this.index+1),this.slideWidth,0)),this.support.transform||(this.slidesContainer[0].style.left=this.index*-this.slideWidth+"px")},unloadSlide:function(t){var e=this.slides[t],i=e.firstChild;null!==i&&e.removeChild(i)},unloadAllSlides:function(){for(var t=0,e=this.slides.length;t<e;t++)this.unloadSlide(t)},toggleControls:function(){var t=this.options.controlsClass;this.container.hasClass(t)?this.container.removeClass(t):this.container.addClass(t)},toggleSlideshow:function(){this.interval?this.pause():this.play()},getNodeIndex:function(t){return parseInt(t.getAttribute("data-index"),10)},getNestedProperty:function(a,t){return t.replace(/\[(?:'([^']+)'|"([^"]+)"|(\d+))\]|(?:(?:^|\.)([^\.\[]+))/g,function(t,e,i,s,n){var o=n||e||i||s&&parseInt(s,10);t&&a&&(a=a[o])}),a},getDataProperty:function(t,e){var i,s;if(t.dataset?(i=e.replace(/-([a-z])/g,function(t,e){return e.toUpperCase()}),s=t.dataset[i]):t.getAttribute&&(s=t.getAttribute("data-"+e.replace(/([A-Z])/g,"-$1").toLowerCase())),"string"==typeof s){if(/^(true|false|null|-?\d+(\.\d+)?|\{[\s\S]*\}|\[[\s\S]*\])$/.test(s))try{return y.parseJSON(s)}catch(n){}return 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this.container=y(this.options.container),this.container.length?(this.slidesContainer=this.container.find(this.options.slidesContainer).first(),this.slidesContainer.length?(this.titleElement=this.container.find(this.options.titleElement).first(),this.playPauseElement=this.container.find("."+this.options.playPauseClass).first(),1===this.num&&this.container.addClass(this.options.singleClass),this.support.svgasimg&&this.container.addClass(this.options.svgasimgClass),this.support.smil&&this.container.addClass(this.options.smilClass),this.options.onopen&&this.options.onopen.call(this),this.support.transition&&this.options.displayTransition?this.container.on(this.support.transition.end,function i(t){t.target===e.container[0]&&(e.container.off(e.support.transition.end,i),e.handleOpen())}):this.handleOpen(),this.options.hidePageScrollbars&&(this.bodyOverflowStyle=document.body.style.overflow,document.body.style.overflow="hidden"),this.container[0].style.display="block",this.initSlides(),void 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define&&define.amd?define(["./blueimp-helper","./blueimp-gallery"],t):t(window.blueimp.helper||window.jQuery,window.blueimp.Gallery)}(function(a,t){"use strict";var e=t.prototype;a.extend(e.options,{indicatorContainer:"ol",activeIndicatorClass:"active",thumbnailProperty:"thumbnail",thumbnailIndicators:!0});var i=e.initSlides,s=e.addSlide,n=e.resetSlides,o=e.handleClick,l=e.handleSlide,r=e.handleClose;return a.extend(e,{createIndicator:function(t){var e,i,s=this.indicatorPrototype.cloneNode(!1),n=this.getItemProperty(t,this.options.titleProperty),o=this.options.thumbnailProperty;return this.options.thumbnailIndicators&&(o&&(e=this.getItemProperty(t,o)),e===undefined&&(i=t.getElementsByTagName&&a(t).find("img")[0])&&(e=i.src),e&&(s.style.backgroundImage='url("'+e+'")')),n&&(s.title=n),s.setAttribute("role","link"),s},addIndicator:function(t){var e;this.indicatorContainer.length&&((e=this.createIndicator(this.list[t])).setAttribute("data-index",t),this.indicatorContainer[0].appendChild(e),this.indicators.push(e))},setActiveIndicator:function(t){this.indicators&&(this.activeIndicator&&this.activeIndicator.removeClass(this.options.activeIndicatorClass),this.activeIndicator=a(this.indicators[t]),this.activeIndicator.addClass(this.options.activeIndicatorClass))},initSlides:function(t){t||(this.indicatorContainer=this.container.find(this.options.indicatorContainer),this.indicatorContainer.length&&(this.indicatorPrototype=document.createElement("li"),this.indicators=this.indicatorContainer[0].children)),i.call(this,t)},addSlide:function(t){s.call(this,t),this.addIndicator(t)},resetSlides:function(){n.call(this),this.indicatorContainer.empty(),this.indicators=[]},handleClick:function(t){var e=t.target||t.srcElement,i=e.parentNode;if(i===this.indicatorContainer[0])this.preventDefault(t),this.slide(this.getNodeIndex(e));else{if(i.parentNode!==this.indicatorContainer[0])return o.call(this,t);this.preventDefault(t),this.slide(this.getNodeIndex(i))}},handleSlide:function(t,e){l.call(this,t,e),this.setActiveIndicator(e)},handleClose:function(){this.activeIndicator&&this.activeIndicator.removeClass(this.options.activeIndicatorClass),r.call(this)}}),t}),function(t){"use strict";"function"==typeof define&&define.amd?define(["./blueimp-helper","./blueimp-gallery"],t):t(window.blueimp.helper||window.jQuery,window.blueimp.Gallery)}(function(C,t){"use strict";var e=t.prototype;C.extend(e.options,{videoContentClass:"video-content",videoLoadingClass:"video-loading",videoPlayingClass:"video-playing",videoIframeClass:"video-iframe",videoCoverClass:"video-cover",videoPlayClass:"video-play",videoPlaysInline:!0,videoPreloadProperty:"preload",videoPosterProperty:"poster"});var i=e.handleSlide;return C.extend(e,{handleSlide:function(t,e){i.call(this,t,e),this.setTimeout(function(){this.activeVideo&&this.activeVideo.pause()})},videoFactory:function(t,e,i){var s,n,o,a=this,l=this.options,r=this.elementPrototype.cloneNode(!1),h=C(r),d=[{type:"error",target:r}],c=i||document.createElement("video"),u=this.elementPrototype.cloneNode(!1),p=document.createElement("a"),m=this.getItemProperty(t,l.urlProperty),y=this.getItemProperty(t,l.sourcesProperty),f=this.getItemProperty(t,l.titleProperty),g=this.getItemProperty(t,l.videoPosterProperty),v=[p];if(h.addClass(l.videoContentClass),C(p).addClass(l.videoPlayClass),C(u).addClass(l.videoCoverClass).hasClass(l.toggleClass)||v.push(u),u.draggable=!1,f&&(r.title=f,p.setAttribute("aria-label",f)),g&&(u.style.backgroundImage='url("'+g+'")'),c.setAttribute?l.videoPlaysInline&&c.setAttribute("playsinline",""):h.addClass(l.videoIframeClass),c.preload=this.getItemProperty(t,l.videoPreloadProperty)||"none",this.support.source&&y)for(o=0;o<y.length;o+=1)c.appendChild(C.extend(this.sourcePrototype.cloneNode(!1),y[o]));return m&&(c.src=m),p.href=m||y&&y.length&&y[0].src,c.play&&c.pause&&((i||C(c)).on("error",function(){a.setTimeout(e,d)}).on("pause",function(){c.seeking||(n=!1,h.removeClass(a.options.videoLoadingClass).removeClass(a.options.videoPlayingClass),s&&a.container.addClass(a.options.controlsClass),c.controls=!1,c===a.activeVideo&&delete a.activeVideo,a.interval&&a.play())}).on("playing",function(){n=!1,u.removeAttribute("style"),h.removeClass(a.options.videoLoadingClass).addClass(a.options.videoPlayingClass)}).on("play",function(){window.clearTimeout(a.timeout),n=!0,h.addClass(a.options.videoLoadingClass),a.container.hasClass(a.options.controlsClass)?(s=!0,a.container.removeClass(a.options.controlsClass)):s=!1,c.controls=!0,a.activeVideo=c}),C(v).on("click",function(t){a.preventDefault(t),a.activeVideo=c,n?c.pause():c.play()}),r.appendChild(i&&i.element||c)),r.appendChild(u),r.appendChild(p),this.setTimeout(e,[{type:"load",target:r}]),r}}),t}),function(t){"use strict";"function"==typeof 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t=this;this.playStatus||(this.listeners.play(),this.playStatus=1),this.ready?!this.hasPlayed&&(this.clickToPlay||window.navigator&&/iP(hone|od|ad)/.test(window.navigator.platform))?this.onPlaying():this.player.api("play"):(this.playOnReady=!0,window.$f?this.player||(this.insertIframe(),this.player=$f(this.element),this.player.addEvent("ready",function(){t.onReady()})):this.loadAPI())},pause:function(){this.ready?this.player.api("pause"):this.playStatus&&(delete this.playOnReady,this.listeners.pause(),delete this.playStatus)}}),l.extend(e,{VimeoPlayer:o,textFactory:function(t,e){var i=this.options,s=this.getItemProperty(t,i.vimeoVideoIdProperty);return s?(this.getItemProperty(t,i.urlProperty)===undefined&&(t[i.urlProperty]="//vimeo.com/"+s),a+=1,this.videoFactory(t,e,new o(i.vimeoPlayerUrl,s,i.vimeoPlayerIdPrefix+a,i.vimeoClickToPlay))):n.call(this,t,e)}}),t}),function(t){"use strict";"function"==typeof define&&define.amd?define(["./blueimp-helper","./blueimp-gallery-video"],t):t(window.blueimp.helper||window.jQuery,window.blueimp.Gallery)}(function(t,e){"use strict";if(!window.postMessage)return e;var i=e.prototype;t.extend(i.options,{youTubeVideoIdProperty:"youtube",youTubePlayerVars:{wmode:"transparent"},youTubeClickToPlay:!1});var n=i.textFactory||i.imageFactory,o=function(t,e,i){this.videoId=t,this.playerVars=e,this.clickToPlay=i,this.element=document.createElement("div"),this.listeners={}};return t.extend(o.prototype,{on:function(t,e){return this.listeners[t]=e,this},loadAPI:function(){var t,e=this,i=window.onYouTubeIframeAPIReady,s="//www.youtube.com/iframe_api",n=document.getElementsByTagName("script"),o=n.length;for(window.onYouTubeIframeAPIReady=function(){i&&i.apply(this),e.playOnReady&&e.play()};o;)if(n[--o].src===s)return;(t=document.createElement("script")).src=s,n[0].parentNode.insertBefore(t,n[0])},onReady:function(){this.ready=!0,this.playOnReady&&this.play()},onPlaying:function(){this.playStatus<2&&(this.listeners.playing(),this.playStatus=2)},onPause:function(){this.listeners.pause(),delete this.playStatus},onStateChange:function(t){switch(window.clearTimeout(this.pauseTimeout),t.data){case YT.PlayerState.PLAYING:this.hasPlayed=!0,this.onPlaying();break;case YT.PlayerState.UNSTARTED:case YT.PlayerState.PAUSED:this.pauseTimeout=i.setTimeout.call(this,this.onPause,null,500);break;case YT.PlayerState.ENDED:this.onPause()}},onError:function(t){this.listeners.error(t)},play:function(){var e=this;this.playStatus||(this.listeners.play(),this.playStatus=1),this.ready?!this.hasPlayed&&(this.clickToPlay||window.navigator&&/iP(hone|od|ad)/.test(window.navigator.platform))?this.onPlaying():this.player.playVideo():(this.playOnReady=!0,window.YT&&YT.Player?this.player||(this.player=new YT.Player(this.element,{videoId:this.videoId,playerVars:this.playerVars,events:{onReady:function(){e.onReady()},onStateChange:function(t){e.onStateChange(t)},onError:function(t){e.onError(t)}}})):this.loadAPI())},pause:function(){this.ready?this.player.pauseVideo():this.playStatus&&(delete this.playOnReady,this.listeners.pause(),delete this.playStatus)}}),t.extend(i,{YouTubePlayer:o,textFactory:function(t,e){var i=this.options,s=this.getItemProperty(t,i.youTubeVideoIdProperty);return s?(this.getItemProperty(t,i.urlProperty)===undefined&&(t[i.urlProperty]="//www.youtube.com/watch?v="+s),this.getItemProperty(t,i.videoPosterProperty)===undefined&&(t[i.videoPosterProperty]="//img.youtube.com/vi/"+s+"/maxresdefault.jpg"),this.videoFactory(t,e,new o(s,i.youTubePlayerVars,i.youTubeClickToPlay))):n.call(this,t,e)}}),e}),function(t){"use strict";"function"==typeof define&&define.amd?define(["jquery","./blueimp-gallery"],t):t(window.jQuery,window.blueimp.Gallery)}(function(l,r){"use strict";l(document).on("click","[data-gallery]",function(t){var e=l(this).data("gallery"),i=l(e),s=i.length&&i||l(r.prototype.options.container),n={onopen:function(){s.data("gallery",this).trigger("open")},onopened:function(){s.trigger("opened")},onslide:function(){s.trigger("slide",arguments)},onslideend:function(){s.trigger("slideend",arguments)},onslidecomplete:function(){s.trigger("slidecomplete",arguments)},onclose:function(){s.trigger("close")},onclosed:function(){s.trigger("closed").removeData("gallery")}},o=l.extend(s.data(),{container:s[0],index:this,event:t},n),a=l(this).closest("[data-gallery-group], body").find('[data-gallery="'+e+'"]');return o.filter&&(a=a.filter(o.filter)),new r(a,o)})});
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diff --git a/docs/pt/js/jquery.blueimp-gallery.min.js.map b/docs/pt/js/jquery.blueimp-gallery.min.js.map
new file mode 100644
index 0000000..aa42508
--- /dev/null
+++ b/docs/pt/js/jquery.blueimp-gallery.min.js.map
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\ No newline at end of file
diff --git a/docs/pt/js/vendor/jquery.js b/docs/pt/js/vendor/jquery.js
new file mode 100644
index 0000000..7fc60fc
--- /dev/null
+++ b/docs/pt/js/vendor/jquery.js
@@ -0,0 +1,11008 @@
+/*!
+ * jQuery JavaScript Library v1.12.4
+ * http://jquery.com/
+ *
+ * Includes Sizzle.js
+ * http://sizzlejs.com/
+ *
+ * Copyright jQuery Foundation and other contributors
+ * Released under the MIT license
+ * http://jquery.org/license
+ *
+ * Date: 2016-05-20T17:17Z
+ */
+
+(function( global, factory ) {
+
+	if ( typeof module === "object" && typeof module.exports === "object" ) {
+		// For CommonJS and CommonJS-like environments where a proper `window`
+		// is present, execute the factory and get jQuery.
+		// For environments that do not have a `window` with a `document`
+		// (such as Node.js), expose a factory as module.exports.
+		// This accentuates the need for the creation of a real `window`.
+		// e.g. var jQuery = require("jquery")(window);
+		// See ticket #14549 for more info.
+		module.exports = global.document ?
+			factory( global, true ) :
+			function( w ) {
+				if ( !w.document ) {
+					throw new Error( "jQuery requires a window with a document" );
+				}
+				return factory( w );
+			};
+	} else {
+		factory( global );
+	}
+
+// Pass this if window is not defined yet
+}(typeof window !== "undefined" ? window : this, function( window, noGlobal ) {
+
+// Support: Firefox 18+
+// Can't be in strict mode, several libs including ASP.NET trace
+// the stack via arguments.caller.callee and Firefox dies if
+// you try to trace through "use strict" call chains. (#13335)
+//"use strict";
+var deletedIds = [];
+
+var document = window.document;
+
+var slice = deletedIds.slice;
+
+var concat = deletedIds.concat;
+
+var push = deletedIds.push;
+
+var indexOf = deletedIds.indexOf;
+
+var class2type = {};
+
+var toString = class2type.toString;
+
+var hasOwn = class2type.hasOwnProperty;
+
+var support = {};
+
+
+
+var
+	version = "1.12.4",
+
+	// Define a local copy of jQuery
+	jQuery = function( selector, context ) {
+
+		// The jQuery object is actually just the init constructor 'enhanced'
+		// Need init if jQuery is called (just allow error to be thrown if not included)
+		return new jQuery.fn.init( selector, context );
+	},
+
+	// Support: Android<4.1, IE<9
+	// Make sure we trim BOM and NBSP
+	rtrim = /^[\s\uFEFF\xA0]+|[\s\uFEFF\xA0]+$/g,
+
+	// Matches dashed string for camelizing
+	rmsPrefix = /^-ms-/,
+	rdashAlpha = /-([\da-z])/gi,
+
+	// Used by jQuery.camelCase as callback to replace()
+	fcamelCase = function( all, letter ) {
+		return letter.toUpperCase();
+	};
+
+jQuery.fn = jQuery.prototype = {
+
+	// The current version of jQuery being used
+	jquery: version,
+
+	constructor: jQuery,
+
+	// Start with an empty selector
+	selector: "",
+
+	// The default length of a jQuery object is 0
+	length: 0,
+
+	toArray: function() {
+		return slice.call( this );
+	},
+
+	// Get the Nth element in the matched element set OR
+	// Get the whole matched element set as a clean array
+	get: function( num ) {
+		return num != null ?
+
+			// Return just the one element from the set
+			( num < 0 ? this[ num + this.length ] : this[ num ] ) :
+
+			// Return all the elements in a clean array
+			slice.call( this );
+	},
+
+	// Take an array of elements and push it onto the stack
+	// (returning the new matched element set)
+	pushStack: function( elems ) {
+
+		// Build a new jQuery matched element set
+		var ret = jQuery.merge( this.constructor(), elems );
+
+		// Add the old object onto the stack (as a reference)
+		ret.prevObject = this;
+		ret.context = this.context;
+
+		// Return the newly-formed element set
+		return ret;
+	},
+
+	// Execute a callback for every element in the matched set.
+	each: function( callback ) {
+		return jQuery.each( this, callback );
+	},
+
+	map: function( callback ) {
+		return this.pushStack( jQuery.map( this, function( elem, i ) {
+			return callback.call( elem, i, elem );
+		} ) );
+	},
+
+	slice: function() {
+		return this.pushStack( slice.apply( this, arguments ) );
+	},
+
+	first: function() {
+		return this.eq( 0 );
+	},
+
+	last: function() {
+		return this.eq( -1 );
+	},
+
+	eq: function( i ) {
+		var len = this.length,
+			j = +i + ( i < 0 ? len : 0 );
+		return this.pushStack( j >= 0 && j < len ? [ this[ j ] ] : [] );
+	},
+
+	end: function() {
+		return this.prevObject || this.constructor();
+	},
+
+	// For internal use only.
+	// Behaves like an Array's method, not like a jQuery method.
+	push: push,
+	sort: deletedIds.sort,
+	splice: deletedIds.splice
+};
+
+jQuery.extend = jQuery.fn.extend = function() {
+	var src, copyIsArray, copy, name, options, clone,
+		target = arguments[ 0 ] || {},
+		i = 1,
+		length = arguments.length,
+		deep = false;
+
+	// Handle a deep copy situation
+	if ( typeof target === "boolean" ) {
+		deep = target;
+
+		// skip the boolean and the target
+		target = arguments[ i ] || {};
+		i++;
+	}
+
+	// Handle case when target is a string or something (possible in deep copy)
+	if ( typeof target !== "object" && !jQuery.isFunction( target ) ) {
+		target = {};
+	}
+
+	// extend jQuery itself if only one argument is passed
+	if ( i === length ) {
+		target = this;
+		i--;
+	}
+
+	for ( ; i < length; i++ ) {
+
+		// Only deal with non-null/undefined values
+		if ( ( options = arguments[ i ] ) != null ) {
+
+			// Extend the base object
+			for ( name in options ) {
+				src = target[ name ];
+				copy = options[ name ];
+
+				// Prevent never-ending loop
+				if ( target === copy ) {
+					continue;
+				}
+
+				// Recurse if we're merging plain objects or arrays
+				if ( deep && copy && ( jQuery.isPlainObject( copy ) ||
+					( copyIsArray = jQuery.isArray( copy ) ) ) ) {
+
+					if ( copyIsArray ) {
+						copyIsArray = false;
+						clone = src && jQuery.isArray( src ) ? src : [];
+
+					} else {
+						clone = src && jQuery.isPlainObject( src ) ? src : {};
+					}
+
+					// Never move original objects, clone them
+					target[ name ] = jQuery.extend( deep, clone, copy );
+
+				// Don't bring in undefined values
+				} else if ( copy !== undefined ) {
+					target[ name ] = copy;
+				}
+			}
+		}
+	}
+
+	// Return the modified object
+	return target;
+};
+
+jQuery.extend( {
+
+	// Unique for each copy of jQuery on the page
+	expando: "jQuery" + ( version + Math.random() ).replace( /\D/g, "" ),
+
+	// Assume jQuery is ready without the ready module
+	isReady: true,
+
+	error: function( msg ) {
+		throw new Error( msg );
+	},
+
+	noop: function() {},
+
+	// See test/unit/core.js for details concerning isFunction.
+	// Since version 1.3, DOM methods and functions like alert
+	// aren't supported. They return false on IE (#2968).
+	isFunction: function( obj ) {
+		return jQuery.type( obj ) === "function";
+	},
+
+	isArray: Array.isArray || function( obj ) {
+		return jQuery.type( obj ) === "array";
+	},
+
+	isWindow: function( obj ) {
+		/* jshint eqeqeq: false */
+		return obj != null && obj == obj.window;
+	},
+
+	isNumeric: function( obj ) {
+
+		// parseFloat NaNs numeric-cast false positives (null|true|false|"")
+		// ...but misinterprets leading-number strings, particularly hex literals ("0x...")
+		// subtraction forces infinities to NaN
+		// adding 1 corrects loss of precision from parseFloat (#15100)
+		var realStringObj = obj && obj.toString();
+		return !jQuery.isArray( obj ) && ( realStringObj - parseFloat( realStringObj ) + 1 ) >= 0;
+	},
+
+	isEmptyObject: function( obj ) {
+		var name;
+		for ( name in obj ) {
+			return false;
+		}
+		return true;
+	},
+
+	isPlainObject: function( obj ) {
+		var key;
+
+		// Must be an Object.
+		// Because of IE, we also have to check the presence of the constructor property.
+		// Make sure that DOM nodes and window objects don't pass through, as well
+		if ( !obj || jQuery.type( obj ) !== "object" || obj.nodeType || jQuery.isWindow( obj ) ) {
+			return false;
+		}
+
+		try {
+
+			// Not own constructor property must be Object
+			if ( obj.constructor &&
+				!hasOwn.call( obj, "constructor" ) &&
+				!hasOwn.call( obj.constructor.prototype, "isPrototypeOf" ) ) {
+				return false;
+			}
+		} catch ( e ) {
+
+			// IE8,9 Will throw exceptions on certain host objects #9897
+			return false;
+		}
+
+		// Support: IE<9
+		// Handle iteration over inherited properties before own properties.
+		if ( !support.ownFirst ) {
+			for ( key in obj ) {
+				return hasOwn.call( obj, key );
+			}
+		}
+
+		// Own properties are enumerated firstly, so to speed up,
+		// if last one is own, then all properties are own.
+		for ( key in obj ) {}
+
+		return key === undefined || hasOwn.call( obj, key );
+	},
+
+	type: function( obj ) {
+		if ( obj == null ) {
+			return obj + "";
+		}
+		return typeof obj === "object" || typeof obj === "function" ?
+			class2type[ toString.call( obj ) ] || "object" :
+			typeof obj;
+	},
+
+	// Workarounds based on findings by Jim Driscoll
+	// http://weblogs.java.net/blog/driscoll/archive/2009/09/08/eval-javascript-global-context
+	globalEval: function( data ) {
+		if ( data && jQuery.trim( data ) ) {
+
+			// We use execScript on Internet Explorer
+			// We use an anonymous function so that context is window
+			// rather than jQuery in Firefox
+			( window.execScript || function( data ) {
+				window[ "eval" ].call( window, data ); // jscs:ignore requireDotNotation
+			} )( data );
+		}
+	},
+
+	// Convert dashed to camelCase; used by the css and data modules
+	// Microsoft forgot to hump their vendor prefix (#9572)
+	camelCase: function( string ) {
+		return string.replace( rmsPrefix, "ms-" ).replace( rdashAlpha, fcamelCase );
+	},
+
+	nodeName: function( elem, name ) {
+		return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase();
+	},
+
+	each: function( obj, callback ) {
+		var length, i = 0;
+
+		if ( isArrayLike( obj ) ) {
+			length = obj.length;
+			for ( ; i < length; i++ ) {
+				if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) {
+					break;
+				}
+			}
+		} else {
+			for ( i in obj ) {
+				if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) {
+					break;
+				}
+			}
+		}
+
+		return obj;
+	},
+
+	// Support: Android<4.1, IE<9
+	trim: function( text ) {
+		return text == null ?
+			"" :
+			( text + "" ).replace( rtrim, "" );
+	},
+
+	// results is for internal usage only
+	makeArray: function( arr, results ) {
+		var ret = results || [];
+
+		if ( arr != null ) {
+			if ( isArrayLike( Object( arr ) ) ) {
+				jQuery.merge( ret,
+					typeof arr === "string" ?
+					[ arr ] : arr
+				);
+			} else {
+				push.call( ret, arr );
+			}
+		}
+
+		return ret;
+	},
+
+	inArray: function( elem, arr, i ) {
+		var len;
+
+		if ( arr ) {
+			if ( indexOf ) {
+				return indexOf.call( arr, elem, i );
+			}
+
+			len = arr.length;
+			i = i ? i < 0 ? Math.max( 0, len + i ) : i : 0;
+
+			for ( ; i < len; i++ ) {
+
+				// Skip accessing in sparse arrays
+				if ( i in arr && arr[ i ] === elem ) {
+					return i;
+				}
+			}
+		}
+
+		return -1;
+	},
+
+	merge: function( first, second ) {
+		var len = +second.length,
+			j = 0,
+			i = first.length;
+
+		while ( j < len ) {
+			first[ i++ ] = second[ j++ ];
+		}
+
+		// Support: IE<9
+		// Workaround casting of .length to NaN on otherwise arraylike objects (e.g., NodeLists)
+		if ( len !== len ) {
+			while ( second[ j ] !== undefined ) {
+				first[ i++ ] = second[ j++ ];
+			}
+		}
+
+		first.length = i;
+
+		return first;
+	},
+
+	grep: function( elems, callback, invert ) {
+		var callbackInverse,
+			matches = [],
+			i = 0,
+			length = elems.length,
+			callbackExpect = !invert;
+
+		// Go through the array, only saving the items
+		// that pass the validator function
+		for ( ; i < length; i++ ) {
+			callbackInverse = !callback( elems[ i ], i );
+			if ( callbackInverse !== callbackExpect ) {
+				matches.push( elems[ i ] );
+			}
+		}
+
+		return matches;
+	},
+
+	// arg is for internal usage only
+	map: function( elems, callback, arg ) {
+		var length, value,
+			i = 0,
+			ret = [];
+
+		// Go through the array, translating each of the items to their new values
+		if ( isArrayLike( elems ) ) {
+			length = elems.length;
+			for ( ; i < length; i++ ) {
+				value = callback( elems[ i ], i, arg );
+
+				if ( value != null ) {
+					ret.push( value );
+				}
+			}
+
+		// Go through every key on the object,
+		} else {
+			for ( i in elems ) {
+				value = callback( elems[ i ], i, arg );
+
+				if ( value != null ) {
+					ret.push( value );
+				}
+			}
+		}
+
+		// Flatten any nested arrays
+		return concat.apply( [], ret );
+	},
+
+	// A global GUID counter for objects
+	guid: 1,
+
+	// Bind a function to a context, optionally partially applying any
+	// arguments.
+	proxy: function( fn, context ) {
+		var args, proxy, tmp;
+
+		if ( typeof context === "string" ) {
+			tmp = fn[ context ];
+			context = fn;
+			fn = tmp;
+		}
+
+		// Quick check to determine if target is callable, in the spec
+		// this throws a TypeError, but we will just return undefined.
+		if ( !jQuery.isFunction( fn ) ) {
+			return undefined;
+		}
+
+		// Simulated bind
+		args = slice.call( arguments, 2 );
+		proxy = function() {
+			return fn.apply( context || this, args.concat( slice.call( arguments ) ) );
+		};
+
+		// Set the guid of unique handler to the same of original handler, so it can be removed
+		proxy.guid = fn.guid = fn.guid || jQuery.guid++;
+
+		return proxy;
+	},
+
+	now: function() {
+		return +( new Date() );
+	},
+
+	// jQuery.support is not used in Core but other projects attach their
+	// properties to it so it needs to exist.
+	support: support
+} );
+
+// JSHint would error on this code due to the Symbol not being defined in ES5.
+// Defining this global in .jshintrc would create a danger of using the global
+// unguarded in another place, it seems safer to just disable JSHint for these
+// three lines.
+/* jshint ignore: start */
+if ( typeof Symbol === "function" ) {
+	jQuery.fn[ Symbol.iterator ] = deletedIds[ Symbol.iterator ];
+}
+/* jshint ignore: end */
+
+// Populate the class2type map
+jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ),
+function( i, name ) {
+	class2type[ "[object " + name + "]" ] = name.toLowerCase();
+} );
+
+function isArrayLike( obj ) {
+
+	// Support: iOS 8.2 (not reproducible in simulator)
+	// `in` check used to prevent JIT error (gh-2145)
+	// hasOwn isn't used here due to false negatives
+	// regarding Nodelist length in IE
+	var length = !!obj && "length" in obj && obj.length,
+		type = jQuery.type( obj );
+
+	if ( type === "function" || jQuery.isWindow( obj ) ) {
+		return false;
+	}
+
+	return type === "array" || length === 0 ||
+		typeof length === "number" && length > 0 && ( length - 1 ) in obj;
+}
+var Sizzle =
+/*!
+ * Sizzle CSS Selector Engine v2.2.1
+ * http://sizzlejs.com/
+ *
+ * Copyright jQuery Foundation and other contributors
+ * Released under the MIT license
+ * http://jquery.org/license
+ *
+ * Date: 2015-10-17
+ */
+(function( window ) {
+
+var i,
+	support,
+	Expr,
+	getText,
+	isXML,
+	tokenize,
+	compile,
+	select,
+	outermostContext,
+	sortInput,
+	hasDuplicate,
+
+	// Local document vars
+	setDocument,
+	document,
+	docElem,
+	documentIsHTML,
+	rbuggyQSA,
+	rbuggyMatches,
+	matches,
+	contains,
+
+	// Instance-specific data
+	expando = "sizzle" + 1 * new Date(),
+	preferredDoc = window.document,
+	dirruns = 0,
+	done = 0,
+	classCache = createCache(),
+	tokenCache = createCache(),
+	compilerCache = createCache(),
+	sortOrder = function( a, b ) {
+		if ( a === b ) {
+			hasDuplicate = true;
+		}
+		return 0;
+	},
+
+	// General-purpose constants
+	MAX_NEGATIVE = 1 << 31,
+
+	// Instance methods
+	hasOwn = ({}).hasOwnProperty,
+	arr = [],
+	pop = arr.pop,
+	push_native = arr.push,
+	push = arr.push,
+	slice = arr.slice,
+	// Use a stripped-down indexOf as it's faster than native
+	// http://jsperf.com/thor-indexof-vs-for/5
+	indexOf = function( list, elem ) {
+		var i = 0,
+			len = list.length;
+		for ( ; i < len; i++ ) {
+			if ( list[i] === elem ) {
+				return i;
+			}
+		}
+		return -1;
+	},
+
+	booleans = "checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|ismap|loop|multiple|open|readonly|required|scoped",
+
+	// Regular expressions
+
+	// http://www.w3.org/TR/css3-selectors/#whitespace
+	whitespace = "[\\x20\\t\\r\\n\\f]",
+
+	// http://www.w3.org/TR/CSS21/syndata.html#value-def-identifier
+	identifier = "(?:\\\\.|[\\w-]|[^\\x00-\\xa0])+",
+
+	// Attribute selectors: http://www.w3.org/TR/selectors/#attribute-selectors
+	attributes = "\\[" + whitespace + "*(" + identifier + ")(?:" + whitespace +
+		// Operator (capture 2)
+		"*([*^$|!~]?=)" + whitespace +
+		// "Attribute values must be CSS identifiers [capture 5] or strings [capture 3 or capture 4]"
+		"*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|(" + identifier + "))|)" + whitespace +
+		"*\\]",
+
+	pseudos = ":(" + identifier + ")(?:\\((" +
+		// To reduce the number of selectors needing tokenize in the preFilter, prefer arguments:
+		// 1. quoted (capture 3; capture 4 or capture 5)
+		"('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|" +
+		// 2. simple (capture 6)
+		"((?:\\\\.|[^\\\\()[\\]]|" + attributes + ")*)|" +
+		// 3. anything else (capture 2)
+		".*" +
+		")\\)|)",
+
+	// Leading and non-escaped trailing whitespace, capturing some non-whitespace characters preceding the latter
+	rwhitespace = new RegExp( whitespace + "+", "g" ),
+	rtrim = new RegExp( "^" + whitespace + "+|((?:^|[^\\\\])(?:\\\\.)*)" + whitespace + "+$", "g" ),
+
+	rcomma = new RegExp( "^" + whitespace + "*," + whitespace + "*" ),
+	rcombinators = new RegExp( "^" + whitespace + "*([>+~]|" + whitespace + ")" + whitespace + "*" ),
+
+	rattributeQuotes = new RegExp( "=" + whitespace + "*([^\\]'\"]*?)" + whitespace + "*\\]", "g" ),
+
+	rpseudo = new RegExp( pseudos ),
+	ridentifier = new RegExp( "^" + identifier + "$" ),
+
+	matchExpr = {
+		"ID": new RegExp( "^#(" + identifier + ")" ),
+		"CLASS": new RegExp( "^\\.(" + identifier + ")" ),
+		"TAG": new RegExp( "^(" + identifier + "|[*])" ),
+		"ATTR": new RegExp( "^" + attributes ),
+		"PSEUDO": new RegExp( "^" + pseudos ),
+		"CHILD": new RegExp( "^:(only|first|last|nth|nth-last)-(child|of-type)(?:\\(" + whitespace +
+			"*(even|odd|(([+-]|)(\\d*)n|)" + whitespace + "*(?:([+-]|)" + whitespace +
+			"*(\\d+)|))" + whitespace + "*\\)|)", "i" ),
+		"bool": new RegExp( "^(?:" + booleans + ")$", "i" ),
+		// For use in libraries implementing .is()
+		// We use this for POS matching in `select`
+		"needsContext": new RegExp( "^" + whitespace + "*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\\(" +
+			whitespace + "*((?:-\\d)?\\d*)" + whitespace + "*\\)|)(?=[^-]|$)", "i" )
+	},
+
+	rinputs = /^(?:input|select|textarea|button)$/i,
+	rheader = /^h\d$/i,
+
+	rnative = /^[^{]+\{\s*\[native \w/,
+
+	// Easily-parseable/retrievable ID or TAG or CLASS selectors
+	rquickExpr = /^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/,
+
+	rsibling = /[+~]/,
+	rescape = /'|\\/g,
+
+	// CSS escapes http://www.w3.org/TR/CSS21/syndata.html#escaped-characters
+	runescape = new RegExp( "\\\\([\\da-f]{1,6}" + whitespace + "?|(" + whitespace + ")|.)", "ig" ),
+	funescape = function( _, escaped, escapedWhitespace ) {
+		var high = "0x" + escaped - 0x10000;
+		// NaN means non-codepoint
+		// Support: Firefox<24
+		// Workaround erroneous numeric interpretation of +"0x"
+		return high !== high || escapedWhitespace ?
+			escaped :
+			high < 0 ?
+				// BMP codepoint
+				String.fromCharCode( high + 0x10000 ) :
+				// Supplemental Plane codepoint (surrogate pair)
+				String.fromCharCode( high >> 10 | 0xD800, high & 0x3FF | 0xDC00 );
+	},
+
+	// Used for iframes
+	// See setDocument()
+	// Removing the function wrapper causes a "Permission Denied"
+	// error in IE
+	unloadHandler = function() {
+		setDocument();
+	};
+
+// Optimize for push.apply( _, NodeList )
+try {
+	push.apply(
+		(arr = slice.call( preferredDoc.childNodes )),
+		preferredDoc.childNodes
+	);
+	// Support: Android<4.0
+	// Detect silently failing push.apply
+	arr[ preferredDoc.childNodes.length ].nodeType;
+} catch ( e ) {
+	push = { apply: arr.length ?
+
+		// Leverage slice if possible
+		function( target, els ) {
+			push_native.apply( target, slice.call(els) );
+		} :
+
+		// Support: IE<9
+		// Otherwise append directly
+		function( target, els ) {
+			var j = target.length,
+				i = 0;
+			// Can't trust NodeList.length
+			while ( (target[j++] = els[i++]) ) {}
+			target.length = j - 1;
+		}
+	};
+}
+
+function Sizzle( selector, context, results, seed ) {
+	var m, i, elem, nid, nidselect, match, groups, newSelector,
+		newContext = context && context.ownerDocument,
+
+		// nodeType defaults to 9, since context defaults to document
+		nodeType = context ? context.nodeType : 9;
+
+	results = results || [];
+
+	// Return early from calls with invalid selector or context
+	if ( typeof selector !== "string" || !selector ||
+		nodeType !== 1 && nodeType !== 9 && nodeType !== 11 ) {
+
+		return results;
+	}
+
+	// Try to shortcut find operations (as opposed to filters) in HTML documents
+	if ( !seed ) {
+
+		if ( ( context ? context.ownerDocument || context : preferredDoc ) !== document ) {
+			setDocument( context );
+		}
+		context = context || document;
+
+		if ( documentIsHTML ) {
+
+			// If the selector is sufficiently simple, try using a "get*By*" DOM method
+			// (excepting DocumentFragment context, where the methods don't exist)
+			if ( nodeType !== 11 && (match = rquickExpr.exec( selector )) ) {
+
+				// ID selector
+				if ( (m = match[1]) ) {
+
+					// Document context
+					if ( nodeType === 9 ) {
+						if ( (elem = context.getElementById( m )) ) {
+
+							// Support: IE, Opera, Webkit
+							// TODO: identify versions
+							// getElementById can match elements by name instead of ID
+							if ( elem.id === m ) {
+								results.push( elem );
+								return results;
+							}
+						} else {
+							return results;
+						}
+
+					// Element context
+					} else {
+
+						// Support: IE, Opera, Webkit
+						// TODO: identify versions
+						// getElementById can match elements by name instead of ID
+						if ( newContext && (elem = newContext.getElementById( m )) &&
+							contains( context, elem ) &&
+							elem.id === m ) {
+
+							results.push( elem );
+							return results;
+						}
+					}
+
+				// Type selector
+				} else if ( match[2] ) {
+					push.apply( results, context.getElementsByTagName( selector ) );
+					return results;
+
+				// Class selector
+				} else if ( (m = match[3]) && support.getElementsByClassName &&
+					context.getElementsByClassName ) {
+
+					push.apply( results, context.getElementsByClassName( m ) );
+					return results;
+				}
+			}
+
+			// Take advantage of querySelectorAll
+			if ( support.qsa &&
+				!compilerCache[ selector + " " ] &&
+				(!rbuggyQSA || !rbuggyQSA.test( selector )) ) {
+
+				if ( nodeType !== 1 ) {
+					newContext = context;
+					newSelector = selector;
+
+				// qSA looks outside Element context, which is not what we want
+				// Thanks to Andrew Dupont for this workaround technique
+				// Support: IE <=8
+				// Exclude object elements
+				} else if ( context.nodeName.toLowerCase() !== "object" ) {
+
+					// Capture the context ID, setting it first if necessary
+					if ( (nid = context.getAttribute( "id" )) ) {
+						nid = nid.replace( rescape, "\\$&" );
+					} else {
+						context.setAttribute( "id", (nid = expando) );
+					}
+
+					// Prefix every selector in the list
+					groups = tokenize( selector );
+					i = groups.length;
+					nidselect = ridentifier.test( nid ) ? "#" + nid : "[id='" + nid + "']";
+					while ( i-- ) {
+						groups[i] = nidselect + " " + toSelector( groups[i] );
+					}
+					newSelector = groups.join( "," );
+
+					// Expand context for sibling selectors
+					newContext = rsibling.test( selector ) && testContext( context.parentNode ) ||
+						context;
+				}
+
+				if ( newSelector ) {
+					try {
+						push.apply( results,
+							newContext.querySelectorAll( newSelector )
+						);
+						return results;
+					} catch ( qsaError ) {
+					} finally {
+						if ( nid === expando ) {
+							context.removeAttribute( "id" );
+						}
+					}
+				}
+			}
+		}
+	}
+
+	// All others
+	return select( selector.replace( rtrim, "$1" ), context, results, seed );
+}
+
+/**
+ * Create key-value caches of limited size
+ * @returns {function(string, object)} Returns the Object data after storing it on itself with
+ *	property name the (space-suffixed) string and (if the cache is larger than Expr.cacheLength)
+ *	deleting the oldest entry
+ */
+function createCache() {
+	var keys = [];
+
+	function cache( key, value ) {
+		// Use (key + " ") to avoid collision with native prototype properties (see Issue #157)
+		if ( keys.push( key + " " ) > Expr.cacheLength ) {
+			// Only keep the most recent entries
+			delete cache[ keys.shift() ];
+		}
+		return (cache[ key + " " ] = value);
+	}
+	return cache;
+}
+
+/**
+ * Mark a function for special use by Sizzle
+ * @param {Function} fn The function to mark
+ */
+function markFunction( fn ) {
+	fn[ expando ] = true;
+	return fn;
+}
+
+/**
+ * Support testing using an element
+ * @param {Function} fn Passed the created div and expects a boolean result
+ */
+function assert( fn ) {
+	var div = document.createElement("div");
+
+	try {
+		return !!fn( div );
+	} catch (e) {
+		return false;
+	} finally {
+		// Remove from its parent by default
+		if ( div.parentNode ) {
+			div.parentNode.removeChild( div );
+		}
+		// release memory in IE
+		div = null;
+	}
+}
+
+/**
+ * Adds the same handler for all of the specified attrs
+ * @param {String} attrs Pipe-separated list of attributes
+ * @param {Function} handler The method that will be applied
+ */
+function addHandle( attrs, handler ) {
+	var arr = attrs.split("|"),
+		i = arr.length;
+
+	while ( i-- ) {
+		Expr.attrHandle[ arr[i] ] = handler;
+	}
+}
+
+/**
+ * Checks document order of two siblings
+ * @param {Element} a
+ * @param {Element} b
+ * @returns {Number} Returns less than 0 if a precedes b, greater than 0 if a follows b
+ */
+function siblingCheck( a, b ) {
+	var cur = b && a,
+		diff = cur && a.nodeType === 1 && b.nodeType === 1 &&
+			( ~b.sourceIndex || MAX_NEGATIVE ) -
+			( ~a.sourceIndex || MAX_NEGATIVE );
+
+	// Use IE sourceIndex if available on both nodes
+	if ( diff ) {
+		return diff;
+	}
+
+	// Check if b follows a
+	if ( cur ) {
+		while ( (cur = cur.nextSibling) ) {
+			if ( cur === b ) {
+				return -1;
+			}
+		}
+	}
+
+	return a ? 1 : -1;
+}
+
+/**
+ * Returns a function to use in pseudos for input types
+ * @param {String} type
+ */
+function createInputPseudo( type ) {
+	return function( elem ) {
+		var name = elem.nodeName.toLowerCase();
+		return name === "input" && elem.type === type;
+	};
+}
+
+/**
+ * Returns a function to use in pseudos for buttons
+ * @param {String} type
+ */
+function createButtonPseudo( type ) {
+	return function( elem ) {
+		var name = elem.nodeName.toLowerCase();
+		return (name === "input" || name === "button") && elem.type === type;
+	};
+}
+
+/**
+ * Returns a function to use in pseudos for positionals
+ * @param {Function} fn
+ */
+function createPositionalPseudo( fn ) {
+	return markFunction(function( argument ) {
+		argument = +argument;
+		return markFunction(function( seed, matches ) {
+			var j,
+				matchIndexes = fn( [], seed.length, argument ),
+				i = matchIndexes.length;
+
+			// Match elements found at the specified indexes
+			while ( i-- ) {
+				if ( seed[ (j = matchIndexes[i]) ] ) {
+					seed[j] = !(matches[j] = seed[j]);
+				}
+			}
+		});
+	});
+}
+
+/**
+ * Checks a node for validity as a Sizzle context
+ * @param {Element|Object=} context
+ * @returns {Element|Object|Boolean} The input node if acceptable, otherwise a falsy value
+ */
+function testContext( context ) {
+	return context && typeof context.getElementsByTagName !== "undefined" && context;
+}
+
+// Expose support vars for convenience
+support = Sizzle.support = {};
+
+/**
+ * Detects XML nodes
+ * @param {Element|Object} elem An element or a document
+ * @returns {Boolean} True iff elem is a non-HTML XML node
+ */
+isXML = Sizzle.isXML = function( elem ) {
+	// documentElement is verified for cases where it doesn't yet exist
+	// (such as loading iframes in IE - #4833)
+	var documentElement = elem && (elem.ownerDocument || elem).documentElement;
+	return documentElement ? documentElement.nodeName !== "HTML" : false;
+};
+
+/**
+ * Sets document-related variables once based on the current document
+ * @param {Element|Object} [doc] An element or document object to use to set the document
+ * @returns {Object} Returns the current document
+ */
+setDocument = Sizzle.setDocument = function( node ) {
+	var hasCompare, parent,
+		doc = node ? node.ownerDocument || node : preferredDoc;
+
+	// Return early if doc is invalid or already selected
+	if ( doc === document || doc.nodeType !== 9 || !doc.documentElement ) {
+		return document;
+	}
+
+	// Update global variables
+	document = doc;
+	docElem = document.documentElement;
+	documentIsHTML = !isXML( document );
+
+	// Support: IE 9-11, Edge
+	// Accessing iframe documents after unload throws "permission denied" errors (jQuery #13936)
+	if ( (parent = document.defaultView) && parent.top !== parent ) {
+		// Support: IE 11
+		if ( parent.addEventListener ) {
+			parent.addEventListener( "unload", unloadHandler, false );
+
+		// Support: IE 9 - 10 only
+		} else if ( parent.attachEvent ) {
+			parent.attachEvent( "onunload", unloadHandler );
+		}
+	}
+
+	/* Attributes
+	---------------------------------------------------------------------- */
+
+	// Support: IE<8
+	// Verify that getAttribute really returns attributes and not properties
+	// (excepting IE8 booleans)
+	support.attributes = assert(function( div ) {
+		div.className = "i";
+		return !div.getAttribute("className");
+	});
+
+	/* getElement(s)By*
+	---------------------------------------------------------------------- */
+
+	// Check if getElementsByTagName("*") returns only elements
+	support.getElementsByTagName = assert(function( div ) {
+		div.appendChild( document.createComment("") );
+		return !div.getElementsByTagName("*").length;
+	});
+
+	// Support: IE<9
+	support.getElementsByClassName = rnative.test( document.getElementsByClassName );
+
+	// Support: IE<10
+	// Check if getElementById returns elements by name
+	// The broken getElementById methods don't pick up programatically-set names,
+	// so use a roundabout getElementsByName test
+	support.getById = assert(function( div ) {
+		docElem.appendChild( div ).id = expando;
+		return !document.getElementsByName || !document.getElementsByName( expando ).length;
+	});
+
+	// ID find and filter
+	if ( support.getById ) {
+		Expr.find["ID"] = function( id, context ) {
+			if ( typeof context.getElementById !== "undefined" && documentIsHTML ) {
+				var m = context.getElementById( id );
+				return m ? [ m ] : [];
+			}
+		};
+		Expr.filter["ID"] = function( id ) {
+			var attrId = id.replace( runescape, funescape );
+			return function( elem ) {
+				return elem.getAttribute("id") === attrId;
+			};
+		};
+	} else {
+		// Support: IE6/7
+		// getElementById is not reliable as a find shortcut
+		delete Expr.find["ID"];
+
+		Expr.filter["ID"] =  function( id ) {
+			var attrId = id.replace( runescape, funescape );
+			return function( elem ) {
+				var node = typeof elem.getAttributeNode !== "undefined" &&
+					elem.getAttributeNode("id");
+				return node && node.value === attrId;
+			};
+		};
+	}
+
+	// Tag
+	Expr.find["TAG"] = support.getElementsByTagName ?
+		function( tag, context ) {
+			if ( typeof context.getElementsByTagName !== "undefined" ) {
+				return context.getElementsByTagName( tag );
+
+			// DocumentFragment nodes don't have gEBTN
+			} else if ( support.qsa ) {
+				return context.querySelectorAll( tag );
+			}
+		} :
+
+		function( tag, context ) {
+			var elem,
+				tmp = [],
+				i = 0,
+				// By happy coincidence, a (broken) gEBTN appears on DocumentFragment nodes too
+				results = context.getElementsByTagName( tag );
+
+			// Filter out possible comments
+			if ( tag === "*" ) {
+				while ( (elem = results[i++]) ) {
+					if ( elem.nodeType === 1 ) {
+						tmp.push( elem );
+					}
+				}
+
+				return tmp;
+			}
+			return results;
+		};
+
+	// Class
+	Expr.find["CLASS"] = support.getElementsByClassName && function( className, context ) {
+		if ( typeof context.getElementsByClassName !== "undefined" && documentIsHTML ) {
+			return context.getElementsByClassName( className );
+		}
+	};
+
+	/* QSA/matchesSelector
+	---------------------------------------------------------------------- */
+
+	// QSA and matchesSelector support
+
+	// matchesSelector(:active) reports false when true (IE9/Opera 11.5)
+	rbuggyMatches = [];
+
+	// qSa(:focus) reports false when true (Chrome 21)
+	// We allow this because of a bug in IE8/9 that throws an error
+	// whenever `document.activeElement` is accessed on an iframe
+	// So, we allow :focus to pass through QSA all the time to avoid the IE error
+	// See http://bugs.jquery.com/ticket/13378
+	rbuggyQSA = [];
+
+	if ( (support.qsa = rnative.test( document.querySelectorAll )) ) {
+		// Build QSA regex
+		// Regex strategy adopted from Diego Perini
+		assert(function( div ) {
+			// Select is set to empty string on purpose
+			// This is to test IE's treatment of not explicitly
+			// setting a boolean content attribute,
+			// since its presence should be enough
+			// http://bugs.jquery.com/ticket/12359
+			docElem.appendChild( div ).innerHTML = "<a id='" + expando + "'></a>" +
+				"<select id='" + expando + "-\r\\' msallowcapture=''>" +
+				"<option selected=''></option></select>";
+
+			// Support: IE8, Opera 11-12.16
+			// Nothing should be selected when empty strings follow ^= or $= or *=
+			// The test attribute must be unknown in Opera but "safe" for WinRT
+			// http://msdn.microsoft.com/en-us/library/ie/hh465388.aspx#attribute_section
+			if ( div.querySelectorAll("[msallowcapture^='']").length ) {
+				rbuggyQSA.push( "[*^$]=" + whitespace + "*(?:''|\"\")" );
+			}
+
+			// Support: IE8
+			// Boolean attributes and "value" are not treated correctly
+			if ( !div.querySelectorAll("[selected]").length ) {
+				rbuggyQSA.push( "\\[" + whitespace + "*(?:value|" + booleans + ")" );
+			}
+
+			// Support: Chrome<29, Android<4.4, Safari<7.0+, iOS<7.0+, PhantomJS<1.9.8+
+			if ( !div.querySelectorAll( "[id~=" + expando + "-]" ).length ) {
+				rbuggyQSA.push("~=");
+			}
+
+			// Webkit/Opera - :checked should return selected option elements
+			// http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked
+			// IE8 throws error here and will not see later tests
+			if ( !div.querySelectorAll(":checked").length ) {
+				rbuggyQSA.push(":checked");
+			}
+
+			// Support: Safari 8+, iOS 8+
+			// https://bugs.webkit.org/show_bug.cgi?id=136851
+			// In-page `selector#id sibing-combinator selector` fails
+			if ( !div.querySelectorAll( "a#" + expando + "+*" ).length ) {
+				rbuggyQSA.push(".#.+[+~]");
+			}
+		});
+
+		assert(function( div ) {
+			// Support: Windows 8 Native Apps
+			// The type and name attributes are restricted during .innerHTML assignment
+			var input = document.createElement("input");
+			input.setAttribute( "type", "hidden" );
+			div.appendChild( input ).setAttribute( "name", "D" );
+
+			// Support: IE8
+			// Enforce case-sensitivity of name attribute
+			if ( div.querySelectorAll("[name=d]").length ) {
+				rbuggyQSA.push( "name" + whitespace + "*[*^$|!~]?=" );
+			}
+
+			// FF 3.5 - :enabled/:disabled and hidden elements (hidden elements are still enabled)
+			// IE8 throws error here and will not see later tests
+			if ( !div.querySelectorAll(":enabled").length ) {
+				rbuggyQSA.push( ":enabled", ":disabled" );
+			}
+
+			// Opera 10-11 does not throw on post-comma invalid pseudos
+			div.querySelectorAll("*,:x");
+			rbuggyQSA.push(",.*:");
+		});
+	}
+
+	if ( (support.matchesSelector = rnative.test( (matches = docElem.matches ||
+		docElem.webkitMatchesSelector ||
+		docElem.mozMatchesSelector ||
+		docElem.oMatchesSelector ||
+		docElem.msMatchesSelector) )) ) {
+
+		assert(function( div ) {
+			// Check to see if it's possible to do matchesSelector
+			// on a disconnected node (IE 9)
+			support.disconnectedMatch = matches.call( div, "div" );
+
+			// This should fail with an exception
+			// Gecko does not error, returns false instead
+			matches.call( div, "[s!='']:x" );
+			rbuggyMatches.push( "!=", pseudos );
+		});
+	}
+
+	rbuggyQSA = rbuggyQSA.length && new RegExp( rbuggyQSA.join("|") );
+	rbuggyMatches = rbuggyMatches.length && new RegExp( rbuggyMatches.join("|") );
+
+	/* Contains
+	---------------------------------------------------------------------- */
+	hasCompare = rnative.test( docElem.compareDocumentPosition );
+
+	// Element contains another
+	// Purposefully self-exclusive
+	// As in, an element does not contain itself
+	contains = hasCompare || rnative.test( docElem.contains ) ?
+		function( a, b ) {
+			var adown = a.nodeType === 9 ? a.documentElement : a,
+				bup = b && b.parentNode;
+			return a === bup || !!( bup && bup.nodeType === 1 && (
+				adown.contains ?
+					adown.contains( bup ) :
+					a.compareDocumentPosition && a.compareDocumentPosition( bup ) & 16
+			));
+		} :
+		function( a, b ) {
+			if ( b ) {
+				while ( (b = b.parentNode) ) {
+					if ( b === a ) {
+						return true;
+					}
+				}
+			}
+			return false;
+		};
+
+	/* Sorting
+	---------------------------------------------------------------------- */
+
+	// Document order sorting
+	sortOrder = hasCompare ?
+	function( a, b ) {
+
+		// Flag for duplicate removal
+		if ( a === b ) {
+			hasDuplicate = true;
+			return 0;
+		}
+
+		// Sort on method existence if only one input has compareDocumentPosition
+		var compare = !a.compareDocumentPosition - !b.compareDocumentPosition;
+		if ( compare ) {
+			return compare;
+		}
+
+		// Calculate position if both inputs belong to the same document
+		compare = ( a.ownerDocument || a ) === ( b.ownerDocument || b ) ?
+			a.compareDocumentPosition( b ) :
+
+			// Otherwise we know they are disconnected
+			1;
+
+		// Disconnected nodes
+		if ( compare & 1 ||
+			(!support.sortDetached && b.compareDocumentPosition( a ) === compare) ) {
+
+			// Choose the first element that is related to our preferred document
+			if ( a === document || a.ownerDocument === preferredDoc && contains(preferredDoc, a) ) {
+				return -1;
+			}
+			if ( b === document || b.ownerDocument === preferredDoc && contains(preferredDoc, b) ) {
+				return 1;
+			}
+
+			// Maintain original order
+			return sortInput ?
+				( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) :
+				0;
+		}
+
+		return compare & 4 ? -1 : 1;
+	} :
+	function( a, b ) {
+		// Exit early if the nodes are identical
+		if ( a === b ) {
+			hasDuplicate = true;
+			return 0;
+		}
+
+		var cur,
+			i = 0,
+			aup = a.parentNode,
+			bup = b.parentNode,
+			ap = [ a ],
+			bp = [ b ];
+
+		// Parentless nodes are either documents or disconnected
+		if ( !aup || !bup ) {
+			return a === document ? -1 :
+				b === document ? 1 :
+				aup ? -1 :
+				bup ? 1 :
+				sortInput ?
+				( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) :
+				0;
+
+		// If the nodes are siblings, we can do a quick check
+		} else if ( aup === bup ) {
+			return siblingCheck( a, b );
+		}
+
+		// Otherwise we need full lists of their ancestors for comparison
+		cur = a;
+		while ( (cur = cur.parentNode) ) {
+			ap.unshift( cur );
+		}
+		cur = b;
+		while ( (cur = cur.parentNode) ) {
+			bp.unshift( cur );
+		}
+
+		// Walk down the tree looking for a discrepancy
+		while ( ap[i] === bp[i] ) {
+			i++;
+		}
+
+		return i ?
+			// Do a sibling check if the nodes have a common ancestor
+			siblingCheck( ap[i], bp[i] ) :
+
+			// Otherwise nodes in our document sort first
+			ap[i] === preferredDoc ? -1 :
+			bp[i] === preferredDoc ? 1 :
+			0;
+	};
+
+	return document;
+};
+
+Sizzle.matches = function( expr, elements ) {
+	return Sizzle( expr, null, null, elements );
+};
+
+Sizzle.matchesSelector = function( elem, expr ) {
+	// Set document vars if needed
+	if ( ( elem.ownerDocument || elem ) !== document ) {
+		setDocument( elem );
+	}
+
+	// Make sure that attribute selectors are quoted
+	expr = expr.replace( rattributeQuotes, "='$1']" );
+
+	if ( support.matchesSelector && documentIsHTML &&
+		!compilerCache[ expr + " " ] &&
+		( !rbuggyMatches || !rbuggyMatches.test( expr ) ) &&
+		( !rbuggyQSA     || !rbuggyQSA.test( expr ) ) ) {
+
+		try {
+			var ret = matches.call( elem, expr );
+
+			// IE 9's matchesSelector returns false on disconnected nodes
+			if ( ret || support.disconnectedMatch ||
+					// As well, disconnected nodes are said to be in a document
+					// fragment in IE 9
+					elem.document && elem.document.nodeType !== 11 ) {
+				return ret;
+			}
+		} catch (e) {}
+	}
+
+	return Sizzle( expr, document, null, [ elem ] ).length > 0;
+};
+
+Sizzle.contains = function( context, elem ) {
+	// Set document vars if needed
+	if ( ( context.ownerDocument || context ) !== document ) {
+		setDocument( context );
+	}
+	return contains( context, elem );
+};
+
+Sizzle.attr = function( elem, name ) {
+	// Set document vars if needed
+	if ( ( elem.ownerDocument || elem ) !== document ) {
+		setDocument( elem );
+	}
+
+	var fn = Expr.attrHandle[ name.toLowerCase() ],
+		// Don't get fooled by Object.prototype properties (jQuery #13807)
+		val = fn && hasOwn.call( Expr.attrHandle, name.toLowerCase() ) ?
+			fn( elem, name, !documentIsHTML ) :
+			undefined;
+
+	return val !== undefined ?
+		val :
+		support.attributes || !documentIsHTML ?
+			elem.getAttribute( name ) :
+			(val = elem.getAttributeNode(name)) && val.specified ?
+				val.value :
+				null;
+};
+
+Sizzle.error = function( msg ) {
+	throw new Error( "Syntax error, unrecognized expression: " + msg );
+};
+
+/**
+ * Document sorting and removing duplicates
+ * @param {ArrayLike} results
+ */
+Sizzle.uniqueSort = function( results ) {
+	var elem,
+		duplicates = [],
+		j = 0,
+		i = 0;
+
+	// Unless we *know* we can detect duplicates, assume their presence
+	hasDuplicate = !support.detectDuplicates;
+	sortInput = !support.sortStable && results.slice( 0 );
+	results.sort( sortOrder );
+
+	if ( hasDuplicate ) {
+		while ( (elem = results[i++]) ) {
+			if ( elem === results[ i ] ) {
+				j = duplicates.push( i );
+			}
+		}
+		while ( j-- ) {
+			results.splice( duplicates[ j ], 1 );
+		}
+	}
+
+	// Clear input after sorting to release objects
+	// See https://github.com/jquery/sizzle/pull/225
+	sortInput = null;
+
+	return results;
+};
+
+/**
+ * Utility function for retrieving the text value of an array of DOM nodes
+ * @param {Array|Element} elem
+ */
+getText = Sizzle.getText = function( elem ) {
+	var node,
+		ret = "",
+		i = 0,
+		nodeType = elem.nodeType;
+
+	if ( !nodeType ) {
+		// If no nodeType, this is expected to be an array
+		while ( (node = elem[i++]) ) {
+			// Do not traverse comment nodes
+			ret += getText( node );
+		}
+	} else if ( nodeType === 1 || nodeType === 9 || nodeType === 11 ) {
+		// Use textContent for elements
+		// innerText usage removed for consistency of new lines (jQuery #11153)
+		if ( typeof elem.textContent === "string" ) {
+			return elem.textContent;
+		} else {
+			// Traverse its children
+			for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) {
+				ret += getText( elem );
+			}
+		}
+	} else if ( nodeType === 3 || nodeType === 4 ) {
+		return elem.nodeValue;
+	}
+	// Do not include comment or processing instruction nodes
+
+	return ret;
+};
+
+Expr = Sizzle.selectors = {
+
+	// Can be adjusted by the user
+	cacheLength: 50,
+
+	createPseudo: markFunction,
+
+	match: matchExpr,
+
+	attrHandle: {},
+
+	find: {},
+
+	relative: {
+		">": { dir: "parentNode", first: true },
+		" ": { dir: "parentNode" },
+		"+": { dir: "previousSibling", first: true },
+		"~": { dir: "previousSibling" }
+	},
+
+	preFilter: {
+		"ATTR": function( match ) {
+			match[1] = match[1].replace( runescape, funescape );
+
+			// Move the given value to match[3] whether quoted or unquoted
+			match[3] = ( match[3] || match[4] || match[5] || "" ).replace( runescape, funescape );
+
+			if ( match[2] === "~=" ) {
+				match[3] = " " + match[3] + " ";
+			}
+
+			return match.slice( 0, 4 );
+		},
+
+		"CHILD": function( match ) {
+			/* matches from matchExpr["CHILD"]
+				1 type (only|nth|...)
+				2 what (child|of-type)
+				3 argument (even|odd|\d*|\d*n([+-]\d+)?|...)
+				4 xn-component of xn+y argument ([+-]?\d*n|)
+				5 sign of xn-component
+				6 x of xn-component
+				7 sign of y-component
+				8 y of y-component
+			*/
+			match[1] = match[1].toLowerCase();
+
+			if ( match[1].slice( 0, 3 ) === "nth" ) {
+				// nth-* requires argument
+				if ( !match[3] ) {
+					Sizzle.error( match[0] );
+				}
+
+				// numeric x and y parameters for Expr.filter.CHILD
+				// remember that false/true cast respectively to 0/1
+				match[4] = +( match[4] ? match[5] + (match[6] || 1) : 2 * ( match[3] === "even" || match[3] === "odd" ) );
+				match[5] = +( ( match[7] + match[8] ) || match[3] === "odd" );
+
+			// other types prohibit arguments
+			} else if ( match[3] ) {
+				Sizzle.error( match[0] );
+			}
+
+			return match;
+		},
+
+		"PSEUDO": function( match ) {
+			var excess,
+				unquoted = !match[6] && match[2];
+
+			if ( matchExpr["CHILD"].test( match[0] ) ) {
+				return null;
+			}
+
+			// Accept quoted arguments as-is
+			if ( match[3] ) {
+				match[2] = match[4] || match[5] || "";
+
+			// Strip excess characters from unquoted arguments
+			} else if ( unquoted && rpseudo.test( unquoted ) &&
+				// Get excess from tokenize (recursively)
+				(excess = tokenize( unquoted, true )) &&
+				// advance to the next closing parenthesis
+				(excess = unquoted.indexOf( ")", unquoted.length - excess ) - unquoted.length) ) {
+
+				// excess is a negative index
+				match[0] = match[0].slice( 0, excess );
+				match[2] = unquoted.slice( 0, excess );
+			}
+
+			// Return only captures needed by the pseudo filter method (type and argument)
+			return match.slice( 0, 3 );
+		}
+	},
+
+	filter: {
+
+		"TAG": function( nodeNameSelector ) {
+			var nodeName = nodeNameSelector.replace( runescape, funescape ).toLowerCase();
+			return nodeNameSelector === "*" ?
+				function() { return true; } :
+				function( elem ) {
+					return elem.nodeName && elem.nodeName.toLowerCase() === nodeName;
+				};
+		},
+
+		"CLASS": function( className ) {
+			var pattern = classCache[ className + " " ];
+
+			return pattern ||
+				(pattern = new RegExp( "(^|" + whitespace + ")" + className + "(" + whitespace + "|$)" )) &&
+				classCache( className, function( elem ) {
+					return pattern.test( typeof elem.className === "string" && elem.className || typeof elem.getAttribute !== "undefined" && elem.getAttribute("class") || "" );
+				});
+		},
+
+		"ATTR": function( name, operator, check ) {
+			return function( elem ) {
+				var result = Sizzle.attr( elem, name );
+
+				if ( result == null ) {
+					return operator === "!=";
+				}
+				if ( !operator ) {
+					return true;
+				}
+
+				result += "";
+
+				return operator === "=" ? result === check :
+					operator === "!=" ? result !== check :
+					operator === "^=" ? check && result.indexOf( check ) === 0 :
+					operator === "*=" ? check && result.indexOf( check ) > -1 :
+					operator === "$=" ? check && result.slice( -check.length ) === check :
+					operator === "~=" ? ( " " + result.replace( rwhitespace, " " ) + " " ).indexOf( check ) > -1 :
+					operator === "|=" ? result === check || result.slice( 0, check.length + 1 ) === check + "-" :
+					false;
+			};
+		},
+
+		"CHILD": function( type, what, argument, first, last ) {
+			var simple = type.slice( 0, 3 ) !== "nth",
+				forward = type.slice( -4 ) !== "last",
+				ofType = what === "of-type";
+
+			return first === 1 && last === 0 ?
+
+				// Shortcut for :nth-*(n)
+				function( elem ) {
+					return !!elem.parentNode;
+				} :
+
+				function( elem, context, xml ) {
+					var cache, uniqueCache, outerCache, node, nodeIndex, start,
+						dir = simple !== forward ? "nextSibling" : "previousSibling",
+						parent = elem.parentNode,
+						name = ofType && elem.nodeName.toLowerCase(),
+						useCache = !xml && !ofType,
+						diff = false;
+
+					if ( parent ) {
+
+						// :(first|last|only)-(child|of-type)
+						if ( simple ) {
+							while ( dir ) {
+								node = elem;
+								while ( (node = node[ dir ]) ) {
+									if ( ofType ?
+										node.nodeName.toLowerCase() === name :
+										node.nodeType === 1 ) {
+
+										return false;
+									}
+								}
+								// Reverse direction for :only-* (if we haven't yet done so)
+								start = dir = type === "only" && !start && "nextSibling";
+							}
+							return true;
+						}
+
+						start = [ forward ? parent.firstChild : parent.lastChild ];
+
+						// non-xml :nth-child(...) stores cache data on `parent`
+						if ( forward && useCache ) {
+
+							// Seek `elem` from a previously-cached index
+
+							// ...in a gzip-friendly way
+							node = parent;
+							outerCache = node[ expando ] || (node[ expando ] = {});
+
+							// Support: IE <9 only
+							// Defend against cloned attroperties (jQuery gh-1709)
+							uniqueCache = outerCache[ node.uniqueID ] ||
+								(outerCache[ node.uniqueID ] = {});
+
+							cache = uniqueCache[ type ] || [];
+							nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ];
+							diff = nodeIndex && cache[ 2 ];
+							node = nodeIndex && parent.childNodes[ nodeIndex ];
+
+							while ( (node = ++nodeIndex && node && node[ dir ] ||
+
+								// Fallback to seeking `elem` from the start
+								(diff = nodeIndex = 0) || start.pop()) ) {
+
+								// When found, cache indexes on `parent` and break
+								if ( node.nodeType === 1 && ++diff && node === elem ) {
+									uniqueCache[ type ] = [ dirruns, nodeIndex, diff ];
+									break;
+								}
+							}
+
+						} else {
+							// Use previously-cached element index if available
+							if ( useCache ) {
+								// ...in a gzip-friendly way
+								node = elem;
+								outerCache = node[ expando ] || (node[ expando ] = {});
+
+								// Support: IE <9 only
+								// Defend against cloned attroperties (jQuery gh-1709)
+								uniqueCache = outerCache[ node.uniqueID ] ||
+									(outerCache[ node.uniqueID ] = {});
+
+								cache = uniqueCache[ type ] || [];
+								nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ];
+								diff = nodeIndex;
+							}
+
+							// xml :nth-child(...)
+							// or :nth-last-child(...) or :nth(-last)?-of-type(...)
+							if ( diff === false ) {
+								// Use the same loop as above to seek `elem` from the start
+								while ( (node = ++nodeIndex && node && node[ dir ] ||
+									(diff = nodeIndex = 0) || start.pop()) ) {
+
+									if ( ( ofType ?
+										node.nodeName.toLowerCase() === name :
+										node.nodeType === 1 ) &&
+										++diff ) {
+
+										// Cache the index of each encountered element
+										if ( useCache ) {
+											outerCache = node[ expando ] || (node[ expando ] = {});
+
+											// Support: IE <9 only
+											// Defend against cloned attroperties (jQuery gh-1709)
+											uniqueCache = outerCache[ node.uniqueID ] ||
+												(outerCache[ node.uniqueID ] = {});
+
+											uniqueCache[ type ] = [ dirruns, diff ];
+										}
+
+										if ( node === elem ) {
+											break;
+										}
+									}
+								}
+							}
+						}
+
+						// Incorporate the offset, then check against cycle size
+						diff -= last;
+						return diff === first || ( diff % first === 0 && diff / first >= 0 );
+					}
+				};
+		},
+
+		"PSEUDO": function( pseudo, argument ) {
+			// pseudo-class names are case-insensitive
+			// http://www.w3.org/TR/selectors/#pseudo-classes
+			// Prioritize by case sensitivity in case custom pseudos are added with uppercase letters
+			// Remember that setFilters inherits from pseudos
+			var args,
+				fn = Expr.pseudos[ pseudo ] || Expr.setFilters[ pseudo.toLowerCase() ] ||
+					Sizzle.error( "unsupported pseudo: " + pseudo );
+
+			// The user may use createPseudo to indicate that
+			// arguments are needed to create the filter function
+			// just as Sizzle does
+			if ( fn[ expando ] ) {
+				return fn( argument );
+			}
+
+			// But maintain support for old signatures
+			if ( fn.length > 1 ) {
+				args = [ pseudo, pseudo, "", argument ];
+				return Expr.setFilters.hasOwnProperty( pseudo.toLowerCase() ) ?
+					markFunction(function( seed, matches ) {
+						var idx,
+							matched = fn( seed, argument ),
+							i = matched.length;
+						while ( i-- ) {
+							idx = indexOf( seed, matched[i] );
+							seed[ idx ] = !( matches[ idx ] = matched[i] );
+						}
+					}) :
+					function( elem ) {
+						return fn( elem, 0, args );
+					};
+			}
+
+			return fn;
+		}
+	},
+
+	pseudos: {
+		// Potentially complex pseudos
+		"not": markFunction(function( selector ) {
+			// Trim the selector passed to compile
+			// to avoid treating leading and trailing
+			// spaces as combinators
+			var input = [],
+				results = [],
+				matcher = compile( selector.replace( rtrim, "$1" ) );
+
+			return matcher[ expando ] ?
+				markFunction(function( seed, matches, context, xml ) {
+					var elem,
+						unmatched = matcher( seed, null, xml, [] ),
+						i = seed.length;
+
+					// Match elements unmatched by `matcher`
+					while ( i-- ) {
+						if ( (elem = unmatched[i]) ) {
+							seed[i] = !(matches[i] = elem);
+						}
+					}
+				}) :
+				function( elem, context, xml ) {
+					input[0] = elem;
+					matcher( input, null, xml, results );
+					// Don't keep the element (issue #299)
+					input[0] = null;
+					return !results.pop();
+				};
+		}),
+
+		"has": markFunction(function( selector ) {
+			return function( elem ) {
+				return Sizzle( selector, elem ).length > 0;
+			};
+		}),
+
+		"contains": markFunction(function( text ) {
+			text = text.replace( runescape, funescape );
+			return function( elem ) {
+				return ( elem.textContent || elem.innerText || getText( elem ) ).indexOf( text ) > -1;
+			};
+		}),
+
+		// "Whether an element is represented by a :lang() selector
+		// is based solely on the element's language value
+		// being equal to the identifier C,
+		// or beginning with the identifier C immediately followed by "-".
+		// The matching of C against the element's language value is performed case-insensitively.
+		// The identifier C does not have to be a valid language name."
+		// http://www.w3.org/TR/selectors/#lang-pseudo
+		"lang": markFunction( function( lang ) {
+			// lang value must be a valid identifier
+			if ( !ridentifier.test(lang || "") ) {
+				Sizzle.error( "unsupported lang: " + lang );
+			}
+			lang = lang.replace( runescape, funescape ).toLowerCase();
+			return function( elem ) {
+				var elemLang;
+				do {
+					if ( (elemLang = documentIsHTML ?
+						elem.lang :
+						elem.getAttribute("xml:lang") || elem.getAttribute("lang")) ) {
+
+						elemLang = elemLang.toLowerCase();
+						return elemLang === lang || elemLang.indexOf( lang + "-" ) === 0;
+					}
+				} while ( (elem = elem.parentNode) && elem.nodeType === 1 );
+				return false;
+			};
+		}),
+
+		// Miscellaneous
+		"target": function( elem ) {
+			var hash = window.location && window.location.hash;
+			return hash && hash.slice( 1 ) === elem.id;
+		},
+
+		"root": function( elem ) {
+			return elem === docElem;
+		},
+
+		"focus": function( elem ) {
+			return elem === document.activeElement && (!document.hasFocus || document.hasFocus()) && !!(elem.type || elem.href || ~elem.tabIndex);
+		},
+
+		// Boolean properties
+		"enabled": function( elem ) {
+			return elem.disabled === false;
+		},
+
+		"disabled": function( elem ) {
+			return elem.disabled === true;
+		},
+
+		"checked": function( elem ) {
+			// In CSS3, :checked should return both checked and selected elements
+			// http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked
+			var nodeName = elem.nodeName.toLowerCase();
+			return (nodeName === "input" && !!elem.checked) || (nodeName === "option" && !!elem.selected);
+		},
+
+		"selected": function( elem ) {
+			// Accessing this property makes selected-by-default
+			// options in Safari work properly
+			if ( elem.parentNode ) {
+				elem.parentNode.selectedIndex;
+			}
+
+			return elem.selected === true;
+		},
+
+		// Contents
+		"empty": function( elem ) {
+			// http://www.w3.org/TR/selectors/#empty-pseudo
+			// :empty is negated by element (1) or content nodes (text: 3; cdata: 4; entity ref: 5),
+			//   but not by others (comment: 8; processing instruction: 7; etc.)
+			// nodeType < 6 works because attributes (2) do not appear as children
+			for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) {
+				if ( elem.nodeType < 6 ) {
+					return false;
+				}
+			}
+			return true;
+		},
+
+		"parent": function( elem ) {
+			return !Expr.pseudos["empty"]( elem );
+		},
+
+		// Element/input types
+		"header": function( elem ) {
+			return rheader.test( elem.nodeName );
+		},
+
+		"input": function( elem ) {
+			return rinputs.test( elem.nodeName );
+		},
+
+		"button": function( elem ) {
+			var name = elem.nodeName.toLowerCase();
+			return name === "input" && elem.type === "button" || name === "button";
+		},
+
+		"text": function( elem ) {
+			var attr;
+			return elem.nodeName.toLowerCase() === "input" &&
+				elem.type === "text" &&
+
+				// Support: IE<8
+				// New HTML5 attribute values (e.g., "search") appear with elem.type === "text"
+				( (attr = elem.getAttribute("type")) == null || attr.toLowerCase() === "text" );
+		},
+
+		// Position-in-collection
+		"first": createPositionalPseudo(function() {
+			return [ 0 ];
+		}),
+
+		"last": createPositionalPseudo(function( matchIndexes, length ) {
+			return [ length - 1 ];
+		}),
+
+		"eq": createPositionalPseudo(function( matchIndexes, length, argument ) {
+			return [ argument < 0 ? argument + length : argument ];
+		}),
+
+		"even": createPositionalPseudo(function( matchIndexes, length ) {
+			var i = 0;
+			for ( ; i < length; i += 2 ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		}),
+
+		"odd": createPositionalPseudo(function( matchIndexes, length ) {
+			var i = 1;
+			for ( ; i < length; i += 2 ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		}),
+
+		"lt": createPositionalPseudo(function( matchIndexes, length, argument ) {
+			var i = argument < 0 ? argument + length : argument;
+			for ( ; --i >= 0; ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		}),
+
+		"gt": createPositionalPseudo(function( matchIndexes, length, argument ) {
+			var i = argument < 0 ? argument + length : argument;
+			for ( ; ++i < length; ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		})
+	}
+};
+
+Expr.pseudos["nth"] = Expr.pseudos["eq"];
+
+// Add button/input type pseudos
+for ( i in { radio: true, checkbox: true, file: true, password: true, image: true } ) {
+	Expr.pseudos[ i ] = createInputPseudo( i );
+}
+for ( i in { submit: true, reset: true } ) {
+	Expr.pseudos[ i ] = createButtonPseudo( i );
+}
+
+// Easy API for creating new setFilters
+function setFilters() {}
+setFilters.prototype = Expr.filters = Expr.pseudos;
+Expr.setFilters = new setFilters();
+
+tokenize = Sizzle.tokenize = function( selector, parseOnly ) {
+	var matched, match, tokens, type,
+		soFar, groups, preFilters,
+		cached = tokenCache[ selector + " " ];
+
+	if ( cached ) {
+		return parseOnly ? 0 : cached.slice( 0 );
+	}
+
+	soFar = selector;
+	groups = [];
+	preFilters = Expr.preFilter;
+
+	while ( soFar ) {
+
+		// Comma and first run
+		if ( !matched || (match = rcomma.exec( soFar )) ) {
+			if ( match ) {
+				// Don't consume trailing commas as valid
+				soFar = soFar.slice( match[0].length ) || soFar;
+			}
+			groups.push( (tokens = []) );
+		}
+
+		matched = false;
+
+		// Combinators
+		if ( (match = rcombinators.exec( soFar )) ) {
+			matched = match.shift();
+			tokens.push({
+				value: matched,
+				// Cast descendant combinators to space
+				type: match[0].replace( rtrim, " " )
+			});
+			soFar = soFar.slice( matched.length );
+		}
+
+		// Filters
+		for ( type in Expr.filter ) {
+			if ( (match = matchExpr[ type ].exec( soFar )) && (!preFilters[ type ] ||
+				(match = preFilters[ type ]( match ))) ) {
+				matched = match.shift();
+				tokens.push({
+					value: matched,
+					type: type,
+					matches: match
+				});
+				soFar = soFar.slice( matched.length );
+			}
+		}
+
+		if ( !matched ) {
+			break;
+		}
+	}
+
+	// Return the length of the invalid excess
+	// if we're just parsing
+	// Otherwise, throw an error or return tokens
+	return parseOnly ?
+		soFar.length :
+		soFar ?
+			Sizzle.error( selector ) :
+			// Cache the tokens
+			tokenCache( selector, groups ).slice( 0 );
+};
+
+function toSelector( tokens ) {
+	var i = 0,
+		len = tokens.length,
+		selector = "";
+	for ( ; i < len; i++ ) {
+		selector += tokens[i].value;
+	}
+	return selector;
+}
+
+function addCombinator( matcher, combinator, base ) {
+	var dir = combinator.dir,
+		checkNonElements = base && dir === "parentNode",
+		doneName = done++;
+
+	return combinator.first ?
+		// Check against closest ancestor/preceding element
+		function( elem, context, xml ) {
+			while ( (elem = elem[ dir ]) ) {
+				if ( elem.nodeType === 1 || checkNonElements ) {
+					return matcher( elem, context, xml );
+				}
+			}
+		} :
+
+		// Check against all ancestor/preceding elements
+		function( elem, context, xml ) {
+			var oldCache, uniqueCache, outerCache,
+				newCache = [ dirruns, doneName ];
+
+			// We can't set arbitrary data on XML nodes, so they don't benefit from combinator caching
+			if ( xml ) {
+				while ( (elem = elem[ dir ]) ) {
+					if ( elem.nodeType === 1 || checkNonElements ) {
+						if ( matcher( elem, context, xml ) ) {
+							return true;
+						}
+					}
+				}
+			} else {
+				while ( (elem = elem[ dir ]) ) {
+					if ( elem.nodeType === 1 || checkNonElements ) {
+						outerCache = elem[ expando ] || (elem[ expando ] = {});
+
+						// Support: IE <9 only
+						// Defend against cloned attroperties (jQuery gh-1709)
+						uniqueCache = outerCache[ elem.uniqueID ] || (outerCache[ elem.uniqueID ] = {});
+
+						if ( (oldCache = uniqueCache[ dir ]) &&
+							oldCache[ 0 ] === dirruns && oldCache[ 1 ] === doneName ) {
+
+							// Assign to newCache so results back-propagate to previous elements
+							return (newCache[ 2 ] = oldCache[ 2 ]);
+						} else {
+							// Reuse newcache so results back-propagate to previous elements
+							uniqueCache[ dir ] = newCache;
+
+							// A match means we're done; a fail means we have to keep checking
+							if ( (newCache[ 2 ] = matcher( elem, context, xml )) ) {
+								return true;
+							}
+						}
+					}
+				}
+			}
+		};
+}
+
+function elementMatcher( matchers ) {
+	return matchers.length > 1 ?
+		function( elem, context, xml ) {
+			var i = matchers.length;
+			while ( i-- ) {
+				if ( !matchers[i]( elem, context, xml ) ) {
+					return false;
+				}
+			}
+			return true;
+		} :
+		matchers[0];
+}
+
+function multipleContexts( selector, contexts, results ) {
+	var i = 0,
+		len = contexts.length;
+	for ( ; i < len; i++ ) {
+		Sizzle( selector, contexts[i], results );
+	}
+	return results;
+}
+
+function condense( unmatched, map, filter, context, xml ) {
+	var elem,
+		newUnmatched = [],
+		i = 0,
+		len = unmatched.length,
+		mapped = map != null;
+
+	for ( ; i < len; i++ ) {
+		if ( (elem = unmatched[i]) ) {
+			if ( !filter || filter( elem, context, xml ) ) {
+				newUnmatched.push( elem );
+				if ( mapped ) {
+					map.push( i );
+				}
+			}
+		}
+	}
+
+	return newUnmatched;
+}
+
+function setMatcher( preFilter, selector, matcher, postFilter, postFinder, postSelector ) {
+	if ( postFilter && !postFilter[ expando ] ) {
+		postFilter = setMatcher( postFilter );
+	}
+	if ( postFinder && !postFinder[ expando ] ) {
+		postFinder = setMatcher( postFinder, postSelector );
+	}
+	return markFunction(function( seed, results, context, xml ) {
+		var temp, i, elem,
+			preMap = [],
+			postMap = [],
+			preexisting = results.length,
+
+			// Get initial elements from seed or context
+			elems = seed || multipleContexts( selector || "*", context.nodeType ? [ context ] : context, [] ),
+
+			// Prefilter to get matcher input, preserving a map for seed-results synchronization
+			matcherIn = preFilter && ( seed || !selector ) ?
+				condense( elems, preMap, preFilter, context, xml ) :
+				elems,
+
+			matcherOut = matcher ?
+				// If we have a postFinder, or filtered seed, or non-seed postFilter or preexisting results,
+				postFinder || ( seed ? preFilter : preexisting || postFilter ) ?
+
+					// ...intermediate processing is necessary
+					[] :
+
+					// ...otherwise use results directly
+					results :
+				matcherIn;
+
+		// Find primary matches
+		if ( matcher ) {
+			matcher( matcherIn, matcherOut, context, xml );
+		}
+
+		// Apply postFilter
+		if ( postFilter ) {
+			temp = condense( matcherOut, postMap );
+			postFilter( temp, [], context, xml );
+
+			// Un-match failing elements by moving them back to matcherIn
+			i = temp.length;
+			while ( i-- ) {
+				if ( (elem = temp[i]) ) {
+					matcherOut[ postMap[i] ] = !(matcherIn[ postMap[i] ] = elem);
+				}
+			}
+		}
+
+		if ( seed ) {
+			if ( postFinder || preFilter ) {
+				if ( postFinder ) {
+					// Get the final matcherOut by condensing this intermediate into postFinder contexts
+					temp = [];
+					i = matcherOut.length;
+					while ( i-- ) {
+						if ( (elem = matcherOut[i]) ) {
+							// Restore matcherIn since elem is not yet a final match
+							temp.push( (matcherIn[i] = elem) );
+						}
+					}
+					postFinder( null, (matcherOut = []), temp, xml );
+				}
+
+				// Move matched elements from seed to results to keep them synchronized
+				i = matcherOut.length;
+				while ( i-- ) {
+					if ( (elem = matcherOut[i]) &&
+						(temp = postFinder ? indexOf( seed, elem ) : preMap[i]) > -1 ) {
+
+						seed[temp] = !(results[temp] = elem);
+					}
+				}
+			}
+
+		// Add elements to results, through postFinder if defined
+		} else {
+			matcherOut = condense(
+				matcherOut === results ?
+					matcherOut.splice( preexisting, matcherOut.length ) :
+					matcherOut
+			);
+			if ( postFinder ) {
+				postFinder( null, results, matcherOut, xml );
+			} else {
+				push.apply( results, matcherOut );
+			}
+		}
+	});
+}
+
+function matcherFromTokens( tokens ) {
+	var checkContext, matcher, j,
+		len = tokens.length,
+		leadingRelative = Expr.relative[ tokens[0].type ],
+		implicitRelative = leadingRelative || Expr.relative[" "],
+		i = leadingRelative ? 1 : 0,
+
+		// The foundational matcher ensures that elements are reachable from top-level context(s)
+		matchContext = addCombinator( function( elem ) {
+			return elem === checkContext;
+		}, implicitRelative, true ),
+		matchAnyContext = addCombinator( function( elem ) {
+			return indexOf( checkContext, elem ) > -1;
+		}, implicitRelative, true ),
+		matchers = [ function( elem, context, xml ) {
+			var ret = ( !leadingRelative && ( xml || context !== outermostContext ) ) || (
+				(checkContext = context).nodeType ?
+					matchContext( elem, context, xml ) :
+					matchAnyContext( elem, context, xml ) );
+			// Avoid hanging onto element (issue #299)
+			checkContext = null;
+			return ret;
+		} ];
+
+	for ( ; i < len; i++ ) {
+		if ( (matcher = Expr.relative[ tokens[i].type ]) ) {
+			matchers = [ addCombinator(elementMatcher( matchers ), matcher) ];
+		} else {
+			matcher = Expr.filter[ tokens[i].type ].apply( null, tokens[i].matches );
+
+			// Return special upon seeing a positional matcher
+			if ( matcher[ expando ] ) {
+				// Find the next relative operator (if any) for proper handling
+				j = ++i;
+				for ( ; j < len; j++ ) {
+					if ( Expr.relative[ tokens[j].type ] ) {
+						break;
+					}
+				}
+				return setMatcher(
+					i > 1 && elementMatcher( matchers ),
+					i > 1 && toSelector(
+						// If the preceding token was a descendant combinator, insert an implicit any-element `*`
+						tokens.slice( 0, i - 1 ).concat({ value: tokens[ i - 2 ].type === " " ? "*" : "" })
+					).replace( rtrim, "$1" ),
+					matcher,
+					i < j && matcherFromTokens( tokens.slice( i, j ) ),
+					j < len && matcherFromTokens( (tokens = tokens.slice( j )) ),
+					j < len && toSelector( tokens )
+				);
+			}
+			matchers.push( matcher );
+		}
+	}
+
+	return elementMatcher( matchers );
+}
+
+function matcherFromGroupMatchers( elementMatchers, setMatchers ) {
+	var bySet = setMatchers.length > 0,
+		byElement = elementMatchers.length > 0,
+		superMatcher = function( seed, context, xml, results, outermost ) {
+			var elem, j, matcher,
+				matchedCount = 0,
+				i = "0",
+				unmatched = seed && [],
+				setMatched = [],
+				contextBackup = outermostContext,
+				// We must always have either seed elements or outermost context
+				elems = seed || byElement && Expr.find["TAG"]( "*", outermost ),
+				// Use integer dirruns iff this is the outermost matcher
+				dirrunsUnique = (dirruns += contextBackup == null ? 1 : Math.random() || 0.1),
+				len = elems.length;
+
+			if ( outermost ) {
+				outermostContext = context === document || context || outermost;
+			}
+
+			// Add elements passing elementMatchers directly to results
+			// Support: IE<9, Safari
+			// Tolerate NodeList properties (IE: "length"; Safari: <number>) matching elements by id
+			for ( ; i !== len && (elem = elems[i]) != null; i++ ) {
+				if ( byElement && elem ) {
+					j = 0;
+					if ( !context && elem.ownerDocument !== document ) {
+						setDocument( elem );
+						xml = !documentIsHTML;
+					}
+					while ( (matcher = elementMatchers[j++]) ) {
+						if ( matcher( elem, context || document, xml) ) {
+							results.push( elem );
+							break;
+						}
+					}
+					if ( outermost ) {
+						dirruns = dirrunsUnique;
+					}
+				}
+
+				// Track unmatched elements for set filters
+				if ( bySet ) {
+					// They will have gone through all possible matchers
+					if ( (elem = !matcher && elem) ) {
+						matchedCount--;
+					}
+
+					// Lengthen the array for every element, matched or not
+					if ( seed ) {
+						unmatched.push( elem );
+					}
+				}
+			}
+
+			// `i` is now the count of elements visited above, and adding it to `matchedCount`
+			// makes the latter nonnegative.
+			matchedCount += i;
+
+			// Apply set filters to unmatched elements
+			// NOTE: This can be skipped if there are no unmatched elements (i.e., `matchedCount`
+			// equals `i`), unless we didn't visit _any_ elements in the above loop because we have
+			// no element matchers and no seed.
+			// Incrementing an initially-string "0" `i` allows `i` to remain a string only in that
+			// case, which will result in a "00" `matchedCount` that differs from `i` but is also
+			// numerically zero.
+			if ( bySet && i !== matchedCount ) {
+				j = 0;
+				while ( (matcher = setMatchers[j++]) ) {
+					matcher( unmatched, setMatched, context, xml );
+				}
+
+				if ( seed ) {
+					// Reintegrate element matches to eliminate the need for sorting
+					if ( matchedCount > 0 ) {
+						while ( i-- ) {
+							if ( !(unmatched[i] || setMatched[i]) ) {
+								setMatched[i] = pop.call( results );
+							}
+						}
+					}
+
+					// Discard index placeholder values to get only actual matches
+					setMatched = condense( setMatched );
+				}
+
+				// Add matches to results
+				push.apply( results, setMatched );
+
+				// Seedless set matches succeeding multiple successful matchers stipulate sorting
+				if ( outermost && !seed && setMatched.length > 0 &&
+					( matchedCount + setMatchers.length ) > 1 ) {
+
+					Sizzle.uniqueSort( results );
+				}
+			}
+
+			// Override manipulation of globals by nested matchers
+			if ( outermost ) {
+				dirruns = dirrunsUnique;
+				outermostContext = contextBackup;
+			}
+
+			return unmatched;
+		};
+
+	return bySet ?
+		markFunction( superMatcher ) :
+		superMatcher;
+}
+
+compile = Sizzle.compile = function( selector, match /* Internal Use Only */ ) {
+	var i,
+		setMatchers = [],
+		elementMatchers = [],
+		cached = compilerCache[ selector + " " ];
+
+	if ( !cached ) {
+		// Generate a function of recursive functions that can be used to check each element
+		if ( !match ) {
+			match = tokenize( selector );
+		}
+		i = match.length;
+		while ( i-- ) {
+			cached = matcherFromTokens( match[i] );
+			if ( cached[ expando ] ) {
+				setMatchers.push( cached );
+			} else {
+				elementMatchers.push( cached );
+			}
+		}
+
+		// Cache the compiled function
+		cached = compilerCache( selector, matcherFromGroupMatchers( elementMatchers, setMatchers ) );
+
+		// Save selector and tokenization
+		cached.selector = selector;
+	}
+	return cached;
+};
+
+/**
+ * A low-level selection function that works with Sizzle's compiled
+ *  selector functions
+ * @param {String|Function} selector A selector or a pre-compiled
+ *  selector function built with Sizzle.compile
+ * @param {Element} context
+ * @param {Array} [results]
+ * @param {Array} [seed] A set of elements to match against
+ */
+select = Sizzle.select = function( selector, context, results, seed ) {
+	var i, tokens, token, type, find,
+		compiled = typeof selector === "function" && selector,
+		match = !seed && tokenize( (selector = compiled.selector || selector) );
+
+	results = results || [];
+
+	// Try to minimize operations if there is only one selector in the list and no seed
+	// (the latter of which guarantees us context)
+	if ( match.length === 1 ) {
+
+		// Reduce context if the leading compound selector is an ID
+		tokens = match[0] = match[0].slice( 0 );
+		if ( tokens.length > 2 && (token = tokens[0]).type === "ID" &&
+				support.getById && context.nodeType === 9 && documentIsHTML &&
+				Expr.relative[ tokens[1].type ] ) {
+
+			context = ( Expr.find["ID"]( token.matches[0].replace(runescape, funescape), context ) || [] )[0];
+			if ( !context ) {
+				return results;
+
+			// Precompiled matchers will still verify ancestry, so step up a level
+			} else if ( compiled ) {
+				context = context.parentNode;
+			}
+
+			selector = selector.slice( tokens.shift().value.length );
+		}
+
+		// Fetch a seed set for right-to-left matching
+		i = matchExpr["needsContext"].test( selector ) ? 0 : tokens.length;
+		while ( i-- ) {
+			token = tokens[i];
+
+			// Abort if we hit a combinator
+			if ( Expr.relative[ (type = token.type) ] ) {
+				break;
+			}
+			if ( (find = Expr.find[ type ]) ) {
+				// Search, expanding context for leading sibling combinators
+				if ( (seed = find(
+					token.matches[0].replace( runescape, funescape ),
+					rsibling.test( tokens[0].type ) && testContext( context.parentNode ) || context
+				)) ) {
+
+					// If seed is empty or no tokens remain, we can return early
+					tokens.splice( i, 1 );
+					selector = seed.length && toSelector( tokens );
+					if ( !selector ) {
+						push.apply( results, seed );
+						return results;
+					}
+
+					break;
+				}
+			}
+		}
+	}
+
+	// Compile and execute a filtering function if one is not provided
+	// Provide `match` to avoid retokenization if we modified the selector above
+	( compiled || compile( selector, match ) )(
+		seed,
+		context,
+		!documentIsHTML,
+		results,
+		!context || rsibling.test( selector ) && testContext( context.parentNode ) || context
+	);
+	return results;
+};
+
+// One-time assignments
+
+// Sort stability
+support.sortStable = expando.split("").sort( sortOrder ).join("") === expando;
+
+// Support: Chrome 14-35+
+// Always assume duplicates if they aren't passed to the comparison function
+support.detectDuplicates = !!hasDuplicate;
+
+// Initialize against the default document
+setDocument();
+
+// Support: Webkit<537.32 - Safari 6.0.3/Chrome 25 (fixed in Chrome 27)
+// Detached nodes confoundingly follow *each other*
+support.sortDetached = assert(function( div1 ) {
+	// Should return 1, but returns 4 (following)
+	return div1.compareDocumentPosition( document.createElement("div") ) & 1;
+});
+
+// Support: IE<8
+// Prevent attribute/property "interpolation"
+// http://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx
+if ( !assert(function( div ) {
+	div.innerHTML = "<a href='#'></a>";
+	return div.firstChild.getAttribute("href") === "#" ;
+}) ) {
+	addHandle( "type|href|height|width", function( elem, name, isXML ) {
+		if ( !isXML ) {
+			return elem.getAttribute( name, name.toLowerCase() === "type" ? 1 : 2 );
+		}
+	});
+}
+
+// Support: IE<9
+// Use defaultValue in place of getAttribute("value")
+if ( !support.attributes || !assert(function( div ) {
+	div.innerHTML = "<input/>";
+	div.firstChild.setAttribute( "value", "" );
+	return div.firstChild.getAttribute( "value" ) === "";
+}) ) {
+	addHandle( "value", function( elem, name, isXML ) {
+		if ( !isXML && elem.nodeName.toLowerCase() === "input" ) {
+			return elem.defaultValue;
+		}
+	});
+}
+
+// Support: IE<9
+// Use getAttributeNode to fetch booleans when getAttribute lies
+if ( !assert(function( div ) {
+	return div.getAttribute("disabled") == null;
+}) ) {
+	addHandle( booleans, function( elem, name, isXML ) {
+		var val;
+		if ( !isXML ) {
+			return elem[ name ] === true ? name.toLowerCase() :
+					(val = elem.getAttributeNode( name )) && val.specified ?
+					val.value :
+				null;
+		}
+	});
+}
+
+return Sizzle;
+
+})( window );
+
+
+
+jQuery.find = Sizzle;
+jQuery.expr = Sizzle.selectors;
+jQuery.expr[ ":" ] = jQuery.expr.pseudos;
+jQuery.uniqueSort = jQuery.unique = Sizzle.uniqueSort;
+jQuery.text = Sizzle.getText;
+jQuery.isXMLDoc = Sizzle.isXML;
+jQuery.contains = Sizzle.contains;
+
+
+
+var dir = function( elem, dir, until ) {
+	var matched = [],
+		truncate = until !== undefined;
+
+	while ( ( elem = elem[ dir ] ) && elem.nodeType !== 9 ) {
+		if ( elem.nodeType === 1 ) {
+			if ( truncate && jQuery( elem ).is( until ) ) {
+				break;
+			}
+			matched.push( elem );
+		}
+	}
+	return matched;
+};
+
+
+var siblings = function( n, elem ) {
+	var matched = [];
+
+	for ( ; n; n = n.nextSibling ) {
+		if ( n.nodeType === 1 && n !== elem ) {
+			matched.push( n );
+		}
+	}
+
+	return matched;
+};
+
+
+var rneedsContext = jQuery.expr.match.needsContext;
+
+var rsingleTag = ( /^<([\w-]+)\s*\/?>(?:<\/\1>|)$/ );
+
+
+
+var risSimple = /^.[^:#\[\.,]*$/;
+
+// Implement the identical functionality for filter and not
+function winnow( elements, qualifier, not ) {
+	if ( jQuery.isFunction( qualifier ) ) {
+		return jQuery.grep( elements, function( elem, i ) {
+			/* jshint -W018 */
+			return !!qualifier.call( elem, i, elem ) !== not;
+		} );
+
+	}
+
+	if ( qualifier.nodeType ) {
+		return jQuery.grep( elements, function( elem ) {
+			return ( elem === qualifier ) !== not;
+		} );
+
+	}
+
+	if ( typeof qualifier === "string" ) {
+		if ( risSimple.test( qualifier ) ) {
+			return jQuery.filter( qualifier, elements, not );
+		}
+
+		qualifier = jQuery.filter( qualifier, elements );
+	}
+
+	return jQuery.grep( elements, function( elem ) {
+		return ( jQuery.inArray( elem, qualifier ) > -1 ) !== not;
+	} );
+}
+
+jQuery.filter = function( expr, elems, not ) {
+	var elem = elems[ 0 ];
+
+	if ( not ) {
+		expr = ":not(" + expr + ")";
+	}
+
+	return elems.length === 1 && elem.nodeType === 1 ?
+		jQuery.find.matchesSelector( elem, expr ) ? [ elem ] : [] :
+		jQuery.find.matches( expr, jQuery.grep( elems, function( elem ) {
+			return elem.nodeType === 1;
+		} ) );
+};
+
+jQuery.fn.extend( {
+	find: function( selector ) {
+		var i,
+			ret = [],
+			self = this,
+			len = self.length;
+
+		if ( typeof selector !== "string" ) {
+			return this.pushStack( jQuery( selector ).filter( function() {
+				for ( i = 0; i < len; i++ ) {
+					if ( jQuery.contains( self[ i ], this ) ) {
+						return true;
+					}
+				}
+			} ) );
+		}
+
+		for ( i = 0; i < len; i++ ) {
+			jQuery.find( selector, self[ i ], ret );
+		}
+
+		// Needed because $( selector, context ) becomes $( context ).find( selector )
+		ret = this.pushStack( len > 1 ? jQuery.unique( ret ) : ret );
+		ret.selector = this.selector ? this.selector + " " + selector : selector;
+		return ret;
+	},
+	filter: function( selector ) {
+		return this.pushStack( winnow( this, selector || [], false ) );
+	},
+	not: function( selector ) {
+		return this.pushStack( winnow( this, selector || [], true ) );
+	},
+	is: function( selector ) {
+		return !!winnow(
+			this,
+
+			// If this is a positional/relative selector, check membership in the returned set
+			// so $("p:first").is("p:last") won't return true for a doc with two "p".
+			typeof selector === "string" && rneedsContext.test( selector ) ?
+				jQuery( selector ) :
+				selector || [],
+			false
+		).length;
+	}
+} );
+
+
+// Initialize a jQuery object
+
+
+// A central reference to the root jQuery(document)
+var rootjQuery,
+
+	// A simple way to check for HTML strings
+	// Prioritize #id over <tag> to avoid XSS via location.hash (#9521)
+	// Strict HTML recognition (#11290: must start with <)
+	rquickExpr = /^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]*))$/,
+
+	init = jQuery.fn.init = function( selector, context, root ) {
+		var match, elem;
+
+		// HANDLE: $(""), $(null), $(undefined), $(false)
+		if ( !selector ) {
+			return this;
+		}
+
+		// init accepts an alternate rootjQuery
+		// so migrate can support jQuery.sub (gh-2101)
+		root = root || rootjQuery;
+
+		// Handle HTML strings
+		if ( typeof selector === "string" ) {
+			if ( selector.charAt( 0 ) === "<" &&
+				selector.charAt( selector.length - 1 ) === ">" &&
+				selector.length >= 3 ) {
+
+				// Assume that strings that start and end with <> are HTML and skip the regex check
+				match = [ null, selector, null ];
+
+			} else {
+				match = rquickExpr.exec( selector );
+			}
+
+			// Match html or make sure no context is specified for #id
+			if ( match && ( match[ 1 ] || !context ) ) {
+
+				// HANDLE: $(html) -> $(array)
+				if ( match[ 1 ] ) {
+					context = context instanceof jQuery ? context[ 0 ] : context;
+
+					// scripts is true for back-compat
+					// Intentionally let the error be thrown if parseHTML is not present
+					jQuery.merge( this, jQuery.parseHTML(
+						match[ 1 ],
+						context && context.nodeType ? context.ownerDocument || context : document,
+						true
+					) );
+
+					// HANDLE: $(html, props)
+					if ( rsingleTag.test( match[ 1 ] ) && jQuery.isPlainObject( context ) ) {
+						for ( match in context ) {
+
+							// Properties of context are called as methods if possible
+							if ( jQuery.isFunction( this[ match ] ) ) {
+								this[ match ]( context[ match ] );
+
+							// ...and otherwise set as attributes
+							} else {
+								this.attr( match, context[ match ] );
+							}
+						}
+					}
+
+					return this;
+
+				// HANDLE: $(#id)
+				} else {
+					elem = document.getElementById( match[ 2 ] );
+
+					// Check parentNode to catch when Blackberry 4.6 returns
+					// nodes that are no longer in the document #6963
+					if ( elem && elem.parentNode ) {
+
+						// Handle the case where IE and Opera return items
+						// by name instead of ID
+						if ( elem.id !== match[ 2 ] ) {
+							return rootjQuery.find( selector );
+						}
+
+						// Otherwise, we inject the element directly into the jQuery object
+						this.length = 1;
+						this[ 0 ] = elem;
+					}
+
+					this.context = document;
+					this.selector = selector;
+					return this;
+				}
+
+			// HANDLE: $(expr, $(...))
+			} else if ( !context || context.jquery ) {
+				return ( context || root ).find( selector );
+
+			// HANDLE: $(expr, context)
+			// (which is just equivalent to: $(context).find(expr)
+			} else {
+				return this.constructor( context ).find( selector );
+			}
+
+		// HANDLE: $(DOMElement)
+		} else if ( selector.nodeType ) {
+			this.context = this[ 0 ] = selector;
+			this.length = 1;
+			return this;
+
+		// HANDLE: $(function)
+		// Shortcut for document ready
+		} else if ( jQuery.isFunction( selector ) ) {
+			return typeof root.ready !== "undefined" ?
+				root.ready( selector ) :
+
+				// Execute immediately if ready is not present
+				selector( jQuery );
+		}
+
+		if ( selector.selector !== undefined ) {
+			this.selector = selector.selector;
+			this.context = selector.context;
+		}
+
+		return jQuery.makeArray( selector, this );
+	};
+
+// Give the init function the jQuery prototype for later instantiation
+init.prototype = jQuery.fn;
+
+// Initialize central reference
+rootjQuery = jQuery( document );
+
+
+var rparentsprev = /^(?:parents|prev(?:Until|All))/,
+
+	// methods guaranteed to produce a unique set when starting from a unique set
+	guaranteedUnique = {
+		children: true,
+		contents: true,
+		next: true,
+		prev: true
+	};
+
+jQuery.fn.extend( {
+	has: function( target ) {
+		var i,
+			targets = jQuery( target, this ),
+			len = targets.length;
+
+		return this.filter( function() {
+			for ( i = 0; i < len; i++ ) {
+				if ( jQuery.contains( this, targets[ i ] ) ) {
+					return true;
+				}
+			}
+		} );
+	},
+
+	closest: function( selectors, context ) {
+		var cur,
+			i = 0,
+			l = this.length,
+			matched = [],
+			pos = rneedsContext.test( selectors ) || typeof selectors !== "string" ?
+				jQuery( selectors, context || this.context ) :
+				0;
+
+		for ( ; i < l; i++ ) {
+			for ( cur = this[ i ]; cur && cur !== context; cur = cur.parentNode ) {
+
+				// Always skip document fragments
+				if ( cur.nodeType < 11 && ( pos ?
+					pos.index( cur ) > -1 :
+
+					// Don't pass non-elements to Sizzle
+					cur.nodeType === 1 &&
+						jQuery.find.matchesSelector( cur, selectors ) ) ) {
+
+					matched.push( cur );
+					break;
+				}
+			}
+		}
+
+		return this.pushStack( matched.length > 1 ? jQuery.uniqueSort( matched ) : matched );
+	},
+
+	// Determine the position of an element within
+	// the matched set of elements
+	index: function( elem ) {
+
+		// No argument, return index in parent
+		if ( !elem ) {
+			return ( this[ 0 ] && this[ 0 ].parentNode ) ? this.first().prevAll().length : -1;
+		}
+
+		// index in selector
+		if ( typeof elem === "string" ) {
+			return jQuery.inArray( this[ 0 ], jQuery( elem ) );
+		}
+
+		// Locate the position of the desired element
+		return jQuery.inArray(
+
+			// If it receives a jQuery object, the first element is used
+			elem.jquery ? elem[ 0 ] : elem, this );
+	},
+
+	add: function( selector, context ) {
+		return this.pushStack(
+			jQuery.uniqueSort(
+				jQuery.merge( this.get(), jQuery( selector, context ) )
+			)
+		);
+	},
+
+	addBack: function( selector ) {
+		return this.add( selector == null ?
+			this.prevObject : this.prevObject.filter( selector )
+		);
+	}
+} );
+
+function sibling( cur, dir ) {
+	do {
+		cur = cur[ dir ];
+	} while ( cur && cur.nodeType !== 1 );
+
+	return cur;
+}
+
+jQuery.each( {
+	parent: function( elem ) {
+		var parent = elem.parentNode;
+		return parent && parent.nodeType !== 11 ? parent : null;
+	},
+	parents: function( elem ) {
+		return dir( elem, "parentNode" );
+	},
+	parentsUntil: function( elem, i, until ) {
+		return dir( elem, "parentNode", until );
+	},
+	next: function( elem ) {
+		return sibling( elem, "nextSibling" );
+	},
+	prev: function( elem ) {
+		return sibling( elem, "previousSibling" );
+	},
+	nextAll: function( elem ) {
+		return dir( elem, "nextSibling" );
+	},
+	prevAll: function( elem ) {
+		return dir( elem, "previousSibling" );
+	},
+	nextUntil: function( elem, i, until ) {
+		return dir( elem, "nextSibling", until );
+	},
+	prevUntil: function( elem, i, until ) {
+		return dir( elem, "previousSibling", until );
+	},
+	siblings: function( elem ) {
+		return siblings( ( elem.parentNode || {} ).firstChild, elem );
+	},
+	children: function( elem ) {
+		return siblings( elem.firstChild );
+	},
+	contents: function( elem ) {
+		return jQuery.nodeName( elem, "iframe" ) ?
+			elem.contentDocument || elem.contentWindow.document :
+			jQuery.merge( [], elem.childNodes );
+	}
+}, function( name, fn ) {
+	jQuery.fn[ name ] = function( until, selector ) {
+		var ret = jQuery.map( this, fn, until );
+
+		if ( name.slice( -5 ) !== "Until" ) {
+			selector = until;
+		}
+
+		if ( selector && typeof selector === "string" ) {
+			ret = jQuery.filter( selector, ret );
+		}
+
+		if ( this.length > 1 ) {
+
+			// Remove duplicates
+			if ( !guaranteedUnique[ name ] ) {
+				ret = jQuery.uniqueSort( ret );
+			}
+
+			// Reverse order for parents* and prev-derivatives
+			if ( rparentsprev.test( name ) ) {
+				ret = ret.reverse();
+			}
+		}
+
+		return this.pushStack( ret );
+	};
+} );
+var rnotwhite = ( /\S+/g );
+
+
+
+// Convert String-formatted options into Object-formatted ones
+function createOptions( options ) {
+	var object = {};
+	jQuery.each( options.match( rnotwhite ) || [], function( _, flag ) {
+		object[ flag ] = true;
+	} );
+	return object;
+}
+
+/*
+ * Create a callback list using the following parameters:
+ *
+ *	options: an optional list of space-separated options that will change how
+ *			the callback list behaves or a more traditional option object
+ *
+ * By default a callback list will act like an event callback list and can be
+ * "fired" multiple times.
+ *
+ * Possible options:
+ *
+ *	once:			will ensure the callback list can only be fired once (like a Deferred)
+ *
+ *	memory:			will keep track of previous values and will call any callback added
+ *					after the list has been fired right away with the latest "memorized"
+ *					values (like a Deferred)
+ *
+ *	unique:			will ensure a callback can only be added once (no duplicate in the list)
+ *
+ *	stopOnFalse:	interrupt callings when a callback returns false
+ *
+ */
+jQuery.Callbacks = function( options ) {
+
+	// Convert options from String-formatted to Object-formatted if needed
+	// (we check in cache first)
+	options = typeof options === "string" ?
+		createOptions( options ) :
+		jQuery.extend( {}, options );
+
+	var // Flag to know if list is currently firing
+		firing,
+
+		// Last fire value for non-forgettable lists
+		memory,
+
+		// Flag to know if list was already fired
+		fired,
+
+		// Flag to prevent firing
+		locked,
+
+		// Actual callback list
+		list = [],
+
+		// Queue of execution data for repeatable lists
+		queue = [],
+
+		// Index of currently firing callback (modified by add/remove as needed)
+		firingIndex = -1,
+
+		// Fire callbacks
+		fire = function() {
+
+			// Enforce single-firing
+			locked = options.once;
+
+			// Execute callbacks for all pending executions,
+			// respecting firingIndex overrides and runtime changes
+			fired = firing = true;
+			for ( ; queue.length; firingIndex = -1 ) {
+				memory = queue.shift();
+				while ( ++firingIndex < list.length ) {
+
+					// Run callback and check for early termination
+					if ( list[ firingIndex ].apply( memory[ 0 ], memory[ 1 ] ) === false &&
+						options.stopOnFalse ) {
+
+						// Jump to end and forget the data so .add doesn't re-fire
+						firingIndex = list.length;
+						memory = false;
+					}
+				}
+			}
+
+			// Forget the data if we're done with it
+			if ( !options.memory ) {
+				memory = false;
+			}
+
+			firing = false;
+
+			// Clean up if we're done firing for good
+			if ( locked ) {
+
+				// Keep an empty list if we have data for future add calls
+				if ( memory ) {
+					list = [];
+
+				// Otherwise, this object is spent
+				} else {
+					list = "";
+				}
+			}
+		},
+
+		// Actual Callbacks object
+		self = {
+
+			// Add a callback or a collection of callbacks to the list
+			add: function() {
+				if ( list ) {
+
+					// If we have memory from a past run, we should fire after adding
+					if ( memory && !firing ) {
+						firingIndex = list.length - 1;
+						queue.push( memory );
+					}
+
+					( function add( args ) {
+						jQuery.each( args, function( _, arg ) {
+							if ( jQuery.isFunction( arg ) ) {
+								if ( !options.unique || !self.has( arg ) ) {
+									list.push( arg );
+								}
+							} else if ( arg && arg.length && jQuery.type( arg ) !== "string" ) {
+
+								// Inspect recursively
+								add( arg );
+							}
+						} );
+					} )( arguments );
+
+					if ( memory && !firing ) {
+						fire();
+					}
+				}
+				return this;
+			},
+
+			// Remove a callback from the list
+			remove: function() {
+				jQuery.each( arguments, function( _, arg ) {
+					var index;
+					while ( ( index = jQuery.inArray( arg, list, index ) ) > -1 ) {
+						list.splice( index, 1 );
+
+						// Handle firing indexes
+						if ( index <= firingIndex ) {
+							firingIndex--;
+						}
+					}
+				} );
+				return this;
+			},
+
+			// Check if a given callback is in the list.
+			// If no argument is given, return whether or not list has callbacks attached.
+			has: function( fn ) {
+				return fn ?
+					jQuery.inArray( fn, list ) > -1 :
+					list.length > 0;
+			},
+
+			// Remove all callbacks from the list
+			empty: function() {
+				if ( list ) {
+					list = [];
+				}
+				return this;
+			},
+
+			// Disable .fire and .add
+			// Abort any current/pending executions
+			// Clear all callbacks and values
+			disable: function() {
+				locked = queue = [];
+				list = memory = "";
+				return this;
+			},
+			disabled: function() {
+				return !list;
+			},
+
+			// Disable .fire
+			// Also disable .add unless we have memory (since it would have no effect)
+			// Abort any pending executions
+			lock: function() {
+				locked = true;
+				if ( !memory ) {
+					self.disable();
+				}
+				return this;
+			},
+			locked: function() {
+				return !!locked;
+			},
+
+			// Call all callbacks with the given context and arguments
+			fireWith: function( context, args ) {
+				if ( !locked ) {
+					args = args || [];
+					args = [ context, args.slice ? args.slice() : args ];
+					queue.push( args );
+					if ( !firing ) {
+						fire();
+					}
+				}
+				return this;
+			},
+
+			// Call all the callbacks with the given arguments
+			fire: function() {
+				self.fireWith( this, arguments );
+				return this;
+			},
+
+			// To know if the callbacks have already been called at least once
+			fired: function() {
+				return !!fired;
+			}
+		};
+
+	return self;
+};
+
+
+jQuery.extend( {
+
+	Deferred: function( func ) {
+		var tuples = [
+
+				// action, add listener, listener list, final state
+				[ "resolve", "done", jQuery.Callbacks( "once memory" ), "resolved" ],
+				[ "reject", "fail", jQuery.Callbacks( "once memory" ), "rejected" ],
+				[ "notify", "progress", jQuery.Callbacks( "memory" ) ]
+			],
+			state = "pending",
+			promise = {
+				state: function() {
+					return state;
+				},
+				always: function() {
+					deferred.done( arguments ).fail( arguments );
+					return this;
+				},
+				then: function( /* fnDone, fnFail, fnProgress */ ) {
+					var fns = arguments;
+					return jQuery.Deferred( function( newDefer ) {
+						jQuery.each( tuples, function( i, tuple ) {
+							var fn = jQuery.isFunction( fns[ i ] ) && fns[ i ];
+
+							// deferred[ done | fail | progress ] for forwarding actions to newDefer
+							deferred[ tuple[ 1 ] ]( function() {
+								var returned = fn && fn.apply( this, arguments );
+								if ( returned && jQuery.isFunction( returned.promise ) ) {
+									returned.promise()
+										.progress( newDefer.notify )
+										.done( newDefer.resolve )
+										.fail( newDefer.reject );
+								} else {
+									newDefer[ tuple[ 0 ] + "With" ](
+										this === promise ? newDefer.promise() : this,
+										fn ? [ returned ] : arguments
+									);
+								}
+							} );
+						} );
+						fns = null;
+					} ).promise();
+				},
+
+				// Get a promise for this deferred
+				// If obj is provided, the promise aspect is added to the object
+				promise: function( obj ) {
+					return obj != null ? jQuery.extend( obj, promise ) : promise;
+				}
+			},
+			deferred = {};
+
+		// Keep pipe for back-compat
+		promise.pipe = promise.then;
+
+		// Add list-specific methods
+		jQuery.each( tuples, function( i, tuple ) {
+			var list = tuple[ 2 ],
+				stateString = tuple[ 3 ];
+
+			// promise[ done | fail | progress ] = list.add
+			promise[ tuple[ 1 ] ] = list.add;
+
+			// Handle state
+			if ( stateString ) {
+				list.add( function() {
+
+					// state = [ resolved | rejected ]
+					state = stateString;
+
+				// [ reject_list | resolve_list ].disable; progress_list.lock
+				}, tuples[ i ^ 1 ][ 2 ].disable, tuples[ 2 ][ 2 ].lock );
+			}
+
+			// deferred[ resolve | reject | notify ]
+			deferred[ tuple[ 0 ] ] = function() {
+				deferred[ tuple[ 0 ] + "With" ]( this === deferred ? promise : this, arguments );
+				return this;
+			};
+			deferred[ tuple[ 0 ] + "With" ] = list.fireWith;
+		} );
+
+		// Make the deferred a promise
+		promise.promise( deferred );
+
+		// Call given func if any
+		if ( func ) {
+			func.call( deferred, deferred );
+		}
+
+		// All done!
+		return deferred;
+	},
+
+	// Deferred helper
+	when: function( subordinate /* , ..., subordinateN */ ) {
+		var i = 0,
+			resolveValues = slice.call( arguments ),
+			length = resolveValues.length,
+
+			// the count of uncompleted subordinates
+			remaining = length !== 1 ||
+				( subordinate && jQuery.isFunction( subordinate.promise ) ) ? length : 0,
+
+			// the master Deferred.
+			// If resolveValues consist of only a single Deferred, just use that.
+			deferred = remaining === 1 ? subordinate : jQuery.Deferred(),
+
+			// Update function for both resolve and progress values
+			updateFunc = function( i, contexts, values ) {
+				return function( value ) {
+					contexts[ i ] = this;
+					values[ i ] = arguments.length > 1 ? slice.call( arguments ) : value;
+					if ( values === progressValues ) {
+						deferred.notifyWith( contexts, values );
+
+					} else if ( !( --remaining ) ) {
+						deferred.resolveWith( contexts, values );
+					}
+				};
+			},
+
+			progressValues, progressContexts, resolveContexts;
+
+		// add listeners to Deferred subordinates; treat others as resolved
+		if ( length > 1 ) {
+			progressValues = new Array( length );
+			progressContexts = new Array( length );
+			resolveContexts = new Array( length );
+			for ( ; i < length; i++ ) {
+				if ( resolveValues[ i ] && jQuery.isFunction( resolveValues[ i ].promise ) ) {
+					resolveValues[ i ].promise()
+						.progress( updateFunc( i, progressContexts, progressValues ) )
+						.done( updateFunc( i, resolveContexts, resolveValues ) )
+						.fail( deferred.reject );
+				} else {
+					--remaining;
+				}
+			}
+		}
+
+		// if we're not waiting on anything, resolve the master
+		if ( !remaining ) {
+			deferred.resolveWith( resolveContexts, resolveValues );
+		}
+
+		return deferred.promise();
+	}
+} );
+
+
+// The deferred used on DOM ready
+var readyList;
+
+jQuery.fn.ready = function( fn ) {
+
+	// Add the callback
+	jQuery.ready.promise().done( fn );
+
+	return this;
+};
+
+jQuery.extend( {
+
+	// Is the DOM ready to be used? Set to true once it occurs.
+	isReady: false,
+
+	// A counter to track how many items to wait for before
+	// the ready event fires. See #6781
+	readyWait: 1,
+
+	// Hold (or release) the ready event
+	holdReady: function( hold ) {
+		if ( hold ) {
+			jQuery.readyWait++;
+		} else {
+			jQuery.ready( true );
+		}
+	},
+
+	// Handle when the DOM is ready
+	ready: function( wait ) {
+
+		// Abort if there are pending holds or we're already ready
+		if ( wait === true ? --jQuery.readyWait : jQuery.isReady ) {
+			return;
+		}
+
+		// Remember that the DOM is ready
+		jQuery.isReady = true;
+
+		// If a normal DOM Ready event fired, decrement, and wait if need be
+		if ( wait !== true && --jQuery.readyWait > 0 ) {
+			return;
+		}
+
+		// If there are functions bound, to execute
+		readyList.resolveWith( document, [ jQuery ] );
+
+		// Trigger any bound ready events
+		if ( jQuery.fn.triggerHandler ) {
+			jQuery( document ).triggerHandler( "ready" );
+			jQuery( document ).off( "ready" );
+		}
+	}
+} );
+
+/**
+ * Clean-up method for dom ready events
+ */
+function detach() {
+	if ( document.addEventListener ) {
+		document.removeEventListener( "DOMContentLoaded", completed );
+		window.removeEventListener( "load", completed );
+
+	} else {
+		document.detachEvent( "onreadystatechange", completed );
+		window.detachEvent( "onload", completed );
+	}
+}
+
+/**
+ * The ready event handler and self cleanup method
+ */
+function completed() {
+
+	// readyState === "complete" is good enough for us to call the dom ready in oldIE
+	if ( document.addEventListener ||
+		window.event.type === "load" ||
+		document.readyState === "complete" ) {
+
+		detach();
+		jQuery.ready();
+	}
+}
+
+jQuery.ready.promise = function( obj ) {
+	if ( !readyList ) {
+
+		readyList = jQuery.Deferred();
+
+		// Catch cases where $(document).ready() is called
+		// after the browser event has already occurred.
+		// Support: IE6-10
+		// Older IE sometimes signals "interactive" too soon
+		if ( document.readyState === "complete" ||
+			( document.readyState !== "loading" && !document.documentElement.doScroll ) ) {
+
+			// Handle it asynchronously to allow scripts the opportunity to delay ready
+			window.setTimeout( jQuery.ready );
+
+		// Standards-based browsers support DOMContentLoaded
+		} else if ( document.addEventListener ) {
+
+			// Use the handy event callback
+			document.addEventListener( "DOMContentLoaded", completed );
+
+			// A fallback to window.onload, that will always work
+			window.addEventListener( "load", completed );
+
+		// If IE event model is used
+		} else {
+
+			// Ensure firing before onload, maybe late but safe also for iframes
+			document.attachEvent( "onreadystatechange", completed );
+
+			// A fallback to window.onload, that will always work
+			window.attachEvent( "onload", completed );
+
+			// If IE and not a frame
+			// continually check to see if the document is ready
+			var top = false;
+
+			try {
+				top = window.frameElement == null && document.documentElement;
+			} catch ( e ) {}
+
+			if ( top && top.doScroll ) {
+				( function doScrollCheck() {
+					if ( !jQuery.isReady ) {
+
+						try {
+
+							// Use the trick by Diego Perini
+							// http://javascript.nwbox.com/IEContentLoaded/
+							top.doScroll( "left" );
+						} catch ( e ) {
+							return window.setTimeout( doScrollCheck, 50 );
+						}
+
+						// detach all dom ready events
+						detach();
+
+						// and execute any waiting functions
+						jQuery.ready();
+					}
+				} )();
+			}
+		}
+	}
+	return readyList.promise( obj );
+};
+
+// Kick off the DOM ready check even if the user does not
+jQuery.ready.promise();
+
+
+
+
+// Support: IE<9
+// Iteration over object's inherited properties before its own
+var i;
+for ( i in jQuery( support ) ) {
+	break;
+}
+support.ownFirst = i === "0";
+
+// Note: most support tests are defined in their respective modules.
+// false until the test is run
+support.inlineBlockNeedsLayout = false;
+
+// Execute ASAP in case we need to set body.style.zoom
+jQuery( function() {
+
+	// Minified: var a,b,c,d
+	var val, div, body, container;
+
+	body = document.getElementsByTagName( "body" )[ 0 ];
+	if ( !body || !body.style ) {
+
+		// Return for frameset docs that don't have a body
+		return;
+	}
+
+	// Setup
+	div = document.createElement( "div" );
+	container = document.createElement( "div" );
+	container.style.cssText = "position:absolute;border:0;width:0;height:0;top:0;left:-9999px";
+	body.appendChild( container ).appendChild( div );
+
+	if ( typeof div.style.zoom !== "undefined" ) {
+
+		// Support: IE<8
+		// Check if natively block-level elements act like inline-block
+		// elements when setting their display to 'inline' and giving
+		// them layout
+		div.style.cssText = "display:inline;margin:0;border:0;padding:1px;width:1px;zoom:1";
+
+		support.inlineBlockNeedsLayout = val = div.offsetWidth === 3;
+		if ( val ) {
+
+			// Prevent IE 6 from affecting layout for positioned elements #11048
+			// Prevent IE from shrinking the body in IE 7 mode #12869
+			// Support: IE<8
+			body.style.zoom = 1;
+		}
+	}
+
+	body.removeChild( container );
+} );
+
+
+( function() {
+	var div = document.createElement( "div" );
+
+	// Support: IE<9
+	support.deleteExpando = true;
+	try {
+		delete div.test;
+	} catch ( e ) {
+		support.deleteExpando = false;
+	}
+
+	// Null elements to avoid leaks in IE.
+	div = null;
+} )();
+var acceptData = function( elem ) {
+	var noData = jQuery.noData[ ( elem.nodeName + " " ).toLowerCase() ],
+		nodeType = +elem.nodeType || 1;
+
+	// Do not set data on non-element DOM nodes because it will not be cleared (#8335).
+	return nodeType !== 1 && nodeType !== 9 ?
+		false :
+
+		// Nodes accept data unless otherwise specified; rejection can be conditional
+		!noData || noData !== true && elem.getAttribute( "classid" ) === noData;
+};
+
+
+
+
+var rbrace = /^(?:\{[\w\W]*\}|\[[\w\W]*\])$/,
+	rmultiDash = /([A-Z])/g;
+
+function dataAttr( elem, key, data ) {
+
+	// If nothing was found internally, try to fetch any
+	// data from the HTML5 data-* attribute
+	if ( data === undefined && elem.nodeType === 1 ) {
+
+		var name = "data-" + key.replace( rmultiDash, "-$1" ).toLowerCase();
+
+		data = elem.getAttribute( name );
+
+		if ( typeof data === "string" ) {
+			try {
+				data = data === "true" ? true :
+					data === "false" ? false :
+					data === "null" ? null :
+
+					// Only convert to a number if it doesn't change the string
+					+data + "" === data ? +data :
+					rbrace.test( data ) ? jQuery.parseJSON( data ) :
+					data;
+			} catch ( e ) {}
+
+			// Make sure we set the data so it isn't changed later
+			jQuery.data( elem, key, data );
+
+		} else {
+			data = undefined;
+		}
+	}
+
+	return data;
+}
+
+// checks a cache object for emptiness
+function isEmptyDataObject( obj ) {
+	var name;
+	for ( name in obj ) {
+
+		// if the public data object is empty, the private is still empty
+		if ( name === "data" && jQuery.isEmptyObject( obj[ name ] ) ) {
+			continue;
+		}
+		if ( name !== "toJSON" ) {
+			return false;
+		}
+	}
+
+	return true;
+}
+
+function internalData( elem, name, data, pvt /* Internal Use Only */ ) {
+	if ( !acceptData( elem ) ) {
+		return;
+	}
+
+	var ret, thisCache,
+		internalKey = jQuery.expando,
+
+		// We have to handle DOM nodes and JS objects differently because IE6-7
+		// can't GC object references properly across the DOM-JS boundary
+		isNode = elem.nodeType,
+
+		// Only DOM nodes need the global jQuery cache; JS object data is
+		// attached directly to the object so GC can occur automatically
+		cache = isNode ? jQuery.cache : elem,
+
+		// Only defining an ID for JS objects if its cache already exists allows
+		// the code to shortcut on the same path as a DOM node with no cache
+		id = isNode ? elem[ internalKey ] : elem[ internalKey ] && internalKey;
+
+	// Avoid doing any more work than we need to when trying to get data on an
+	// object that has no data at all
+	if ( ( !id || !cache[ id ] || ( !pvt && !cache[ id ].data ) ) &&
+		data === undefined && typeof name === "string" ) {
+		return;
+	}
+
+	if ( !id ) {
+
+		// Only DOM nodes need a new unique ID for each element since their data
+		// ends up in the global cache
+		if ( isNode ) {
+			id = elem[ internalKey ] = deletedIds.pop() || jQuery.guid++;
+		} else {
+			id = internalKey;
+		}
+	}
+
+	if ( !cache[ id ] ) {
+
+		// Avoid exposing jQuery metadata on plain JS objects when the object
+		// is serialized using JSON.stringify
+		cache[ id ] = isNode ? {} : { toJSON: jQuery.noop };
+	}
+
+	// An object can be passed to jQuery.data instead of a key/value pair; this gets
+	// shallow copied over onto the existing cache
+	if ( typeof name === "object" || typeof name === "function" ) {
+		if ( pvt ) {
+			cache[ id ] = jQuery.extend( cache[ id ], name );
+		} else {
+			cache[ id ].data = jQuery.extend( cache[ id ].data, name );
+		}
+	}
+
+	thisCache = cache[ id ];
+
+	// jQuery data() is stored in a separate object inside the object's internal data
+	// cache in order to avoid key collisions between internal data and user-defined
+	// data.
+	if ( !pvt ) {
+		if ( !thisCache.data ) {
+			thisCache.data = {};
+		}
+
+		thisCache = thisCache.data;
+	}
+
+	if ( data !== undefined ) {
+		thisCache[ jQuery.camelCase( name ) ] = data;
+	}
+
+	// Check for both converted-to-camel and non-converted data property names
+	// If a data property was specified
+	if ( typeof name === "string" ) {
+
+		// First Try to find as-is property data
+		ret = thisCache[ name ];
+
+		// Test for null|undefined property data
+		if ( ret == null ) {
+
+			// Try to find the camelCased property
+			ret = thisCache[ jQuery.camelCase( name ) ];
+		}
+	} else {
+		ret = thisCache;
+	}
+
+	return ret;
+}
+
+function internalRemoveData( elem, name, pvt ) {
+	if ( !acceptData( elem ) ) {
+		return;
+	}
+
+	var thisCache, i,
+		isNode = elem.nodeType,
+
+		// See jQuery.data for more information
+		cache = isNode ? jQuery.cache : elem,
+		id = isNode ? elem[ jQuery.expando ] : jQuery.expando;
+
+	// If there is already no cache entry for this object, there is no
+	// purpose in continuing
+	if ( !cache[ id ] ) {
+		return;
+	}
+
+	if ( name ) {
+
+		thisCache = pvt ? cache[ id ] : cache[ id ].data;
+
+		if ( thisCache ) {
+
+			// Support array or space separated string names for data keys
+			if ( !jQuery.isArray( name ) ) {
+
+				// try the string as a key before any manipulation
+				if ( name in thisCache ) {
+					name = [ name ];
+				} else {
+
+					// split the camel cased version by spaces unless a key with the spaces exists
+					name = jQuery.camelCase( name );
+					if ( name in thisCache ) {
+						name = [ name ];
+					} else {
+						name = name.split( " " );
+					}
+				}
+			} else {
+
+				// If "name" is an array of keys...
+				// When data is initially created, via ("key", "val") signature,
+				// keys will be converted to camelCase.
+				// Since there is no way to tell _how_ a key was added, remove
+				// both plain key and camelCase key. #12786
+				// This will only penalize the array argument path.
+				name = name.concat( jQuery.map( name, jQuery.camelCase ) );
+			}
+
+			i = name.length;
+			while ( i-- ) {
+				delete thisCache[ name[ i ] ];
+			}
+
+			// If there is no data left in the cache, we want to continue
+			// and let the cache object itself get destroyed
+			if ( pvt ? !isEmptyDataObject( thisCache ) : !jQuery.isEmptyObject( thisCache ) ) {
+				return;
+			}
+		}
+	}
+
+	// See jQuery.data for more information
+	if ( !pvt ) {
+		delete cache[ id ].data;
+
+		// Don't destroy the parent cache unless the internal data object
+		// had been the only thing left in it
+		if ( !isEmptyDataObject( cache[ id ] ) ) {
+			return;
+		}
+	}
+
+	// Destroy the cache
+	if ( isNode ) {
+		jQuery.cleanData( [ elem ], true );
+
+	// Use delete when supported for expandos or `cache` is not a window per isWindow (#10080)
+	/* jshint eqeqeq: false */
+	} else if ( support.deleteExpando || cache != cache.window ) {
+		/* jshint eqeqeq: true */
+		delete cache[ id ];
+
+	// When all else fails, undefined
+	} else {
+		cache[ id ] = undefined;
+	}
+}
+
+jQuery.extend( {
+	cache: {},
+
+	// The following elements (space-suffixed to avoid Object.prototype collisions)
+	// throw uncatchable exceptions if you attempt to set expando properties
+	noData: {
+		"applet ": true,
+		"embed ": true,
+
+		// ...but Flash objects (which have this classid) *can* handle expandos
+		"object ": "clsid:D27CDB6E-AE6D-11cf-96B8-444553540000"
+	},
+
+	hasData: function( elem ) {
+		elem = elem.nodeType ? jQuery.cache[ elem[ jQuery.expando ] ] : elem[ jQuery.expando ];
+		return !!elem && !isEmptyDataObject( elem );
+	},
+
+	data: function( elem, name, data ) {
+		return internalData( elem, name, data );
+	},
+
+	removeData: function( elem, name ) {
+		return internalRemoveData( elem, name );
+	},
+
+	// For internal use only.
+	_data: function( elem, name, data ) {
+		return internalData( elem, name, data, true );
+	},
+
+	_removeData: function( elem, name ) {
+		return internalRemoveData( elem, name, true );
+	}
+} );
+
+jQuery.fn.extend( {
+	data: function( key, value ) {
+		var i, name, data,
+			elem = this[ 0 ],
+			attrs = elem && elem.attributes;
+
+		// Special expections of .data basically thwart jQuery.access,
+		// so implement the relevant behavior ourselves
+
+		// Gets all values
+		if ( key === undefined ) {
+			if ( this.length ) {
+				data = jQuery.data( elem );
+
+				if ( elem.nodeType === 1 && !jQuery._data( elem, "parsedAttrs" ) ) {
+					i = attrs.length;
+					while ( i-- ) {
+
+						// Support: IE11+
+						// The attrs elements can be null (#14894)
+						if ( attrs[ i ] ) {
+							name = attrs[ i ].name;
+							if ( name.indexOf( "data-" ) === 0 ) {
+								name = jQuery.camelCase( name.slice( 5 ) );
+								dataAttr( elem, name, data[ name ] );
+							}
+						}
+					}
+					jQuery._data( elem, "parsedAttrs", true );
+				}
+			}
+
+			return data;
+		}
+
+		// Sets multiple values
+		if ( typeof key === "object" ) {
+			return this.each( function() {
+				jQuery.data( this, key );
+			} );
+		}
+
+		return arguments.length > 1 ?
+
+			// Sets one value
+			this.each( function() {
+				jQuery.data( this, key, value );
+			} ) :
+
+			// Gets one value
+			// Try to fetch any internally stored data first
+			elem ? dataAttr( elem, key, jQuery.data( elem, key ) ) : undefined;
+	},
+
+	removeData: function( key ) {
+		return this.each( function() {
+			jQuery.removeData( this, key );
+		} );
+	}
+} );
+
+
+jQuery.extend( {
+	queue: function( elem, type, data ) {
+		var queue;
+
+		if ( elem ) {
+			type = ( type || "fx" ) + "queue";
+			queue = jQuery._data( elem, type );
+
+			// Speed up dequeue by getting out quickly if this is just a lookup
+			if ( data ) {
+				if ( !queue || jQuery.isArray( data ) ) {
+					queue = jQuery._data( elem, type, jQuery.makeArray( data ) );
+				} else {
+					queue.push( data );
+				}
+			}
+			return queue || [];
+		}
+	},
+
+	dequeue: function( elem, type ) {
+		type = type || "fx";
+
+		var queue = jQuery.queue( elem, type ),
+			startLength = queue.length,
+			fn = queue.shift(),
+			hooks = jQuery._queueHooks( elem, type ),
+			next = function() {
+				jQuery.dequeue( elem, type );
+			};
+
+		// If the fx queue is dequeued, always remove the progress sentinel
+		if ( fn === "inprogress" ) {
+			fn = queue.shift();
+			startLength--;
+		}
+
+		if ( fn ) {
+
+			// Add a progress sentinel to prevent the fx queue from being
+			// automatically dequeued
+			if ( type === "fx" ) {
+				queue.unshift( "inprogress" );
+			}
+
+			// clear up the last queue stop function
+			delete hooks.stop;
+			fn.call( elem, next, hooks );
+		}
+
+		if ( !startLength && hooks ) {
+			hooks.empty.fire();
+		}
+	},
+
+	// not intended for public consumption - generates a queueHooks object,
+	// or returns the current one
+	_queueHooks: function( elem, type ) {
+		var key = type + "queueHooks";
+		return jQuery._data( elem, key ) || jQuery._data( elem, key, {
+			empty: jQuery.Callbacks( "once memory" ).add( function() {
+				jQuery._removeData( elem, type + "queue" );
+				jQuery._removeData( elem, key );
+			} )
+		} );
+	}
+} );
+
+jQuery.fn.extend( {
+	queue: function( type, data ) {
+		var setter = 2;
+
+		if ( typeof type !== "string" ) {
+			data = type;
+			type = "fx";
+			setter--;
+		}
+
+		if ( arguments.length < setter ) {
+			return jQuery.queue( this[ 0 ], type );
+		}
+
+		return data === undefined ?
+			this :
+			this.each( function() {
+				var queue = jQuery.queue( this, type, data );
+
+				// ensure a hooks for this queue
+				jQuery._queueHooks( this, type );
+
+				if ( type === "fx" && queue[ 0 ] !== "inprogress" ) {
+					jQuery.dequeue( this, type );
+				}
+			} );
+	},
+	dequeue: function( type ) {
+		return this.each( function() {
+			jQuery.dequeue( this, type );
+		} );
+	},
+	clearQueue: function( type ) {
+		return this.queue( type || "fx", [] );
+	},
+
+	// Get a promise resolved when queues of a certain type
+	// are emptied (fx is the type by default)
+	promise: function( type, obj ) {
+		var tmp,
+			count = 1,
+			defer = jQuery.Deferred(),
+			elements = this,
+			i = this.length,
+			resolve = function() {
+				if ( !( --count ) ) {
+					defer.resolveWith( elements, [ elements ] );
+				}
+			};
+
+		if ( typeof type !== "string" ) {
+			obj = type;
+			type = undefined;
+		}
+		type = type || "fx";
+
+		while ( i-- ) {
+			tmp = jQuery._data( elements[ i ], type + "queueHooks" );
+			if ( tmp && tmp.empty ) {
+				count++;
+				tmp.empty.add( resolve );
+			}
+		}
+		resolve();
+		return defer.promise( obj );
+	}
+} );
+
+
+( function() {
+	var shrinkWrapBlocksVal;
+
+	support.shrinkWrapBlocks = function() {
+		if ( shrinkWrapBlocksVal != null ) {
+			return shrinkWrapBlocksVal;
+		}
+
+		// Will be changed later if needed.
+		shrinkWrapBlocksVal = false;
+
+		// Minified: var b,c,d
+		var div, body, container;
+
+		body = document.getElementsByTagName( "body" )[ 0 ];
+		if ( !body || !body.style ) {
+
+			// Test fired too early or in an unsupported environment, exit.
+			return;
+		}
+
+		// Setup
+		div = document.createElement( "div" );
+		container = document.createElement( "div" );
+		container.style.cssText = "position:absolute;border:0;width:0;height:0;top:0;left:-9999px";
+		body.appendChild( container ).appendChild( div );
+
+		// Support: IE6
+		// Check if elements with layout shrink-wrap their children
+		if ( typeof div.style.zoom !== "undefined" ) {
+
+			// Reset CSS: box-sizing; display; margin; border
+			div.style.cssText =
+
+				// Support: Firefox<29, Android 2.3
+				// Vendor-prefix box-sizing
+				"-webkit-box-sizing:content-box;-moz-box-sizing:content-box;" +
+				"box-sizing:content-box;display:block;margin:0;border:0;" +
+				"padding:1px;width:1px;zoom:1";
+			div.appendChild( document.createElement( "div" ) ).style.width = "5px";
+			shrinkWrapBlocksVal = div.offsetWidth !== 3;
+		}
+
+		body.removeChild( container );
+
+		return shrinkWrapBlocksVal;
+	};
+
+} )();
+var pnum = ( /[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/ ).source;
+
+var rcssNum = new RegExp( "^(?:([+-])=|)(" + pnum + ")([a-z%]*)$", "i" );
+
+
+var cssExpand = [ "Top", "Right", "Bottom", "Left" ];
+
+var isHidden = function( elem, el ) {
+
+		// isHidden might be called from jQuery#filter function;
+		// in that case, element will be second argument
+		elem = el || elem;
+		return jQuery.css( elem, "display" ) === "none" ||
+			!jQuery.contains( elem.ownerDocument, elem );
+	};
+
+
+
+function adjustCSS( elem, prop, valueParts, tween ) {
+	var adjusted,
+		scale = 1,
+		maxIterations = 20,
+		currentValue = tween ?
+			function() { return tween.cur(); } :
+			function() { return jQuery.css( elem, prop, "" ); },
+		initial = currentValue(),
+		unit = valueParts && valueParts[ 3 ] || ( jQuery.cssNumber[ prop ] ? "" : "px" ),
+
+		// Starting value computation is required for potential unit mismatches
+		initialInUnit = ( jQuery.cssNumber[ prop ] || unit !== "px" && +initial ) &&
+			rcssNum.exec( jQuery.css( elem, prop ) );
+
+	if ( initialInUnit && initialInUnit[ 3 ] !== unit ) {
+
+		// Trust units reported by jQuery.css
+		unit = unit || initialInUnit[ 3 ];
+
+		// Make sure we update the tween properties later on
+		valueParts = valueParts || [];
+
+		// Iteratively approximate from a nonzero starting point
+		initialInUnit = +initial || 1;
+
+		do {
+
+			// If previous iteration zeroed out, double until we get *something*.
+			// Use string for doubling so we don't accidentally see scale as unchanged below
+			scale = scale || ".5";
+
+			// Adjust and apply
+			initialInUnit = initialInUnit / scale;
+			jQuery.style( elem, prop, initialInUnit + unit );
+
+		// Update scale, tolerating zero or NaN from tween.cur()
+		// Break the loop if scale is unchanged or perfect, or if we've just had enough.
+		} while (
+			scale !== ( scale = currentValue() / initial ) && scale !== 1 && --maxIterations
+		);
+	}
+
+	if ( valueParts ) {
+		initialInUnit = +initialInUnit || +initial || 0;
+
+		// Apply relative offset (+=/-=) if specified
+		adjusted = valueParts[ 1 ] ?
+			initialInUnit + ( valueParts[ 1 ] + 1 ) * valueParts[ 2 ] :
+			+valueParts[ 2 ];
+		if ( tween ) {
+			tween.unit = unit;
+			tween.start = initialInUnit;
+			tween.end = adjusted;
+		}
+	}
+	return adjusted;
+}
+
+
+// Multifunctional method to get and set values of a collection
+// The value/s can optionally be executed if it's a function
+var access = function( elems, fn, key, value, chainable, emptyGet, raw ) {
+	var i = 0,
+		length = elems.length,
+		bulk = key == null;
+
+	// Sets many values
+	if ( jQuery.type( key ) === "object" ) {
+		chainable = true;
+		for ( i in key ) {
+			access( elems, fn, i, key[ i ], true, emptyGet, raw );
+		}
+
+	// Sets one value
+	} else if ( value !== undefined ) {
+		chainable = true;
+
+		if ( !jQuery.isFunction( value ) ) {
+			raw = true;
+		}
+
+		if ( bulk ) {
+
+			// Bulk operations run against the entire set
+			if ( raw ) {
+				fn.call( elems, value );
+				fn = null;
+
+			// ...except when executing function values
+			} else {
+				bulk = fn;
+				fn = function( elem, key, value ) {
+					return bulk.call( jQuery( elem ), value );
+				};
+			}
+		}
+
+		if ( fn ) {
+			for ( ; i < length; i++ ) {
+				fn(
+					elems[ i ],
+					key,
+					raw ? value : value.call( elems[ i ], i, fn( elems[ i ], key ) )
+				);
+			}
+		}
+	}
+
+	return chainable ?
+		elems :
+
+		// Gets
+		bulk ?
+			fn.call( elems ) :
+			length ? fn( elems[ 0 ], key ) : emptyGet;
+};
+var rcheckableType = ( /^(?:checkbox|radio)$/i );
+
+var rtagName = ( /<([\w:-]+)/ );
+
+var rscriptType = ( /^$|\/(?:java|ecma)script/i );
+
+var rleadingWhitespace = ( /^\s+/ );
+
+var nodeNames = "abbr|article|aside|audio|bdi|canvas|data|datalist|" +
+		"details|dialog|figcaption|figure|footer|header|hgroup|main|" +
+		"mark|meter|nav|output|picture|progress|section|summary|template|time|video";
+
+
+
+function createSafeFragment( document ) {
+	var list = nodeNames.split( "|" ),
+		safeFrag = document.createDocumentFragment();
+
+	if ( safeFrag.createElement ) {
+		while ( list.length ) {
+			safeFrag.createElement(
+				list.pop()
+			);
+		}
+	}
+	return safeFrag;
+}
+
+
+( function() {
+	var div = document.createElement( "div" ),
+		fragment = document.createDocumentFragment(),
+		input = document.createElement( "input" );
+
+	// Setup
+	div.innerHTML = "  <link/><table></table><a href='/a'>a</a><input type='checkbox'/>";
+
+	// IE strips leading whitespace when .innerHTML is used
+	support.leadingWhitespace = div.firstChild.nodeType === 3;
+
+	// Make sure that tbody elements aren't automatically inserted
+	// IE will insert them into empty tables
+	support.tbody = !div.getElementsByTagName( "tbody" ).length;
+
+	// Make sure that link elements get serialized correctly by innerHTML
+	// This requires a wrapper element in IE
+	support.htmlSerialize = !!div.getElementsByTagName( "link" ).length;
+
+	// Makes sure cloning an html5 element does not cause problems
+	// Where outerHTML is undefined, this still works
+	support.html5Clone =
+		document.createElement( "nav" ).cloneNode( true ).outerHTML !== "<:nav></:nav>";
+
+	// Check if a disconnected checkbox will retain its checked
+	// value of true after appended to the DOM (IE6/7)
+	input.type = "checkbox";
+	input.checked = true;
+	fragment.appendChild( input );
+	support.appendChecked = input.checked;
+
+	// Make sure textarea (and checkbox) defaultValue is properly cloned
+	// Support: IE6-IE11+
+	div.innerHTML = "<textarea>x</textarea>";
+	support.noCloneChecked = !!div.cloneNode( true ).lastChild.defaultValue;
+
+	// #11217 - WebKit loses check when the name is after the checked attribute
+	fragment.appendChild( div );
+
+	// Support: Windows Web Apps (WWA)
+	// `name` and `type` must use .setAttribute for WWA (#14901)
+	input = document.createElement( "input" );
+	input.setAttribute( "type", "radio" );
+	input.setAttribute( "checked", "checked" );
+	input.setAttribute( "name", "t" );
+
+	div.appendChild( input );
+
+	// Support: Safari 5.1, iOS 5.1, Android 4.x, Android 2.3
+	// old WebKit doesn't clone checked state correctly in fragments
+	support.checkClone = div.cloneNode( true ).cloneNode( true ).lastChild.checked;
+
+	// Support: IE<9
+	// Cloned elements keep attachEvent handlers, we use addEventListener on IE9+
+	support.noCloneEvent = !!div.addEventListener;
+
+	// Support: IE<9
+	// Since attributes and properties are the same in IE,
+	// cleanData must set properties to undefined rather than use removeAttribute
+	div[ jQuery.expando ] = 1;
+	support.attributes = !div.getAttribute( jQuery.expando );
+} )();
+
+
+// We have to close these tags to support XHTML (#13200)
+var wrapMap = {
+	option: [ 1, "<select multiple='multiple'>", "</select>" ],
+	legend: [ 1, "<fieldset>", "</fieldset>" ],
+	area: [ 1, "<map>", "</map>" ],
+
+	// Support: IE8
+	param: [ 1, "<object>", "</object>" ],
+	thead: [ 1, "<table>", "</table>" ],
+	tr: [ 2, "<table><tbody>", "</tbody></table>" ],
+	col: [ 2, "<table><tbody></tbody><colgroup>", "</colgroup></table>" ],
+	td: [ 3, "<table><tbody><tr>", "</tr></tbody></table>" ],
+
+	// IE6-8 can't serialize link, script, style, or any html5 (NoScope) tags,
+	// unless wrapped in a div with non-breaking characters in front of it.
+	_default: support.htmlSerialize ? [ 0, "", "" ] : [ 1, "X<div>", "</div>" ]
+};
+
+// Support: IE8-IE9
+wrapMap.optgroup = wrapMap.option;
+
+wrapMap.tbody = wrapMap.tfoot = wrapMap.colgroup = wrapMap.caption = wrapMap.thead;
+wrapMap.th = wrapMap.td;
+
+
+function getAll( context, tag ) {
+	var elems, elem,
+		i = 0,
+		found = typeof context.getElementsByTagName !== "undefined" ?
+			context.getElementsByTagName( tag || "*" ) :
+			typeof context.querySelectorAll !== "undefined" ?
+				context.querySelectorAll( tag || "*" ) :
+				undefined;
+
+	if ( !found ) {
+		for ( found = [], elems = context.childNodes || context;
+			( elem = elems[ i ] ) != null;
+			i++
+		) {
+			if ( !tag || jQuery.nodeName( elem, tag ) ) {
+				found.push( elem );
+			} else {
+				jQuery.merge( found, getAll( elem, tag ) );
+			}
+		}
+	}
+
+	return tag === undefined || tag && jQuery.nodeName( context, tag ) ?
+		jQuery.merge( [ context ], found ) :
+		found;
+}
+
+
+// Mark scripts as having already been evaluated
+function setGlobalEval( elems, refElements ) {
+	var elem,
+		i = 0;
+	for ( ; ( elem = elems[ i ] ) != null; i++ ) {
+		jQuery._data(
+			elem,
+			"globalEval",
+			!refElements || jQuery._data( refElements[ i ], "globalEval" )
+		);
+	}
+}
+
+
+var rhtml = /<|&#?\w+;/,
+	rtbody = /<tbody/i;
+
+function fixDefaultChecked( elem ) {
+	if ( rcheckableType.test( elem.type ) ) {
+		elem.defaultChecked = elem.checked;
+	}
+}
+
+function buildFragment( elems, context, scripts, selection, ignored ) {
+	var j, elem, contains,
+		tmp, tag, tbody, wrap,
+		l = elems.length,
+
+		// Ensure a safe fragment
+		safe = createSafeFragment( context ),
+
+		nodes = [],
+		i = 0;
+
+	for ( ; i < l; i++ ) {
+		elem = elems[ i ];
+
+		if ( elem || elem === 0 ) {
+
+			// Add nodes directly
+			if ( jQuery.type( elem ) === "object" ) {
+				jQuery.merge( nodes, elem.nodeType ? [ elem ] : elem );
+
+			// Convert non-html into a text node
+			} else if ( !rhtml.test( elem ) ) {
+				nodes.push( context.createTextNode( elem ) );
+
+			// Convert html into DOM nodes
+			} else {
+				tmp = tmp || safe.appendChild( context.createElement( "div" ) );
+
+				// Deserialize a standard representation
+				tag = ( rtagName.exec( elem ) || [ "", "" ] )[ 1 ].toLowerCase();
+				wrap = wrapMap[ tag ] || wrapMap._default;
+
+				tmp.innerHTML = wrap[ 1 ] + jQuery.htmlPrefilter( elem ) + wrap[ 2 ];
+
+				// Descend through wrappers to the right content
+				j = wrap[ 0 ];
+				while ( j-- ) {
+					tmp = tmp.lastChild;
+				}
+
+				// Manually add leading whitespace removed by IE
+				if ( !support.leadingWhitespace && rleadingWhitespace.test( elem ) ) {
+					nodes.push( context.createTextNode( rleadingWhitespace.exec( elem )[ 0 ] ) );
+				}
+
+				// Remove IE's autoinserted <tbody> from table fragments
+				if ( !support.tbody ) {
+
+					// String was a <table>, *may* have spurious <tbody>
+					elem = tag === "table" && !rtbody.test( elem ) ?
+						tmp.firstChild :
+
+						// String was a bare <thead> or <tfoot>
+						wrap[ 1 ] === "<table>" && !rtbody.test( elem ) ?
+							tmp :
+							0;
+
+					j = elem && elem.childNodes.length;
+					while ( j-- ) {
+						if ( jQuery.nodeName( ( tbody = elem.childNodes[ j ] ), "tbody" ) &&
+							!tbody.childNodes.length ) {
+
+							elem.removeChild( tbody );
+						}
+					}
+				}
+
+				jQuery.merge( nodes, tmp.childNodes );
+
+				// Fix #12392 for WebKit and IE > 9
+				tmp.textContent = "";
+
+				// Fix #12392 for oldIE
+				while ( tmp.firstChild ) {
+					tmp.removeChild( tmp.firstChild );
+				}
+
+				// Remember the top-level container for proper cleanup
+				tmp = safe.lastChild;
+			}
+		}
+	}
+
+	// Fix #11356: Clear elements from fragment
+	if ( tmp ) {
+		safe.removeChild( tmp );
+	}
+
+	// Reset defaultChecked for any radios and checkboxes
+	// about to be appended to the DOM in IE 6/7 (#8060)
+	if ( !support.appendChecked ) {
+		jQuery.grep( getAll( nodes, "input" ), fixDefaultChecked );
+	}
+
+	i = 0;
+	while ( ( elem = nodes[ i++ ] ) ) {
+
+		// Skip elements already in the context collection (trac-4087)
+		if ( selection && jQuery.inArray( elem, selection ) > -1 ) {
+			if ( ignored ) {
+				ignored.push( elem );
+			}
+
+			continue;
+		}
+
+		contains = jQuery.contains( elem.ownerDocument, elem );
+
+		// Append to fragment
+		tmp = getAll( safe.appendChild( elem ), "script" );
+
+		// Preserve script evaluation history
+		if ( contains ) {
+			setGlobalEval( tmp );
+		}
+
+		// Capture executables
+		if ( scripts ) {
+			j = 0;
+			while ( ( elem = tmp[ j++ ] ) ) {
+				if ( rscriptType.test( elem.type || "" ) ) {
+					scripts.push( elem );
+				}
+			}
+		}
+	}
+
+	tmp = null;
+
+	return safe;
+}
+
+
+( function() {
+	var i, eventName,
+		div = document.createElement( "div" );
+
+	// Support: IE<9 (lack submit/change bubble), Firefox (lack focus(in | out) events)
+	for ( i in { submit: true, change: true, focusin: true } ) {
+		eventName = "on" + i;
+
+		if ( !( support[ i ] = eventName in window ) ) {
+
+			// Beware of CSP restrictions (https://developer.mozilla.org/en/Security/CSP)
+			div.setAttribute( eventName, "t" );
+			support[ i ] = div.attributes[ eventName ].expando === false;
+		}
+	}
+
+	// Null elements to avoid leaks in IE.
+	div = null;
+} )();
+
+
+var rformElems = /^(?:input|select|textarea)$/i,
+	rkeyEvent = /^key/,
+	rmouseEvent = /^(?:mouse|pointer|contextmenu|drag|drop)|click/,
+	rfocusMorph = /^(?:focusinfocus|focusoutblur)$/,
+	rtypenamespace = /^([^.]*)(?:\.(.+)|)/;
+
+function returnTrue() {
+	return true;
+}
+
+function returnFalse() {
+	return false;
+}
+
+// Support: IE9
+// See #13393 for more info
+function safeActiveElement() {
+	try {
+		return document.activeElement;
+	} catch ( err ) { }
+}
+
+function on( elem, types, selector, data, fn, one ) {
+	var origFn, type;
+
+	// Types can be a map of types/handlers
+	if ( typeof types === "object" ) {
+
+		// ( types-Object, selector, data )
+		if ( typeof selector !== "string" ) {
+
+			// ( types-Object, data )
+			data = data || selector;
+			selector = undefined;
+		}
+		for ( type in types ) {
+			on( elem, type, selector, data, types[ type ], one );
+		}
+		return elem;
+	}
+
+	if ( data == null && fn == null ) {
+
+		// ( types, fn )
+		fn = selector;
+		data = selector = undefined;
+	} else if ( fn == null ) {
+		if ( typeof selector === "string" ) {
+
+			// ( types, selector, fn )
+			fn = data;
+			data = undefined;
+		} else {
+
+			// ( types, data, fn )
+			fn = data;
+			data = selector;
+			selector = undefined;
+		}
+	}
+	if ( fn === false ) {
+		fn = returnFalse;
+	} else if ( !fn ) {
+		return elem;
+	}
+
+	if ( one === 1 ) {
+		origFn = fn;
+		fn = function( event ) {
+
+			// Can use an empty set, since event contains the info
+			jQuery().off( event );
+			return origFn.apply( this, arguments );
+		};
+
+		// Use same guid so caller can remove using origFn
+		fn.guid = origFn.guid || ( origFn.guid = jQuery.guid++ );
+	}
+	return elem.each( function() {
+		jQuery.event.add( this, types, fn, data, selector );
+	} );
+}
+
+/*
+ * Helper functions for managing events -- not part of the public interface.
+ * Props to Dean Edwards' addEvent library for many of the ideas.
+ */
+jQuery.event = {
+
+	global: {},
+
+	add: function( elem, types, handler, data, selector ) {
+		var tmp, events, t, handleObjIn,
+			special, eventHandle, handleObj,
+			handlers, type, namespaces, origType,
+			elemData = jQuery._data( elem );
+
+		// Don't attach events to noData or text/comment nodes (but allow plain objects)
+		if ( !elemData ) {
+			return;
+		}
+
+		// Caller can pass in an object of custom data in lieu of the handler
+		if ( handler.handler ) {
+			handleObjIn = handler;
+			handler = handleObjIn.handler;
+			selector = handleObjIn.selector;
+		}
+
+		// Make sure that the handler has a unique ID, used to find/remove it later
+		if ( !handler.guid ) {
+			handler.guid = jQuery.guid++;
+		}
+
+		// Init the element's event structure and main handler, if this is the first
+		if ( !( events = elemData.events ) ) {
+			events = elemData.events = {};
+		}
+		if ( !( eventHandle = elemData.handle ) ) {
+			eventHandle = elemData.handle = function( e ) {
+
+				// Discard the second event of a jQuery.event.trigger() and
+				// when an event is called after a page has unloaded
+				return typeof jQuery !== "undefined" &&
+					( !e || jQuery.event.triggered !== e.type ) ?
+					jQuery.event.dispatch.apply( eventHandle.elem, arguments ) :
+					undefined;
+			};
+
+			// Add elem as a property of the handle fn to prevent a memory leak
+			// with IE non-native events
+			eventHandle.elem = elem;
+		}
+
+		// Handle multiple events separated by a space
+		types = ( types || "" ).match( rnotwhite ) || [ "" ];
+		t = types.length;
+		while ( t-- ) {
+			tmp = rtypenamespace.exec( types[ t ] ) || [];
+			type = origType = tmp[ 1 ];
+			namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort();
+
+			// There *must* be a type, no attaching namespace-only handlers
+			if ( !type ) {
+				continue;
+			}
+
+			// If event changes its type, use the special event handlers for the changed type
+			special = jQuery.event.special[ type ] || {};
+
+			// If selector defined, determine special event api type, otherwise given type
+			type = ( selector ? special.delegateType : special.bindType ) || type;
+
+			// Update special based on newly reset type
+			special = jQuery.event.special[ type ] || {};
+
+			// handleObj is passed to all event handlers
+			handleObj = jQuery.extend( {
+				type: type,
+				origType: origType,
+				data: data,
+				handler: handler,
+				guid: handler.guid,
+				selector: selector,
+				needsContext: selector && jQuery.expr.match.needsContext.test( selector ),
+				namespace: namespaces.join( "." )
+			}, handleObjIn );
+
+			// Init the event handler queue if we're the first
+			if ( !( handlers = events[ type ] ) ) {
+				handlers = events[ type ] = [];
+				handlers.delegateCount = 0;
+
+				// Only use addEventListener/attachEvent if the special events handler returns false
+				if ( !special.setup ||
+					special.setup.call( elem, data, namespaces, eventHandle ) === false ) {
+
+					// Bind the global event handler to the element
+					if ( elem.addEventListener ) {
+						elem.addEventListener( type, eventHandle, false );
+
+					} else if ( elem.attachEvent ) {
+						elem.attachEvent( "on" + type, eventHandle );
+					}
+				}
+			}
+
+			if ( special.add ) {
+				special.add.call( elem, handleObj );
+
+				if ( !handleObj.handler.guid ) {
+					handleObj.handler.guid = handler.guid;
+				}
+			}
+
+			// Add to the element's handler list, delegates in front
+			if ( selector ) {
+				handlers.splice( handlers.delegateCount++, 0, handleObj );
+			} else {
+				handlers.push( handleObj );
+			}
+
+			// Keep track of which events have ever been used, for event optimization
+			jQuery.event.global[ type ] = true;
+		}
+
+		// Nullify elem to prevent memory leaks in IE
+		elem = null;
+	},
+
+	// Detach an event or set of events from an element
+	remove: function( elem, types, handler, selector, mappedTypes ) {
+		var j, handleObj, tmp,
+			origCount, t, events,
+			special, handlers, type,
+			namespaces, origType,
+			elemData = jQuery.hasData( elem ) && jQuery._data( elem );
+
+		if ( !elemData || !( events = elemData.events ) ) {
+			return;
+		}
+
+		// Once for each type.namespace in types; type may be omitted
+		types = ( types || "" ).match( rnotwhite ) || [ "" ];
+		t = types.length;
+		while ( t-- ) {
+			tmp = rtypenamespace.exec( types[ t ] ) || [];
+			type = origType = tmp[ 1 ];
+			namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort();
+
+			// Unbind all events (on this namespace, if provided) for the element
+			if ( !type ) {
+				for ( type in events ) {
+					jQuery.event.remove( elem, type + types[ t ], handler, selector, true );
+				}
+				continue;
+			}
+
+			special = jQuery.event.special[ type ] || {};
+			type = ( selector ? special.delegateType : special.bindType ) || type;
+			handlers = events[ type ] || [];
+			tmp = tmp[ 2 ] &&
+				new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" );
+
+			// Remove matching events
+			origCount = j = handlers.length;
+			while ( j-- ) {
+				handleObj = handlers[ j ];
+
+				if ( ( mappedTypes || origType === handleObj.origType ) &&
+					( !handler || handler.guid === handleObj.guid ) &&
+					( !tmp || tmp.test( handleObj.namespace ) ) &&
+					( !selector || selector === handleObj.selector ||
+						selector === "**" && handleObj.selector ) ) {
+					handlers.splice( j, 1 );
+
+					if ( handleObj.selector ) {
+						handlers.delegateCount--;
+					}
+					if ( special.remove ) {
+						special.remove.call( elem, handleObj );
+					}
+				}
+			}
+
+			// Remove generic event handler if we removed something and no more handlers exist
+			// (avoids potential for endless recursion during removal of special event handlers)
+			if ( origCount && !handlers.length ) {
+				if ( !special.teardown ||
+					special.teardown.call( elem, namespaces, elemData.handle ) === false ) {
+
+					jQuery.removeEvent( elem, type, elemData.handle );
+				}
+
+				delete events[ type ];
+			}
+		}
+
+		// Remove the expando if it's no longer used
+		if ( jQuery.isEmptyObject( events ) ) {
+			delete elemData.handle;
+
+			// removeData also checks for emptiness and clears the expando if empty
+			// so use it instead of delete
+			jQuery._removeData( elem, "events" );
+		}
+	},
+
+	trigger: function( event, data, elem, onlyHandlers ) {
+		var handle, ontype, cur,
+			bubbleType, special, tmp, i,
+			eventPath = [ elem || document ],
+			type = hasOwn.call( event, "type" ) ? event.type : event,
+			namespaces = hasOwn.call( event, "namespace" ) ? event.namespace.split( "." ) : [];
+
+		cur = tmp = elem = elem || document;
+
+		// Don't do events on text and comment nodes
+		if ( elem.nodeType === 3 || elem.nodeType === 8 ) {
+			return;
+		}
+
+		// focus/blur morphs to focusin/out; ensure we're not firing them right now
+		if ( rfocusMorph.test( type + jQuery.event.triggered ) ) {
+			return;
+		}
+
+		if ( type.indexOf( "." ) > -1 ) {
+
+			// Namespaced trigger; create a regexp to match event type in handle()
+			namespaces = type.split( "." );
+			type = namespaces.shift();
+			namespaces.sort();
+		}
+		ontype = type.indexOf( ":" ) < 0 && "on" + type;
+
+		// Caller can pass in a jQuery.Event object, Object, or just an event type string
+		event = event[ jQuery.expando ] ?
+			event :
+			new jQuery.Event( type, typeof event === "object" && event );
+
+		// Trigger bitmask: & 1 for native handlers; & 2 for jQuery (always true)
+		event.isTrigger = onlyHandlers ? 2 : 3;
+		event.namespace = namespaces.join( "." );
+		event.rnamespace = event.namespace ?
+			new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ) :
+			null;
+
+		// Clean up the event in case it is being reused
+		event.result = undefined;
+		if ( !event.target ) {
+			event.target = elem;
+		}
+
+		// Clone any incoming data and prepend the event, creating the handler arg list
+		data = data == null ?
+			[ event ] :
+			jQuery.makeArray( data, [ event ] );
+
+		// Allow special events to draw outside the lines
+		special = jQuery.event.special[ type ] || {};
+		if ( !onlyHandlers && special.trigger && special.trigger.apply( elem, data ) === false ) {
+			return;
+		}
+
+		// Determine event propagation path in advance, per W3C events spec (#9951)
+		// Bubble up to document, then to window; watch for a global ownerDocument var (#9724)
+		if ( !onlyHandlers && !special.noBubble && !jQuery.isWindow( elem ) ) {
+
+			bubbleType = special.delegateType || type;
+			if ( !rfocusMorph.test( bubbleType + type ) ) {
+				cur = cur.parentNode;
+			}
+			for ( ; cur; cur = cur.parentNode ) {
+				eventPath.push( cur );
+				tmp = cur;
+			}
+
+			// Only add window if we got to document (e.g., not plain obj or detached DOM)
+			if ( tmp === ( elem.ownerDocument || document ) ) {
+				eventPath.push( tmp.defaultView || tmp.parentWindow || window );
+			}
+		}
+
+		// Fire handlers on the event path
+		i = 0;
+		while ( ( cur = eventPath[ i++ ] ) && !event.isPropagationStopped() ) {
+
+			event.type = i > 1 ?
+				bubbleType :
+				special.bindType || type;
+
+			// jQuery handler
+			handle = ( jQuery._data( cur, "events" ) || {} )[ event.type ] &&
+				jQuery._data( cur, "handle" );
+
+			if ( handle ) {
+				handle.apply( cur, data );
+			}
+
+			// Native handler
+			handle = ontype && cur[ ontype ];
+			if ( handle && handle.apply && acceptData( cur ) ) {
+				event.result = handle.apply( cur, data );
+				if ( event.result === false ) {
+					event.preventDefault();
+				}
+			}
+		}
+		event.type = type;
+
+		// If nobody prevented the default action, do it now
+		if ( !onlyHandlers && !event.isDefaultPrevented() ) {
+
+			if (
+				( !special._default ||
+				 special._default.apply( eventPath.pop(), data ) === false
+				) && acceptData( elem )
+			) {
+
+				// Call a native DOM method on the target with the same name name as the event.
+				// Can't use an .isFunction() check here because IE6/7 fails that test.
+				// Don't do default actions on window, that's where global variables be (#6170)
+				if ( ontype && elem[ type ] && !jQuery.isWindow( elem ) ) {
+
+					// Don't re-trigger an onFOO event when we call its FOO() method
+					tmp = elem[ ontype ];
+
+					if ( tmp ) {
+						elem[ ontype ] = null;
+					}
+
+					// Prevent re-triggering of the same event, since we already bubbled it above
+					jQuery.event.triggered = type;
+					try {
+						elem[ type ]();
+					} catch ( e ) {
+
+						// IE<9 dies on focus/blur to hidden element (#1486,#12518)
+						// only reproducible on winXP IE8 native, not IE9 in IE8 mode
+					}
+					jQuery.event.triggered = undefined;
+
+					if ( tmp ) {
+						elem[ ontype ] = tmp;
+					}
+				}
+			}
+		}
+
+		return event.result;
+	},
+
+	dispatch: function( event ) {
+
+		// Make a writable jQuery.Event from the native event object
+		event = jQuery.event.fix( event );
+
+		var i, j, ret, matched, handleObj,
+			handlerQueue = [],
+			args = slice.call( arguments ),
+			handlers = ( jQuery._data( this, "events" ) || {} )[ event.type ] || [],
+			special = jQuery.event.special[ event.type ] || {};
+
+		// Use the fix-ed jQuery.Event rather than the (read-only) native event
+		args[ 0 ] = event;
+		event.delegateTarget = this;
+
+		// Call the preDispatch hook for the mapped type, and let it bail if desired
+		if ( special.preDispatch && special.preDispatch.call( this, event ) === false ) {
+			return;
+		}
+
+		// Determine handlers
+		handlerQueue = jQuery.event.handlers.call( this, event, handlers );
+
+		// Run delegates first; they may want to stop propagation beneath us
+		i = 0;
+		while ( ( matched = handlerQueue[ i++ ] ) && !event.isPropagationStopped() ) {
+			event.currentTarget = matched.elem;
+
+			j = 0;
+			while ( ( handleObj = matched.handlers[ j++ ] ) &&
+				!event.isImmediatePropagationStopped() ) {
+
+				// Triggered event must either 1) have no namespace, or 2) have namespace(s)
+				// a subset or equal to those in the bound event (both can have no namespace).
+				if ( !event.rnamespace || event.rnamespace.test( handleObj.namespace ) ) {
+
+					event.handleObj = handleObj;
+					event.data = handleObj.data;
+
+					ret = ( ( jQuery.event.special[ handleObj.origType ] || {} ).handle ||
+						handleObj.handler ).apply( matched.elem, args );
+
+					if ( ret !== undefined ) {
+						if ( ( event.result = ret ) === false ) {
+							event.preventDefault();
+							event.stopPropagation();
+						}
+					}
+				}
+			}
+		}
+
+		// Call the postDispatch hook for the mapped type
+		if ( special.postDispatch ) {
+			special.postDispatch.call( this, event );
+		}
+
+		return event.result;
+	},
+
+	handlers: function( event, handlers ) {
+		var i, matches, sel, handleObj,
+			handlerQueue = [],
+			delegateCount = handlers.delegateCount,
+			cur = event.target;
+
+		// Support (at least): Chrome, IE9
+		// Find delegate handlers
+		// Black-hole SVG <use> instance trees (#13180)
+		//
+		// Support: Firefox<=42+
+		// Avoid non-left-click in FF but don't block IE radio events (#3861, gh-2343)
+		if ( delegateCount && cur.nodeType &&
+			( event.type !== "click" || isNaN( event.button ) || event.button < 1 ) ) {
+
+			/* jshint eqeqeq: false */
+			for ( ; cur != this; cur = cur.parentNode || this ) {
+				/* jshint eqeqeq: true */
+
+				// Don't check non-elements (#13208)
+				// Don't process clicks on disabled elements (#6911, #8165, #11382, #11764)
+				if ( cur.nodeType === 1 && ( cur.disabled !== true || event.type !== "click" ) ) {
+					matches = [];
+					for ( i = 0; i < delegateCount; i++ ) {
+						handleObj = handlers[ i ];
+
+						// Don't conflict with Object.prototype properties (#13203)
+						sel = handleObj.selector + " ";
+
+						if ( matches[ sel ] === undefined ) {
+							matches[ sel ] = handleObj.needsContext ?
+								jQuery( sel, this ).index( cur ) > -1 :
+								jQuery.find( sel, this, null, [ cur ] ).length;
+						}
+						if ( matches[ sel ] ) {
+							matches.push( handleObj );
+						}
+					}
+					if ( matches.length ) {
+						handlerQueue.push( { elem: cur, handlers: matches } );
+					}
+				}
+			}
+		}
+
+		// Add the remaining (directly-bound) handlers
+		if ( delegateCount < handlers.length ) {
+			handlerQueue.push( { elem: this, handlers: handlers.slice( delegateCount ) } );
+		}
+
+		return handlerQueue;
+	},
+
+	fix: function( event ) {
+		if ( event[ jQuery.expando ] ) {
+			return event;
+		}
+
+		// Create a writable copy of the event object and normalize some properties
+		var i, prop, copy,
+			type = event.type,
+			originalEvent = event,
+			fixHook = this.fixHooks[ type ];
+
+		if ( !fixHook ) {
+			this.fixHooks[ type ] = fixHook =
+				rmouseEvent.test( type ) ? this.mouseHooks :
+				rkeyEvent.test( type ) ? this.keyHooks :
+				{};
+		}
+		copy = fixHook.props ? this.props.concat( fixHook.props ) : this.props;
+
+		event = new jQuery.Event( originalEvent );
+
+		i = copy.length;
+		while ( i-- ) {
+			prop = copy[ i ];
+			event[ prop ] = originalEvent[ prop ];
+		}
+
+		// Support: IE<9
+		// Fix target property (#1925)
+		if ( !event.target ) {
+			event.target = originalEvent.srcElement || document;
+		}
+
+		// Support: Safari 6-8+
+		// Target should not be a text node (#504, #13143)
+		if ( event.target.nodeType === 3 ) {
+			event.target = event.target.parentNode;
+		}
+
+		// Support: IE<9
+		// For mouse/key events, metaKey==false if it's undefined (#3368, #11328)
+		event.metaKey = !!event.metaKey;
+
+		return fixHook.filter ? fixHook.filter( event, originalEvent ) : event;
+	},
+
+	// Includes some event props shared by KeyEvent and MouseEvent
+	props: ( "altKey bubbles cancelable ctrlKey currentTarget detail eventPhase " +
+		"metaKey relatedTarget shiftKey target timeStamp view which" ).split( " " ),
+
+	fixHooks: {},
+
+	keyHooks: {
+		props: "char charCode key keyCode".split( " " ),
+		filter: function( event, original ) {
+
+			// Add which for key events
+			if ( event.which == null ) {
+				event.which = original.charCode != null ? original.charCode : original.keyCode;
+			}
+
+			return event;
+		}
+	},
+
+	mouseHooks: {
+		props: ( "button buttons clientX clientY fromElement offsetX offsetY " +
+			"pageX pageY screenX screenY toElement" ).split( " " ),
+		filter: function( event, original ) {
+			var body, eventDoc, doc,
+				button = original.button,
+				fromElement = original.fromElement;
+
+			// Calculate pageX/Y if missing and clientX/Y available
+			if ( event.pageX == null && original.clientX != null ) {
+				eventDoc = event.target.ownerDocument || document;
+				doc = eventDoc.documentElement;
+				body = eventDoc.body;
+
+				event.pageX = original.clientX +
+					( doc && doc.scrollLeft || body && body.scrollLeft || 0 ) -
+					( doc && doc.clientLeft || body && body.clientLeft || 0 );
+				event.pageY = original.clientY +
+					( doc && doc.scrollTop  || body && body.scrollTop  || 0 ) -
+					( doc && doc.clientTop  || body && body.clientTop  || 0 );
+			}
+
+			// Add relatedTarget, if necessary
+			if ( !event.relatedTarget && fromElement ) {
+				event.relatedTarget = fromElement === event.target ?
+					original.toElement :
+					fromElement;
+			}
+
+			// Add which for click: 1 === left; 2 === middle; 3 === right
+			// Note: button is not normalized, so don't use it
+			if ( !event.which && button !== undefined ) {
+				event.which = ( button & 1 ? 1 : ( button & 2 ? 3 : ( button & 4 ? 2 : 0 ) ) );
+			}
+
+			return event;
+		}
+	},
+
+	special: {
+		load: {
+
+			// Prevent triggered image.load events from bubbling to window.load
+			noBubble: true
+		},
+		focus: {
+
+			// Fire native event if possible so blur/focus sequence is correct
+			trigger: function() {
+				if ( this !== safeActiveElement() && this.focus ) {
+					try {
+						this.focus();
+						return false;
+					} catch ( e ) {
+
+						// Support: IE<9
+						// If we error on focus to hidden element (#1486, #12518),
+						// let .trigger() run the handlers
+					}
+				}
+			},
+			delegateType: "focusin"
+		},
+		blur: {
+			trigger: function() {
+				if ( this === safeActiveElement() && this.blur ) {
+					this.blur();
+					return false;
+				}
+			},
+			delegateType: "focusout"
+		},
+		click: {
+
+			// For checkbox, fire native event so checked state will be right
+			trigger: function() {
+				if ( jQuery.nodeName( this, "input" ) && this.type === "checkbox" && this.click ) {
+					this.click();
+					return false;
+				}
+			},
+
+			// For cross-browser consistency, don't fire native .click() on links
+			_default: function( event ) {
+				return jQuery.nodeName( event.target, "a" );
+			}
+		},
+
+		beforeunload: {
+			postDispatch: function( event ) {
+
+				// Support: Firefox 20+
+				// Firefox doesn't alert if the returnValue field is not set.
+				if ( event.result !== undefined && event.originalEvent ) {
+					event.originalEvent.returnValue = event.result;
+				}
+			}
+		}
+	},
+
+	// Piggyback on a donor event to simulate a different one
+	simulate: function( type, elem, event ) {
+		var e = jQuery.extend(
+			new jQuery.Event(),
+			event,
+			{
+				type: type,
+				isSimulated: true
+
+				// Previously, `originalEvent: {}` was set here, so stopPropagation call
+				// would not be triggered on donor event, since in our own
+				// jQuery.event.stopPropagation function we had a check for existence of
+				// originalEvent.stopPropagation method, so, consequently it would be a noop.
+				//
+				// Guard for simulated events was moved to jQuery.event.stopPropagation function
+				// since `originalEvent` should point to the original event for the
+				// constancy with other events and for more focused logic
+			}
+		);
+
+		jQuery.event.trigger( e, null, elem );
+
+		if ( e.isDefaultPrevented() ) {
+			event.preventDefault();
+		}
+	}
+};
+
+jQuery.removeEvent = document.removeEventListener ?
+	function( elem, type, handle ) {
+
+		// This "if" is needed for plain objects
+		if ( elem.removeEventListener ) {
+			elem.removeEventListener( type, handle );
+		}
+	} :
+	function( elem, type, handle ) {
+		var name = "on" + type;
+
+		if ( elem.detachEvent ) {
+
+			// #8545, #7054, preventing memory leaks for custom events in IE6-8
+			// detachEvent needed property on element, by name of that event,
+			// to properly expose it to GC
+			if ( typeof elem[ name ] === "undefined" ) {
+				elem[ name ] = null;
+			}
+
+			elem.detachEvent( name, handle );
+		}
+	};
+
+jQuery.Event = function( src, props ) {
+
+	// Allow instantiation without the 'new' keyword
+	if ( !( this instanceof jQuery.Event ) ) {
+		return new jQuery.Event( src, props );
+	}
+
+	// Event object
+	if ( src && src.type ) {
+		this.originalEvent = src;
+		this.type = src.type;
+
+		// Events bubbling up the document may have been marked as prevented
+		// by a handler lower down the tree; reflect the correct value.
+		this.isDefaultPrevented = src.defaultPrevented ||
+				src.defaultPrevented === undefined &&
+
+				// Support: IE < 9, Android < 4.0
+				src.returnValue === false ?
+			returnTrue :
+			returnFalse;
+
+	// Event type
+	} else {
+		this.type = src;
+	}
+
+	// Put explicitly provided properties onto the event object
+	if ( props ) {
+		jQuery.extend( this, props );
+	}
+
+	// Create a timestamp if incoming event doesn't have one
+	this.timeStamp = src && src.timeStamp || jQuery.now();
+
+	// Mark it as fixed
+	this[ jQuery.expando ] = true;
+};
+
+// jQuery.Event is based on DOM3 Events as specified by the ECMAScript Language Binding
+// http://www.w3.org/TR/2003/WD-DOM-Level-3-Events-20030331/ecma-script-binding.html
+jQuery.Event.prototype = {
+	constructor: jQuery.Event,
+	isDefaultPrevented: returnFalse,
+	isPropagationStopped: returnFalse,
+	isImmediatePropagationStopped: returnFalse,
+
+	preventDefault: function() {
+		var e = this.originalEvent;
+
+		this.isDefaultPrevented = returnTrue;
+		if ( !e ) {
+			return;
+		}
+
+		// If preventDefault exists, run it on the original event
+		if ( e.preventDefault ) {
+			e.preventDefault();
+
+		// Support: IE
+		// Otherwise set the returnValue property of the original event to false
+		} else {
+			e.returnValue = false;
+		}
+	},
+	stopPropagation: function() {
+		var e = this.originalEvent;
+
+		this.isPropagationStopped = returnTrue;
+
+		if ( !e || this.isSimulated ) {
+			return;
+		}
+
+		// If stopPropagation exists, run it on the original event
+		if ( e.stopPropagation ) {
+			e.stopPropagation();
+		}
+
+		// Support: IE
+		// Set the cancelBubble property of the original event to true
+		e.cancelBubble = true;
+	},
+	stopImmediatePropagation: function() {
+		var e = this.originalEvent;
+
+		this.isImmediatePropagationStopped = returnTrue;
+
+		if ( e && e.stopImmediatePropagation ) {
+			e.stopImmediatePropagation();
+		}
+
+		this.stopPropagation();
+	}
+};
+
+// Create mouseenter/leave events using mouseover/out and event-time checks
+// so that event delegation works in jQuery.
+// Do the same for pointerenter/pointerleave and pointerover/pointerout
+//
+// Support: Safari 7 only
+// Safari sends mouseenter too often; see:
+// https://code.google.com/p/chromium/issues/detail?id=470258
+// for the description of the bug (it existed in older Chrome versions as well).
+jQuery.each( {
+	mouseenter: "mouseover",
+	mouseleave: "mouseout",
+	pointerenter: "pointerover",
+	pointerleave: "pointerout"
+}, function( orig, fix ) {
+	jQuery.event.special[ orig ] = {
+		delegateType: fix,
+		bindType: fix,
+
+		handle: function( event ) {
+			var ret,
+				target = this,
+				related = event.relatedTarget,
+				handleObj = event.handleObj;
+
+			// For mouseenter/leave call the handler if related is outside the target.
+			// NB: No relatedTarget if the mouse left/entered the browser window
+			if ( !related || ( related !== target && !jQuery.contains( target, related ) ) ) {
+				event.type = handleObj.origType;
+				ret = handleObj.handler.apply( this, arguments );
+				event.type = fix;
+			}
+			return ret;
+		}
+	};
+} );
+
+// IE submit delegation
+if ( !support.submit ) {
+
+	jQuery.event.special.submit = {
+		setup: function() {
+
+			// Only need this for delegated form submit events
+			if ( jQuery.nodeName( this, "form" ) ) {
+				return false;
+			}
+
+			// Lazy-add a submit handler when a descendant form may potentially be submitted
+			jQuery.event.add( this, "click._submit keypress._submit", function( e ) {
+
+				// Node name check avoids a VML-related crash in IE (#9807)
+				var elem = e.target,
+					form = jQuery.nodeName( elem, "input" ) || jQuery.nodeName( elem, "button" ) ?
+
+						// Support: IE <=8
+						// We use jQuery.prop instead of elem.form
+						// to allow fixing the IE8 delegated submit issue (gh-2332)
+						// by 3rd party polyfills/workarounds.
+						jQuery.prop( elem, "form" ) :
+						undefined;
+
+				if ( form && !jQuery._data( form, "submit" ) ) {
+					jQuery.event.add( form, "submit._submit", function( event ) {
+						event._submitBubble = true;
+					} );
+					jQuery._data( form, "submit", true );
+				}
+			} );
+
+			// return undefined since we don't need an event listener
+		},
+
+		postDispatch: function( event ) {
+
+			// If form was submitted by the user, bubble the event up the tree
+			if ( event._submitBubble ) {
+				delete event._submitBubble;
+				if ( this.parentNode && !event.isTrigger ) {
+					jQuery.event.simulate( "submit", this.parentNode, event );
+				}
+			}
+		},
+
+		teardown: function() {
+
+			// Only need this for delegated form submit events
+			if ( jQuery.nodeName( this, "form" ) ) {
+				return false;
+			}
+
+			// Remove delegated handlers; cleanData eventually reaps submit handlers attached above
+			jQuery.event.remove( this, "._submit" );
+		}
+	};
+}
+
+// IE change delegation and checkbox/radio fix
+if ( !support.change ) {
+
+	jQuery.event.special.change = {
+
+		setup: function() {
+
+			if ( rformElems.test( this.nodeName ) ) {
+
+				// IE doesn't fire change on a check/radio until blur; trigger it on click
+				// after a propertychange. Eat the blur-change in special.change.handle.
+				// This still fires onchange a second time for check/radio after blur.
+				if ( this.type === "checkbox" || this.type === "radio" ) {
+					jQuery.event.add( this, "propertychange._change", function( event ) {
+						if ( event.originalEvent.propertyName === "checked" ) {
+							this._justChanged = true;
+						}
+					} );
+					jQuery.event.add( this, "click._change", function( event ) {
+						if ( this._justChanged && !event.isTrigger ) {
+							this._justChanged = false;
+						}
+
+						// Allow triggered, simulated change events (#11500)
+						jQuery.event.simulate( "change", this, event );
+					} );
+				}
+				return false;
+			}
+
+			// Delegated event; lazy-add a change handler on descendant inputs
+			jQuery.event.add( this, "beforeactivate._change", function( e ) {
+				var elem = e.target;
+
+				if ( rformElems.test( elem.nodeName ) && !jQuery._data( elem, "change" ) ) {
+					jQuery.event.add( elem, "change._change", function( event ) {
+						if ( this.parentNode && !event.isSimulated && !event.isTrigger ) {
+							jQuery.event.simulate( "change", this.parentNode, event );
+						}
+					} );
+					jQuery._data( elem, "change", true );
+				}
+			} );
+		},
+
+		handle: function( event ) {
+			var elem = event.target;
+
+			// Swallow native change events from checkbox/radio, we already triggered them above
+			if ( this !== elem || event.isSimulated || event.isTrigger ||
+				( elem.type !== "radio" && elem.type !== "checkbox" ) ) {
+
+				return event.handleObj.handler.apply( this, arguments );
+			}
+		},
+
+		teardown: function() {
+			jQuery.event.remove( this, "._change" );
+
+			return !rformElems.test( this.nodeName );
+		}
+	};
+}
+
+// Support: Firefox
+// Firefox doesn't have focus(in | out) events
+// Related ticket - https://bugzilla.mozilla.org/show_bug.cgi?id=687787
+//
+// Support: Chrome, Safari
+// focus(in | out) events fire after focus & blur events,
+// which is spec violation - http://www.w3.org/TR/DOM-Level-3-Events/#events-focusevent-event-order
+// Related ticket - https://code.google.com/p/chromium/issues/detail?id=449857
+if ( !support.focusin ) {
+	jQuery.each( { focus: "focusin", blur: "focusout" }, function( orig, fix ) {
+
+		// Attach a single capturing handler on the document while someone wants focusin/focusout
+		var handler = function( event ) {
+			jQuery.event.simulate( fix, event.target, jQuery.event.fix( event ) );
+		};
+
+		jQuery.event.special[ fix ] = {
+			setup: function() {
+				var doc = this.ownerDocument || this,
+					attaches = jQuery._data( doc, fix );
+
+				if ( !attaches ) {
+					doc.addEventListener( orig, handler, true );
+				}
+				jQuery._data( doc, fix, ( attaches || 0 ) + 1 );
+			},
+			teardown: function() {
+				var doc = this.ownerDocument || this,
+					attaches = jQuery._data( doc, fix ) - 1;
+
+				if ( !attaches ) {
+					doc.removeEventListener( orig, handler, true );
+					jQuery._removeData( doc, fix );
+				} else {
+					jQuery._data( doc, fix, attaches );
+				}
+			}
+		};
+	} );
+}
+
+jQuery.fn.extend( {
+
+	on: function( types, selector, data, fn ) {
+		return on( this, types, selector, data, fn );
+	},
+	one: function( types, selector, data, fn ) {
+		return on( this, types, selector, data, fn, 1 );
+	},
+	off: function( types, selector, fn ) {
+		var handleObj, type;
+		if ( types && types.preventDefault && types.handleObj ) {
+
+			// ( event )  dispatched jQuery.Event
+			handleObj = types.handleObj;
+			jQuery( types.delegateTarget ).off(
+				handleObj.namespace ?
+					handleObj.origType + "." + handleObj.namespace :
+					handleObj.origType,
+				handleObj.selector,
+				handleObj.handler
+			);
+			return this;
+		}
+		if ( typeof types === "object" ) {
+
+			// ( types-object [, selector] )
+			for ( type in types ) {
+				this.off( type, selector, types[ type ] );
+			}
+			return this;
+		}
+		if ( selector === false || typeof selector === "function" ) {
+
+			// ( types [, fn] )
+			fn = selector;
+			selector = undefined;
+		}
+		if ( fn === false ) {
+			fn = returnFalse;
+		}
+		return this.each( function() {
+			jQuery.event.remove( this, types, fn, selector );
+		} );
+	},
+
+	trigger: function( type, data ) {
+		return this.each( function() {
+			jQuery.event.trigger( type, data, this );
+		} );
+	},
+	triggerHandler: function( type, data ) {
+		var elem = this[ 0 ];
+		if ( elem ) {
+			return jQuery.event.trigger( type, data, elem, true );
+		}
+	}
+} );
+
+
+var rinlinejQuery = / jQuery\d+="(?:null|\d+)"/g,
+	rnoshimcache = new RegExp( "<(?:" + nodeNames + ")[\\s/>]", "i" ),
+	rxhtmlTag = /<(?!area|br|col|embed|hr|img|input|link|meta|param)(([\w:-]+)[^>]*)\/>/gi,
+
+	// Support: IE 10-11, Edge 10240+
+	// In IE/Edge using regex groups here causes severe slowdowns.
+	// See https://connect.microsoft.com/IE/feedback/details/1736512/
+	rnoInnerhtml = /<script|<style|<link/i,
+
+	// checked="checked" or checked
+	rchecked = /checked\s*(?:[^=]|=\s*.checked.)/i,
+	rscriptTypeMasked = /^true\/(.*)/,
+	rcleanScript = /^\s*<!(?:\[CDATA\[|--)|(?:\]\]|--)>\s*$/g,
+	safeFragment = createSafeFragment( document ),
+	fragmentDiv = safeFragment.appendChild( document.createElement( "div" ) );
+
+// Support: IE<8
+// Manipulating tables requires a tbody
+function manipulationTarget( elem, content ) {
+	return jQuery.nodeName( elem, "table" ) &&
+		jQuery.nodeName( content.nodeType !== 11 ? content : content.firstChild, "tr" ) ?
+
+		elem.getElementsByTagName( "tbody" )[ 0 ] ||
+			elem.appendChild( elem.ownerDocument.createElement( "tbody" ) ) :
+		elem;
+}
+
+// Replace/restore the type attribute of script elements for safe DOM manipulation
+function disableScript( elem ) {
+	elem.type = ( jQuery.find.attr( elem, "type" ) !== null ) + "/" + elem.type;
+	return elem;
+}
+function restoreScript( elem ) {
+	var match = rscriptTypeMasked.exec( elem.type );
+	if ( match ) {
+		elem.type = match[ 1 ];
+	} else {
+		elem.removeAttribute( "type" );
+	}
+	return elem;
+}
+
+function cloneCopyEvent( src, dest ) {
+	if ( dest.nodeType !== 1 || !jQuery.hasData( src ) ) {
+		return;
+	}
+
+	var type, i, l,
+		oldData = jQuery._data( src ),
+		curData = jQuery._data( dest, oldData ),
+		events = oldData.events;
+
+	if ( events ) {
+		delete curData.handle;
+		curData.events = {};
+
+		for ( type in events ) {
+			for ( i = 0, l = events[ type ].length; i < l; i++ ) {
+				jQuery.event.add( dest, type, events[ type ][ i ] );
+			}
+		}
+	}
+
+	// make the cloned public data object a copy from the original
+	if ( curData.data ) {
+		curData.data = jQuery.extend( {}, curData.data );
+	}
+}
+
+function fixCloneNodeIssues( src, dest ) {
+	var nodeName, e, data;
+
+	// We do not need to do anything for non-Elements
+	if ( dest.nodeType !== 1 ) {
+		return;
+	}
+
+	nodeName = dest.nodeName.toLowerCase();
+
+	// IE6-8 copies events bound via attachEvent when using cloneNode.
+	if ( !support.noCloneEvent && dest[ jQuery.expando ] ) {
+		data = jQuery._data( dest );
+
+		for ( e in data.events ) {
+			jQuery.removeEvent( dest, e, data.handle );
+		}
+
+		// Event data gets referenced instead of copied if the expando gets copied too
+		dest.removeAttribute( jQuery.expando );
+	}
+
+	// IE blanks contents when cloning scripts, and tries to evaluate newly-set text
+	if ( nodeName === "script" && dest.text !== src.text ) {
+		disableScript( dest ).text = src.text;
+		restoreScript( dest );
+
+	// IE6-10 improperly clones children of object elements using classid.
+	// IE10 throws NoModificationAllowedError if parent is null, #12132.
+	} else if ( nodeName === "object" ) {
+		if ( dest.parentNode ) {
+			dest.outerHTML = src.outerHTML;
+		}
+
+		// This path appears unavoidable for IE9. When cloning an object
+		// element in IE9, the outerHTML strategy above is not sufficient.
+		// If the src has innerHTML and the destination does not,
+		// copy the src.innerHTML into the dest.innerHTML. #10324
+		if ( support.html5Clone && ( src.innerHTML && !jQuery.trim( dest.innerHTML ) ) ) {
+			dest.innerHTML = src.innerHTML;
+		}
+
+	} else if ( nodeName === "input" && rcheckableType.test( src.type ) ) {
+
+		// IE6-8 fails to persist the checked state of a cloned checkbox
+		// or radio button. Worse, IE6-7 fail to give the cloned element
+		// a checked appearance if the defaultChecked value isn't also set
+
+		dest.defaultChecked = dest.checked = src.checked;
+
+		// IE6-7 get confused and end up setting the value of a cloned
+		// checkbox/radio button to an empty string instead of "on"
+		if ( dest.value !== src.value ) {
+			dest.value = src.value;
+		}
+
+	// IE6-8 fails to return the selected option to the default selected
+	// state when cloning options
+	} else if ( nodeName === "option" ) {
+		dest.defaultSelected = dest.selected = src.defaultSelected;
+
+	// IE6-8 fails to set the defaultValue to the correct value when
+	// cloning other types of input fields
+	} else if ( nodeName === "input" || nodeName === "textarea" ) {
+		dest.defaultValue = src.defaultValue;
+	}
+}
+
+function domManip( collection, args, callback, ignored ) {
+
+	// Flatten any nested arrays
+	args = concat.apply( [], args );
+
+	var first, node, hasScripts,
+		scripts, doc, fragment,
+		i = 0,
+		l = collection.length,
+		iNoClone = l - 1,
+		value = args[ 0 ],
+		isFunction = jQuery.isFunction( value );
+
+	// We can't cloneNode fragments that contain checked, in WebKit
+	if ( isFunction ||
+			( l > 1 && typeof value === "string" &&
+				!support.checkClone && rchecked.test( value ) ) ) {
+		return collection.each( function( index ) {
+			var self = collection.eq( index );
+			if ( isFunction ) {
+				args[ 0 ] = value.call( this, index, self.html() );
+			}
+			domManip( self, args, callback, ignored );
+		} );
+	}
+
+	if ( l ) {
+		fragment = buildFragment( args, collection[ 0 ].ownerDocument, false, collection, ignored );
+		first = fragment.firstChild;
+
+		if ( fragment.childNodes.length === 1 ) {
+			fragment = first;
+		}
+
+		// Require either new content or an interest in ignored elements to invoke the callback
+		if ( first || ignored ) {
+			scripts = jQuery.map( getAll( fragment, "script" ), disableScript );
+			hasScripts = scripts.length;
+
+			// Use the original fragment for the last item
+			// instead of the first because it can end up
+			// being emptied incorrectly in certain situations (#8070).
+			for ( ; i < l; i++ ) {
+				node = fragment;
+
+				if ( i !== iNoClone ) {
+					node = jQuery.clone( node, true, true );
+
+					// Keep references to cloned scripts for later restoration
+					if ( hasScripts ) {
+
+						// Support: Android<4.1, PhantomJS<2
+						// push.apply(_, arraylike) throws on ancient WebKit
+						jQuery.merge( scripts, getAll( node, "script" ) );
+					}
+				}
+
+				callback.call( collection[ i ], node, i );
+			}
+
+			if ( hasScripts ) {
+				doc = scripts[ scripts.length - 1 ].ownerDocument;
+
+				// Reenable scripts
+				jQuery.map( scripts, restoreScript );
+
+				// Evaluate executable scripts on first document insertion
+				for ( i = 0; i < hasScripts; i++ ) {
+					node = scripts[ i ];
+					if ( rscriptType.test( node.type || "" ) &&
+						!jQuery._data( node, "globalEval" ) &&
+						jQuery.contains( doc, node ) ) {
+
+						if ( node.src ) {
+
+							// Optional AJAX dependency, but won't run scripts if not present
+							if ( jQuery._evalUrl ) {
+								jQuery._evalUrl( node.src );
+							}
+						} else {
+							jQuery.globalEval(
+								( node.text || node.textContent || node.innerHTML || "" )
+									.replace( rcleanScript, "" )
+							);
+						}
+					}
+				}
+			}
+
+			// Fix #11809: Avoid leaking memory
+			fragment = first = null;
+		}
+	}
+
+	return collection;
+}
+
+function remove( elem, selector, keepData ) {
+	var node,
+		elems = selector ? jQuery.filter( selector, elem ) : elem,
+		i = 0;
+
+	for ( ; ( node = elems[ i ] ) != null; i++ ) {
+
+		if ( !keepData && node.nodeType === 1 ) {
+			jQuery.cleanData( getAll( node ) );
+		}
+
+		if ( node.parentNode ) {
+			if ( keepData && jQuery.contains( node.ownerDocument, node ) ) {
+				setGlobalEval( getAll( node, "script" ) );
+			}
+			node.parentNode.removeChild( node );
+		}
+	}
+
+	return elem;
+}
+
+jQuery.extend( {
+	htmlPrefilter: function( html ) {
+		return html.replace( rxhtmlTag, "<$1></$2>" );
+	},
+
+	clone: function( elem, dataAndEvents, deepDataAndEvents ) {
+		var destElements, node, clone, i, srcElements,
+			inPage = jQuery.contains( elem.ownerDocument, elem );
+
+		if ( support.html5Clone || jQuery.isXMLDoc( elem ) ||
+			!rnoshimcache.test( "<" + elem.nodeName + ">" ) ) {
+
+			clone = elem.cloneNode( true );
+
+		// IE<=8 does not properly clone detached, unknown element nodes
+		} else {
+			fragmentDiv.innerHTML = elem.outerHTML;
+			fragmentDiv.removeChild( clone = fragmentDiv.firstChild );
+		}
+
+		if ( ( !support.noCloneEvent || !support.noCloneChecked ) &&
+				( elem.nodeType === 1 || elem.nodeType === 11 ) && !jQuery.isXMLDoc( elem ) ) {
+
+			// We eschew Sizzle here for performance reasons: http://jsperf.com/getall-vs-sizzle/2
+			destElements = getAll( clone );
+			srcElements = getAll( elem );
+
+			// Fix all IE cloning issues
+			for ( i = 0; ( node = srcElements[ i ] ) != null; ++i ) {
+
+				// Ensure that the destination node is not null; Fixes #9587
+				if ( destElements[ i ] ) {
+					fixCloneNodeIssues( node, destElements[ i ] );
+				}
+			}
+		}
+
+		// Copy the events from the original to the clone
+		if ( dataAndEvents ) {
+			if ( deepDataAndEvents ) {
+				srcElements = srcElements || getAll( elem );
+				destElements = destElements || getAll( clone );
+
+				for ( i = 0; ( node = srcElements[ i ] ) != null; i++ ) {
+					cloneCopyEvent( node, destElements[ i ] );
+				}
+			} else {
+				cloneCopyEvent( elem, clone );
+			}
+		}
+
+		// Preserve script evaluation history
+		destElements = getAll( clone, "script" );
+		if ( destElements.length > 0 ) {
+			setGlobalEval( destElements, !inPage && getAll( elem, "script" ) );
+		}
+
+		destElements = srcElements = node = null;
+
+		// Return the cloned set
+		return clone;
+	},
+
+	cleanData: function( elems, /* internal */ forceAcceptData ) {
+		var elem, type, id, data,
+			i = 0,
+			internalKey = jQuery.expando,
+			cache = jQuery.cache,
+			attributes = support.attributes,
+			special = jQuery.event.special;
+
+		for ( ; ( elem = elems[ i ] ) != null; i++ ) {
+			if ( forceAcceptData || acceptData( elem ) ) {
+
+				id = elem[ internalKey ];
+				data = id && cache[ id ];
+
+				if ( data ) {
+					if ( data.events ) {
+						for ( type in data.events ) {
+							if ( special[ type ] ) {
+								jQuery.event.remove( elem, type );
+
+							// This is a shortcut to avoid jQuery.event.remove's overhead
+							} else {
+								jQuery.removeEvent( elem, type, data.handle );
+							}
+						}
+					}
+
+					// Remove cache only if it was not already removed by jQuery.event.remove
+					if ( cache[ id ] ) {
+
+						delete cache[ id ];
+
+						// Support: IE<9
+						// IE does not allow us to delete expando properties from nodes
+						// IE creates expando attributes along with the property
+						// IE does not have a removeAttribute function on Document nodes
+						if ( !attributes && typeof elem.removeAttribute !== "undefined" ) {
+							elem.removeAttribute( internalKey );
+
+						// Webkit & Blink performance suffers when deleting properties
+						// from DOM nodes, so set to undefined instead
+						// https://code.google.com/p/chromium/issues/detail?id=378607
+						} else {
+							elem[ internalKey ] = undefined;
+						}
+
+						deletedIds.push( id );
+					}
+				}
+			}
+		}
+	}
+} );
+
+jQuery.fn.extend( {
+
+	// Keep domManip exposed until 3.0 (gh-2225)
+	domManip: domManip,
+
+	detach: function( selector ) {
+		return remove( this, selector, true );
+	},
+
+	remove: function( selector ) {
+		return remove( this, selector );
+	},
+
+	text: function( value ) {
+		return access( this, function( value ) {
+			return value === undefined ?
+				jQuery.text( this ) :
+				this.empty().append(
+					( this[ 0 ] && this[ 0 ].ownerDocument || document ).createTextNode( value )
+				);
+		}, null, value, arguments.length );
+	},
+
+	append: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) {
+				var target = manipulationTarget( this, elem );
+				target.appendChild( elem );
+			}
+		} );
+	},
+
+	prepend: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) {
+				var target = manipulationTarget( this, elem );
+				target.insertBefore( elem, target.firstChild );
+			}
+		} );
+	},
+
+	before: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.parentNode ) {
+				this.parentNode.insertBefore( elem, this );
+			}
+		} );
+	},
+
+	after: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.parentNode ) {
+				this.parentNode.insertBefore( elem, this.nextSibling );
+			}
+		} );
+	},
+
+	empty: function() {
+		var elem,
+			i = 0;
+
+		for ( ; ( elem = this[ i ] ) != null; i++ ) {
+
+			// Remove element nodes and prevent memory leaks
+			if ( elem.nodeType === 1 ) {
+				jQuery.cleanData( getAll( elem, false ) );
+			}
+
+			// Remove any remaining nodes
+			while ( elem.firstChild ) {
+				elem.removeChild( elem.firstChild );
+			}
+
+			// If this is a select, ensure that it displays empty (#12336)
+			// Support: IE<9
+			if ( elem.options && jQuery.nodeName( elem, "select" ) ) {
+				elem.options.length = 0;
+			}
+		}
+
+		return this;
+	},
+
+	clone: function( dataAndEvents, deepDataAndEvents ) {
+		dataAndEvents = dataAndEvents == null ? false : dataAndEvents;
+		deepDataAndEvents = deepDataAndEvents == null ? dataAndEvents : deepDataAndEvents;
+
+		return this.map( function() {
+			return jQuery.clone( this, dataAndEvents, deepDataAndEvents );
+		} );
+	},
+
+	html: function( value ) {
+		return access( this, function( value ) {
+			var elem = this[ 0 ] || {},
+				i = 0,
+				l = this.length;
+
+			if ( value === undefined ) {
+				return elem.nodeType === 1 ?
+					elem.innerHTML.replace( rinlinejQuery, "" ) :
+					undefined;
+			}
+
+			// See if we can take a shortcut and just use innerHTML
+			if ( typeof value === "string" && !rnoInnerhtml.test( value ) &&
+				( support.htmlSerialize || !rnoshimcache.test( value )  ) &&
+				( support.leadingWhitespace || !rleadingWhitespace.test( value ) ) &&
+				!wrapMap[ ( rtagName.exec( value ) || [ "", "" ] )[ 1 ].toLowerCase() ] ) {
+
+				value = jQuery.htmlPrefilter( value );
+
+				try {
+					for ( ; i < l; i++ ) {
+
+						// Remove element nodes and prevent memory leaks
+						elem = this[ i ] || {};
+						if ( elem.nodeType === 1 ) {
+							jQuery.cleanData( getAll( elem, false ) );
+							elem.innerHTML = value;
+						}
+					}
+
+					elem = 0;
+
+				// If using innerHTML throws an exception, use the fallback method
+				} catch ( e ) {}
+			}
+
+			if ( elem ) {
+				this.empty().append( value );
+			}
+		}, null, value, arguments.length );
+	},
+
+	replaceWith: function() {
+		var ignored = [];
+
+		// Make the changes, replacing each non-ignored context element with the new content
+		return domManip( this, arguments, function( elem ) {
+			var parent = this.parentNode;
+
+			if ( jQuery.inArray( this, ignored ) < 0 ) {
+				jQuery.cleanData( getAll( this ) );
+				if ( parent ) {
+					parent.replaceChild( elem, this );
+				}
+			}
+
+		// Force callback invocation
+		}, ignored );
+	}
+} );
+
+jQuery.each( {
+	appendTo: "append",
+	prependTo: "prepend",
+	insertBefore: "before",
+	insertAfter: "after",
+	replaceAll: "replaceWith"
+}, function( name, original ) {
+	jQuery.fn[ name ] = function( selector ) {
+		var elems,
+			i = 0,
+			ret = [],
+			insert = jQuery( selector ),
+			last = insert.length - 1;
+
+		for ( ; i <= last; i++ ) {
+			elems = i === last ? this : this.clone( true );
+			jQuery( insert[ i ] )[ original ]( elems );
+
+			// Modern browsers can apply jQuery collections as arrays, but oldIE needs a .get()
+			push.apply( ret, elems.get() );
+		}
+
+		return this.pushStack( ret );
+	};
+} );
+
+
+var iframe,
+	elemdisplay = {
+
+		// Support: Firefox
+		// We have to pre-define these values for FF (#10227)
+		HTML: "block",
+		BODY: "block"
+	};
+
+/**
+ * Retrieve the actual display of a element
+ * @param {String} name nodeName of the element
+ * @param {Object} doc Document object
+ */
+
+// Called only from within defaultDisplay
+function actualDisplay( name, doc ) {
+	var elem = jQuery( doc.createElement( name ) ).appendTo( doc.body ),
+
+		display = jQuery.css( elem[ 0 ], "display" );
+
+	// We don't have any data stored on the element,
+	// so use "detach" method as fast way to get rid of the element
+	elem.detach();
+
+	return display;
+}
+
+/**
+ * Try to determine the default display value of an element
+ * @param {String} nodeName
+ */
+function defaultDisplay( nodeName ) {
+	var doc = document,
+		display = elemdisplay[ nodeName ];
+
+	if ( !display ) {
+		display = actualDisplay( nodeName, doc );
+
+		// If the simple way fails, read from inside an iframe
+		if ( display === "none" || !display ) {
+
+			// Use the already-created iframe if possible
+			iframe = ( iframe || jQuery( "<iframe frameborder='0' width='0' height='0'/>" ) )
+				.appendTo( doc.documentElement );
+
+			// Always write a new HTML skeleton so Webkit and Firefox don't choke on reuse
+			doc = ( iframe[ 0 ].contentWindow || iframe[ 0 ].contentDocument ).document;
+
+			// Support: IE
+			doc.write();
+			doc.close();
+
+			display = actualDisplay( nodeName, doc );
+			iframe.detach();
+		}
+
+		// Store the correct default display
+		elemdisplay[ nodeName ] = display;
+	}
+
+	return display;
+}
+var rmargin = ( /^margin/ );
+
+var rnumnonpx = new RegExp( "^(" + pnum + ")(?!px)[a-z%]+$", "i" );
+
+var swap = function( elem, options, callback, args ) {
+	var ret, name,
+		old = {};
+
+	// Remember the old values, and insert the new ones
+	for ( name in options ) {
+		old[ name ] = elem.style[ name ];
+		elem.style[ name ] = options[ name ];
+	}
+
+	ret = callback.apply( elem, args || [] );
+
+	// Revert the old values
+	for ( name in options ) {
+		elem.style[ name ] = old[ name ];
+	}
+
+	return ret;
+};
+
+
+var documentElement = document.documentElement;
+
+
+
+( function() {
+	var pixelPositionVal, pixelMarginRightVal, boxSizingReliableVal,
+		reliableHiddenOffsetsVal, reliableMarginRightVal, reliableMarginLeftVal,
+		container = document.createElement( "div" ),
+		div = document.createElement( "div" );
+
+	// Finish early in limited (non-browser) environments
+	if ( !div.style ) {
+		return;
+	}
+
+	div.style.cssText = "float:left;opacity:.5";
+
+	// Support: IE<9
+	// Make sure that element opacity exists (as opposed to filter)
+	support.opacity = div.style.opacity === "0.5";
+
+	// Verify style float existence
+	// (IE uses styleFloat instead of cssFloat)
+	support.cssFloat = !!div.style.cssFloat;
+
+	div.style.backgroundClip = "content-box";
+	div.cloneNode( true ).style.backgroundClip = "";
+	support.clearCloneStyle = div.style.backgroundClip === "content-box";
+
+	container = document.createElement( "div" );
+	container.style.cssText = "border:0;width:8px;height:0;top:0;left:-9999px;" +
+		"padding:0;margin-top:1px;position:absolute";
+	div.innerHTML = "";
+	container.appendChild( div );
+
+	// Support: Firefox<29, Android 2.3
+	// Vendor-prefix box-sizing
+	support.boxSizing = div.style.boxSizing === "" || div.style.MozBoxSizing === "" ||
+		div.style.WebkitBoxSizing === "";
+
+	jQuery.extend( support, {
+		reliableHiddenOffsets: function() {
+			if ( pixelPositionVal == null ) {
+				computeStyleTests();
+			}
+			return reliableHiddenOffsetsVal;
+		},
+
+		boxSizingReliable: function() {
+
+			// We're checking for pixelPositionVal here instead of boxSizingReliableVal
+			// since that compresses better and they're computed together anyway.
+			if ( pixelPositionVal == null ) {
+				computeStyleTests();
+			}
+			return boxSizingReliableVal;
+		},
+
+		pixelMarginRight: function() {
+
+			// Support: Android 4.0-4.3
+			if ( pixelPositionVal == null ) {
+				computeStyleTests();
+			}
+			return pixelMarginRightVal;
+		},
+
+		pixelPosition: function() {
+			if ( pixelPositionVal == null ) {
+				computeStyleTests();
+			}
+			return pixelPositionVal;
+		},
+
+		reliableMarginRight: function() {
+
+			// Support: Android 2.3
+			if ( pixelPositionVal == null ) {
+				computeStyleTests();
+			}
+			return reliableMarginRightVal;
+		},
+
+		reliableMarginLeft: function() {
+
+			// Support: IE <=8 only, Android 4.0 - 4.3 only, Firefox <=3 - 37
+			if ( pixelPositionVal == null ) {
+				computeStyleTests();
+			}
+			return reliableMarginLeftVal;
+		}
+	} );
+
+	function computeStyleTests() {
+		var contents, divStyle,
+			documentElement = document.documentElement;
+
+		// Setup
+		documentElement.appendChild( container );
+
+		div.style.cssText =
+
+			// Support: Android 2.3
+			// Vendor-prefix box-sizing
+			"-webkit-box-sizing:border-box;box-sizing:border-box;" +
+			"position:relative;display:block;" +
+			"margin:auto;border:1px;padding:1px;" +
+			"top:1%;width:50%";
+
+		// Support: IE<9
+		// Assume reasonable values in the absence of getComputedStyle
+		pixelPositionVal = boxSizingReliableVal = reliableMarginLeftVal = false;
+		pixelMarginRightVal = reliableMarginRightVal = true;
+
+		// Check for getComputedStyle so that this code is not run in IE<9.
+		if ( window.getComputedStyle ) {
+			divStyle = window.getComputedStyle( div );
+			pixelPositionVal = ( divStyle || {} ).top !== "1%";
+			reliableMarginLeftVal = ( divStyle || {} ).marginLeft === "2px";
+			boxSizingReliableVal = ( divStyle || { width: "4px" } ).width === "4px";
+
+			// Support: Android 4.0 - 4.3 only
+			// Some styles come back with percentage values, even though they shouldn't
+			div.style.marginRight = "50%";
+			pixelMarginRightVal = ( divStyle || { marginRight: "4px" } ).marginRight === "4px";
+
+			// Support: Android 2.3 only
+			// Div with explicit width and no margin-right incorrectly
+			// gets computed margin-right based on width of container (#3333)
+			// WebKit Bug 13343 - getComputedStyle returns wrong value for margin-right
+			contents = div.appendChild( document.createElement( "div" ) );
+
+			// Reset CSS: box-sizing; display; margin; border; padding
+			contents.style.cssText = div.style.cssText =
+
+				// Support: Android 2.3
+				// Vendor-prefix box-sizing
+				"-webkit-box-sizing:content-box;-moz-box-sizing:content-box;" +
+				"box-sizing:content-box;display:block;margin:0;border:0;padding:0";
+			contents.style.marginRight = contents.style.width = "0";
+			div.style.width = "1px";
+
+			reliableMarginRightVal =
+				!parseFloat( ( window.getComputedStyle( contents ) || {} ).marginRight );
+
+			div.removeChild( contents );
+		}
+
+		// Support: IE6-8
+		// First check that getClientRects works as expected
+		// Check if table cells still have offsetWidth/Height when they are set
+		// to display:none and there are still other visible table cells in a
+		// table row; if so, offsetWidth/Height are not reliable for use when
+		// determining if an element has been hidden directly using
+		// display:none (it is still safe to use offsets if a parent element is
+		// hidden; don safety goggles and see bug #4512 for more information).
+		div.style.display = "none";
+		reliableHiddenOffsetsVal = div.getClientRects().length === 0;
+		if ( reliableHiddenOffsetsVal ) {
+			div.style.display = "";
+			div.innerHTML = "<table><tr><td></td><td>t</td></tr></table>";
+			div.childNodes[ 0 ].style.borderCollapse = "separate";
+			contents = div.getElementsByTagName( "td" );
+			contents[ 0 ].style.cssText = "margin:0;border:0;padding:0;display:none";
+			reliableHiddenOffsetsVal = contents[ 0 ].offsetHeight === 0;
+			if ( reliableHiddenOffsetsVal ) {
+				contents[ 0 ].style.display = "";
+				contents[ 1 ].style.display = "none";
+				reliableHiddenOffsetsVal = contents[ 0 ].offsetHeight === 0;
+			}
+		}
+
+		// Teardown
+		documentElement.removeChild( container );
+	}
+
+} )();
+
+
+var getStyles, curCSS,
+	rposition = /^(top|right|bottom|left)$/;
+
+if ( window.getComputedStyle ) {
+	getStyles = function( elem ) {
+
+		// Support: IE<=11+, Firefox<=30+ (#15098, #14150)
+		// IE throws on elements created in popups
+		// FF meanwhile throws on frame elements through "defaultView.getComputedStyle"
+		var view = elem.ownerDocument.defaultView;
+
+		if ( !view || !view.opener ) {
+			view = window;
+		}
+
+		return view.getComputedStyle( elem );
+	};
+
+	curCSS = function( elem, name, computed ) {
+		var width, minWidth, maxWidth, ret,
+			style = elem.style;
+
+		computed = computed || getStyles( elem );
+
+		// getPropertyValue is only needed for .css('filter') in IE9, see #12537
+		ret = computed ? computed.getPropertyValue( name ) || computed[ name ] : undefined;
+
+		// Support: Opera 12.1x only
+		// Fall back to style even without computed
+		// computed is undefined for elems on document fragments
+		if ( ( ret === "" || ret === undefined ) && !jQuery.contains( elem.ownerDocument, elem ) ) {
+			ret = jQuery.style( elem, name );
+		}
+
+		if ( computed ) {
+
+			// A tribute to the "awesome hack by Dean Edwards"
+			// Chrome < 17 and Safari 5.0 uses "computed value"
+			// instead of "used value" for margin-right
+			// Safari 5.1.7 (at least) returns percentage for a larger set of values,
+			// but width seems to be reliably pixels
+			// this is against the CSSOM draft spec:
+			// http://dev.w3.org/csswg/cssom/#resolved-values
+			if ( !support.pixelMarginRight() && rnumnonpx.test( ret ) && rmargin.test( name ) ) {
+
+				// Remember the original values
+				width = style.width;
+				minWidth = style.minWidth;
+				maxWidth = style.maxWidth;
+
+				// Put in the new values to get a computed value out
+				style.minWidth = style.maxWidth = style.width = ret;
+				ret = computed.width;
+
+				// Revert the changed values
+				style.width = width;
+				style.minWidth = minWidth;
+				style.maxWidth = maxWidth;
+			}
+		}
+
+		// Support: IE
+		// IE returns zIndex value as an integer.
+		return ret === undefined ?
+			ret :
+			ret + "";
+	};
+} else if ( documentElement.currentStyle ) {
+	getStyles = function( elem ) {
+		return elem.currentStyle;
+	};
+
+	curCSS = function( elem, name, computed ) {
+		var left, rs, rsLeft, ret,
+			style = elem.style;
+
+		computed = computed || getStyles( elem );
+		ret = computed ? computed[ name ] : undefined;
+
+		// Avoid setting ret to empty string here
+		// so we don't default to auto
+		if ( ret == null && style && style[ name ] ) {
+			ret = style[ name ];
+		}
+
+		// From the awesome hack by Dean Edwards
+		// http://erik.eae.net/archives/2007/07/27/18.54.15/#comment-102291
+
+		// If we're not dealing with a regular pixel number
+		// but a number that has a weird ending, we need to convert it to pixels
+		// but not position css attributes, as those are
+		// proportional to the parent element instead
+		// and we can't measure the parent instead because it
+		// might trigger a "stacking dolls" problem
+		if ( rnumnonpx.test( ret ) && !rposition.test( name ) ) {
+
+			// Remember the original values
+			left = style.left;
+			rs = elem.runtimeStyle;
+			rsLeft = rs && rs.left;
+
+			// Put in the new values to get a computed value out
+			if ( rsLeft ) {
+				rs.left = elem.currentStyle.left;
+			}
+			style.left = name === "fontSize" ? "1em" : ret;
+			ret = style.pixelLeft + "px";
+
+			// Revert the changed values
+			style.left = left;
+			if ( rsLeft ) {
+				rs.left = rsLeft;
+			}
+		}
+
+		// Support: IE
+		// IE returns zIndex value as an integer.
+		return ret === undefined ?
+			ret :
+			ret + "" || "auto";
+	};
+}
+
+
+
+
+function addGetHookIf( conditionFn, hookFn ) {
+
+	// Define the hook, we'll check on the first run if it's really needed.
+	return {
+		get: function() {
+			if ( conditionFn() ) {
+
+				// Hook not needed (or it's not possible to use it due
+				// to missing dependency), remove it.
+				delete this.get;
+				return;
+			}
+
+			// Hook needed; redefine it so that the support test is not executed again.
+			return ( this.get = hookFn ).apply( this, arguments );
+		}
+	};
+}
+
+
+var
+
+		ralpha = /alpha\([^)]*\)/i,
+	ropacity = /opacity\s*=\s*([^)]*)/i,
+
+	// swappable if display is none or starts with table except
+	// "table", "table-cell", or "table-caption"
+	// see here for display values:
+	// https://developer.mozilla.org/en-US/docs/CSS/display
+	rdisplayswap = /^(none|table(?!-c[ea]).+)/,
+	rnumsplit = new RegExp( "^(" + pnum + ")(.*)$", "i" ),
+
+	cssShow = { position: "absolute", visibility: "hidden", display: "block" },
+	cssNormalTransform = {
+		letterSpacing: "0",
+		fontWeight: "400"
+	},
+
+	cssPrefixes = [ "Webkit", "O", "Moz", "ms" ],
+	emptyStyle = document.createElement( "div" ).style;
+
+
+// return a css property mapped to a potentially vendor prefixed property
+function vendorPropName( name ) {
+
+	// shortcut for names that are not vendor prefixed
+	if ( name in emptyStyle ) {
+		return name;
+	}
+
+	// check for vendor prefixed names
+	var capName = name.charAt( 0 ).toUpperCase() + name.slice( 1 ),
+		i = cssPrefixes.length;
+
+	while ( i-- ) {
+		name = cssPrefixes[ i ] + capName;
+		if ( name in emptyStyle ) {
+			return name;
+		}
+	}
+}
+
+function showHide( elements, show ) {
+	var display, elem, hidden,
+		values = [],
+		index = 0,
+		length = elements.length;
+
+	for ( ; index < length; index++ ) {
+		elem = elements[ index ];
+		if ( !elem.style ) {
+			continue;
+		}
+
+		values[ index ] = jQuery._data( elem, "olddisplay" );
+		display = elem.style.display;
+		if ( show ) {
+
+			// Reset the inline display of this element to learn if it is
+			// being hidden by cascaded rules or not
+			if ( !values[ index ] && display === "none" ) {
+				elem.style.display = "";
+			}
+
+			// Set elements which have been overridden with display: none
+			// in a stylesheet to whatever the default browser style is
+			// for such an element
+			if ( elem.style.display === "" && isHidden( elem ) ) {
+				values[ index ] =
+					jQuery._data( elem, "olddisplay", defaultDisplay( elem.nodeName ) );
+			}
+		} else {
+			hidden = isHidden( elem );
+
+			if ( display && display !== "none" || !hidden ) {
+				jQuery._data(
+					elem,
+					"olddisplay",
+					hidden ? display : jQuery.css( elem, "display" )
+				);
+			}
+		}
+	}
+
+	// Set the display of most of the elements in a second loop
+	// to avoid the constant reflow
+	for ( index = 0; index < length; index++ ) {
+		elem = elements[ index ];
+		if ( !elem.style ) {
+			continue;
+		}
+		if ( !show || elem.style.display === "none" || elem.style.display === "" ) {
+			elem.style.display = show ? values[ index ] || "" : "none";
+		}
+	}
+
+	return elements;
+}
+
+function setPositiveNumber( elem, value, subtract ) {
+	var matches = rnumsplit.exec( value );
+	return matches ?
+
+		// Guard against undefined "subtract", e.g., when used as in cssHooks
+		Math.max( 0, matches[ 1 ] - ( subtract || 0 ) ) + ( matches[ 2 ] || "px" ) :
+		value;
+}
+
+function augmentWidthOrHeight( elem, name, extra, isBorderBox, styles ) {
+	var i = extra === ( isBorderBox ? "border" : "content" ) ?
+
+		// If we already have the right measurement, avoid augmentation
+		4 :
+
+		// Otherwise initialize for horizontal or vertical properties
+		name === "width" ? 1 : 0,
+
+		val = 0;
+
+	for ( ; i < 4; i += 2 ) {
+
+		// both box models exclude margin, so add it if we want it
+		if ( extra === "margin" ) {
+			val += jQuery.css( elem, extra + cssExpand[ i ], true, styles );
+		}
+
+		if ( isBorderBox ) {
+
+			// border-box includes padding, so remove it if we want content
+			if ( extra === "content" ) {
+				val -= jQuery.css( elem, "padding" + cssExpand[ i ], true, styles );
+			}
+
+			// at this point, extra isn't border nor margin, so remove border
+			if ( extra !== "margin" ) {
+				val -= jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles );
+			}
+		} else {
+
+			// at this point, extra isn't content, so add padding
+			val += jQuery.css( elem, "padding" + cssExpand[ i ], true, styles );
+
+			// at this point, extra isn't content nor padding, so add border
+			if ( extra !== "padding" ) {
+				val += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles );
+			}
+		}
+	}
+
+	return val;
+}
+
+function getWidthOrHeight( elem, name, extra ) {
+
+	// Start with offset property, which is equivalent to the border-box value
+	var valueIsBorderBox = true,
+		val = name === "width" ? elem.offsetWidth : elem.offsetHeight,
+		styles = getStyles( elem ),
+		isBorderBox = support.boxSizing &&
+			jQuery.css( elem, "boxSizing", false, styles ) === "border-box";
+
+	// some non-html elements return undefined for offsetWidth, so check for null/undefined
+	// svg - https://bugzilla.mozilla.org/show_bug.cgi?id=649285
+	// MathML - https://bugzilla.mozilla.org/show_bug.cgi?id=491668
+	if ( val <= 0 || val == null ) {
+
+		// Fall back to computed then uncomputed css if necessary
+		val = curCSS( elem, name, styles );
+		if ( val < 0 || val == null ) {
+			val = elem.style[ name ];
+		}
+
+		// Computed unit is not pixels. Stop here and return.
+		if ( rnumnonpx.test( val ) ) {
+			return val;
+		}
+
+		// we need the check for style in case a browser which returns unreliable values
+		// for getComputedStyle silently falls back to the reliable elem.style
+		valueIsBorderBox = isBorderBox &&
+			( support.boxSizingReliable() || val === elem.style[ name ] );
+
+		// Normalize "", auto, and prepare for extra
+		val = parseFloat( val ) || 0;
+	}
+
+	// use the active box-sizing model to add/subtract irrelevant styles
+	return ( val +
+		augmentWidthOrHeight(
+			elem,
+			name,
+			extra || ( isBorderBox ? "border" : "content" ),
+			valueIsBorderBox,
+			styles
+		)
+	) + "px";
+}
+
+jQuery.extend( {
+
+	// Add in style property hooks for overriding the default
+	// behavior of getting and setting a style property
+	cssHooks: {
+		opacity: {
+			get: function( elem, computed ) {
+				if ( computed ) {
+
+					// We should always get a number back from opacity
+					var ret = curCSS( elem, "opacity" );
+					return ret === "" ? "1" : ret;
+				}
+			}
+		}
+	},
+
+	// Don't automatically add "px" to these possibly-unitless properties
+	cssNumber: {
+		"animationIterationCount": true,
+		"columnCount": true,
+		"fillOpacity": true,
+		"flexGrow": true,
+		"flexShrink": true,
+		"fontWeight": true,
+		"lineHeight": true,
+		"opacity": true,
+		"order": true,
+		"orphans": true,
+		"widows": true,
+		"zIndex": true,
+		"zoom": true
+	},
+
+	// Add in properties whose names you wish to fix before
+	// setting or getting the value
+	cssProps: {
+
+		// normalize float css property
+		"float": support.cssFloat ? "cssFloat" : "styleFloat"
+	},
+
+	// Get and set the style property on a DOM Node
+	style: function( elem, name, value, extra ) {
+
+		// Don't set styles on text and comment nodes
+		if ( !elem || elem.nodeType === 3 || elem.nodeType === 8 || !elem.style ) {
+			return;
+		}
+
+		// Make sure that we're working with the right name
+		var ret, type, hooks,
+			origName = jQuery.camelCase( name ),
+			style = elem.style;
+
+		name = jQuery.cssProps[ origName ] ||
+			( jQuery.cssProps[ origName ] = vendorPropName( origName ) || origName );
+
+		// gets hook for the prefixed version
+		// followed by the unprefixed version
+		hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ];
+
+		// Check if we're setting a value
+		if ( value !== undefined ) {
+			type = typeof value;
+
+			// Convert "+=" or "-=" to relative numbers (#7345)
+			if ( type === "string" && ( ret = rcssNum.exec( value ) ) && ret[ 1 ] ) {
+				value = adjustCSS( elem, name, ret );
+
+				// Fixes bug #9237
+				type = "number";
+			}
+
+			// Make sure that null and NaN values aren't set. See: #7116
+			if ( value == null || value !== value ) {
+				return;
+			}
+
+			// If a number was passed in, add the unit (except for certain CSS properties)
+			if ( type === "number" ) {
+				value += ret && ret[ 3 ] || ( jQuery.cssNumber[ origName ] ? "" : "px" );
+			}
+
+			// Fixes #8908, it can be done more correctly by specifing setters in cssHooks,
+			// but it would mean to define eight
+			// (for every problematic property) identical functions
+			if ( !support.clearCloneStyle && value === "" && name.indexOf( "background" ) === 0 ) {
+				style[ name ] = "inherit";
+			}
+
+			// If a hook was provided, use that value, otherwise just set the specified value
+			if ( !hooks || !( "set" in hooks ) ||
+				( value = hooks.set( elem, value, extra ) ) !== undefined ) {
+
+				// Support: IE
+				// Swallow errors from 'invalid' CSS values (#5509)
+				try {
+					style[ name ] = value;
+				} catch ( e ) {}
+			}
+
+		} else {
+
+			// If a hook was provided get the non-computed value from there
+			if ( hooks && "get" in hooks &&
+				( ret = hooks.get( elem, false, extra ) ) !== undefined ) {
+
+				return ret;
+			}
+
+			// Otherwise just get the value from the style object
+			return style[ name ];
+		}
+	},
+
+	css: function( elem, name, extra, styles ) {
+		var num, val, hooks,
+			origName = jQuery.camelCase( name );
+
+		// Make sure that we're working with the right name
+		name = jQuery.cssProps[ origName ] ||
+			( jQuery.cssProps[ origName ] = vendorPropName( origName ) || origName );
+
+		// gets hook for the prefixed version
+		// followed by the unprefixed version
+		hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ];
+
+		// If a hook was provided get the computed value from there
+		if ( hooks && "get" in hooks ) {
+			val = hooks.get( elem, true, extra );
+		}
+
+		// Otherwise, if a way to get the computed value exists, use that
+		if ( val === undefined ) {
+			val = curCSS( elem, name, styles );
+		}
+
+		//convert "normal" to computed value
+		if ( val === "normal" && name in cssNormalTransform ) {
+			val = cssNormalTransform[ name ];
+		}
+
+		// Return, converting to number if forced or a qualifier was provided and val looks numeric
+		if ( extra === "" || extra ) {
+			num = parseFloat( val );
+			return extra === true || isFinite( num ) ? num || 0 : val;
+		}
+		return val;
+	}
+} );
+
+jQuery.each( [ "height", "width" ], function( i, name ) {
+	jQuery.cssHooks[ name ] = {
+		get: function( elem, computed, extra ) {
+			if ( computed ) {
+
+				// certain elements can have dimension info if we invisibly show them
+				// however, it must have a current display style that would benefit from this
+				return rdisplayswap.test( jQuery.css( elem, "display" ) ) &&
+					elem.offsetWidth === 0 ?
+						swap( elem, cssShow, function() {
+							return getWidthOrHeight( elem, name, extra );
+						} ) :
+						getWidthOrHeight( elem, name, extra );
+			}
+		},
+
+		set: function( elem, value, extra ) {
+			var styles = extra && getStyles( elem );
+			return setPositiveNumber( elem, value, extra ?
+				augmentWidthOrHeight(
+					elem,
+					name,
+					extra,
+					support.boxSizing &&
+						jQuery.css( elem, "boxSizing", false, styles ) === "border-box",
+					styles
+				) : 0
+			);
+		}
+	};
+} );
+
+if ( !support.opacity ) {
+	jQuery.cssHooks.opacity = {
+		get: function( elem, computed ) {
+
+			// IE uses filters for opacity
+			return ropacity.test( ( computed && elem.currentStyle ?
+				elem.currentStyle.filter :
+				elem.style.filter ) || "" ) ?
+					( 0.01 * parseFloat( RegExp.$1 ) ) + "" :
+					computed ? "1" : "";
+		},
+
+		set: function( elem, value ) {
+			var style = elem.style,
+				currentStyle = elem.currentStyle,
+				opacity = jQuery.isNumeric( value ) ? "alpha(opacity=" + value * 100 + ")" : "",
+				filter = currentStyle && currentStyle.filter || style.filter || "";
+
+			// IE has trouble with opacity if it does not have layout
+			// Force it by setting the zoom level
+			style.zoom = 1;
+
+			// if setting opacity to 1, and no other filters exist -
+			// attempt to remove filter attribute #6652
+			// if value === "", then remove inline opacity #12685
+			if ( ( value >= 1 || value === "" ) &&
+					jQuery.trim( filter.replace( ralpha, "" ) ) === "" &&
+					style.removeAttribute ) {
+
+				// Setting style.filter to null, "" & " " still leave "filter:" in the cssText
+				// if "filter:" is present at all, clearType is disabled, we want to avoid this
+				// style.removeAttribute is IE Only, but so apparently is this code path...
+				style.removeAttribute( "filter" );
+
+				// if there is no filter style applied in a css rule
+				// or unset inline opacity, we are done
+				if ( value === "" || currentStyle && !currentStyle.filter ) {
+					return;
+				}
+			}
+
+			// otherwise, set new filter values
+			style.filter = ralpha.test( filter ) ?
+				filter.replace( ralpha, opacity ) :
+				filter + " " + opacity;
+		}
+	};
+}
+
+jQuery.cssHooks.marginRight = addGetHookIf( support.reliableMarginRight,
+	function( elem, computed ) {
+		if ( computed ) {
+			return swap( elem, { "display": "inline-block" },
+				curCSS, [ elem, "marginRight" ] );
+		}
+	}
+);
+
+jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft,
+	function( elem, computed ) {
+		if ( computed ) {
+			return (
+				parseFloat( curCSS( elem, "marginLeft" ) ) ||
+
+				// Support: IE<=11+
+				// Running getBoundingClientRect on a disconnected node in IE throws an error
+				// Support: IE8 only
+				// getClientRects() errors on disconnected elems
+				( jQuery.contains( elem.ownerDocument, elem ) ?
+					elem.getBoundingClientRect().left -
+						swap( elem, { marginLeft: 0 }, function() {
+							return elem.getBoundingClientRect().left;
+						} ) :
+					0
+				)
+			) + "px";
+		}
+	}
+);
+
+// These hooks are used by animate to expand properties
+jQuery.each( {
+	margin: "",
+	padding: "",
+	border: "Width"
+}, function( prefix, suffix ) {
+	jQuery.cssHooks[ prefix + suffix ] = {
+		expand: function( value ) {
+			var i = 0,
+				expanded = {},
+
+				// assumes a single number if not a string
+				parts = typeof value === "string" ? value.split( " " ) : [ value ];
+
+			for ( ; i < 4; i++ ) {
+				expanded[ prefix + cssExpand[ i ] + suffix ] =
+					parts[ i ] || parts[ i - 2 ] || parts[ 0 ];
+			}
+
+			return expanded;
+		}
+	};
+
+	if ( !rmargin.test( prefix ) ) {
+		jQuery.cssHooks[ prefix + suffix ].set = setPositiveNumber;
+	}
+} );
+
+jQuery.fn.extend( {
+	css: function( name, value ) {
+		return access( this, function( elem, name, value ) {
+			var styles, len,
+				map = {},
+				i = 0;
+
+			if ( jQuery.isArray( name ) ) {
+				styles = getStyles( elem );
+				len = name.length;
+
+				for ( ; i < len; i++ ) {
+					map[ name[ i ] ] = jQuery.css( elem, name[ i ], false, styles );
+				}
+
+				return map;
+			}
+
+			return value !== undefined ?
+				jQuery.style( elem, name, value ) :
+				jQuery.css( elem, name );
+		}, name, value, arguments.length > 1 );
+	},
+	show: function() {
+		return showHide( this, true );
+	},
+	hide: function() {
+		return showHide( this );
+	},
+	toggle: function( state ) {
+		if ( typeof state === "boolean" ) {
+			return state ? this.show() : this.hide();
+		}
+
+		return this.each( function() {
+			if ( isHidden( this ) ) {
+				jQuery( this ).show();
+			} else {
+				jQuery( this ).hide();
+			}
+		} );
+	}
+} );
+
+
+function Tween( elem, options, prop, end, easing ) {
+	return new Tween.prototype.init( elem, options, prop, end, easing );
+}
+jQuery.Tween = Tween;
+
+Tween.prototype = {
+	constructor: Tween,
+	init: function( elem, options, prop, end, easing, unit ) {
+		this.elem = elem;
+		this.prop = prop;
+		this.easing = easing || jQuery.easing._default;
+		this.options = options;
+		this.start = this.now = this.cur();
+		this.end = end;
+		this.unit = unit || ( jQuery.cssNumber[ prop ] ? "" : "px" );
+	},
+	cur: function() {
+		var hooks = Tween.propHooks[ this.prop ];
+
+		return hooks && hooks.get ?
+			hooks.get( this ) :
+			Tween.propHooks._default.get( this );
+	},
+	run: function( percent ) {
+		var eased,
+			hooks = Tween.propHooks[ this.prop ];
+
+		if ( this.options.duration ) {
+			this.pos = eased = jQuery.easing[ this.easing ](
+				percent, this.options.duration * percent, 0, 1, this.options.duration
+			);
+		} else {
+			this.pos = eased = percent;
+		}
+		this.now = ( this.end - this.start ) * eased + this.start;
+
+		if ( this.options.step ) {
+			this.options.step.call( this.elem, this.now, this );
+		}
+
+		if ( hooks && hooks.set ) {
+			hooks.set( this );
+		} else {
+			Tween.propHooks._default.set( this );
+		}
+		return this;
+	}
+};
+
+Tween.prototype.init.prototype = Tween.prototype;
+
+Tween.propHooks = {
+	_default: {
+		get: function( tween ) {
+			var result;
+
+			// Use a property on the element directly when it is not a DOM element,
+			// or when there is no matching style property that exists.
+			if ( tween.elem.nodeType !== 1 ||
+				tween.elem[ tween.prop ] != null && tween.elem.style[ tween.prop ] == null ) {
+				return tween.elem[ tween.prop ];
+			}
+
+			// passing an empty string as a 3rd parameter to .css will automatically
+			// attempt a parseFloat and fallback to a string if the parse fails
+			// so, simple values such as "10px" are parsed to Float.
+			// complex values such as "rotate(1rad)" are returned as is.
+			result = jQuery.css( tween.elem, tween.prop, "" );
+
+			// Empty strings, null, undefined and "auto" are converted to 0.
+			return !result || result === "auto" ? 0 : result;
+		},
+		set: function( tween ) {
+
+			// use step hook for back compat - use cssHook if its there - use .style if its
+			// available and use plain properties where available
+			if ( jQuery.fx.step[ tween.prop ] ) {
+				jQuery.fx.step[ tween.prop ]( tween );
+			} else if ( tween.elem.nodeType === 1 &&
+				( tween.elem.style[ jQuery.cssProps[ tween.prop ] ] != null ||
+					jQuery.cssHooks[ tween.prop ] ) ) {
+				jQuery.style( tween.elem, tween.prop, tween.now + tween.unit );
+			} else {
+				tween.elem[ tween.prop ] = tween.now;
+			}
+		}
+	}
+};
+
+// Support: IE <=9
+// Panic based approach to setting things on disconnected nodes
+
+Tween.propHooks.scrollTop = Tween.propHooks.scrollLeft = {
+	set: function( tween ) {
+		if ( tween.elem.nodeType && tween.elem.parentNode ) {
+			tween.elem[ tween.prop ] = tween.now;
+		}
+	}
+};
+
+jQuery.easing = {
+	linear: function( p ) {
+		return p;
+	},
+	swing: function( p ) {
+		return 0.5 - Math.cos( p * Math.PI ) / 2;
+	},
+	_default: "swing"
+};
+
+jQuery.fx = Tween.prototype.init;
+
+// Back Compat <1.8 extension point
+jQuery.fx.step = {};
+
+
+
+
+var
+	fxNow, timerId,
+	rfxtypes = /^(?:toggle|show|hide)$/,
+	rrun = /queueHooks$/;
+
+// Animations created synchronously will run synchronously
+function createFxNow() {
+	window.setTimeout( function() {
+		fxNow = undefined;
+	} );
+	return ( fxNow = jQuery.now() );
+}
+
+// Generate parameters to create a standard animation
+function genFx( type, includeWidth ) {
+	var which,
+		attrs = { height: type },
+		i = 0;
+
+	// if we include width, step value is 1 to do all cssExpand values,
+	// if we don't include width, step value is 2 to skip over Left and Right
+	includeWidth = includeWidth ? 1 : 0;
+	for ( ; i < 4 ; i += 2 - includeWidth ) {
+		which = cssExpand[ i ];
+		attrs[ "margin" + which ] = attrs[ "padding" + which ] = type;
+	}
+
+	if ( includeWidth ) {
+		attrs.opacity = attrs.width = type;
+	}
+
+	return attrs;
+}
+
+function createTween( value, prop, animation ) {
+	var tween,
+		collection = ( Animation.tweeners[ prop ] || [] ).concat( Animation.tweeners[ "*" ] ),
+		index = 0,
+		length = collection.length;
+	for ( ; index < length; index++ ) {
+		if ( ( tween = collection[ index ].call( animation, prop, value ) ) ) {
+
+			// we're done with this property
+			return tween;
+		}
+	}
+}
+
+function defaultPrefilter( elem, props, opts ) {
+	/* jshint validthis: true */
+	var prop, value, toggle, tween, hooks, oldfire, display, checkDisplay,
+		anim = this,
+		orig = {},
+		style = elem.style,
+		hidden = elem.nodeType && isHidden( elem ),
+		dataShow = jQuery._data( elem, "fxshow" );
+
+	// handle queue: false promises
+	if ( !opts.queue ) {
+		hooks = jQuery._queueHooks( elem, "fx" );
+		if ( hooks.unqueued == null ) {
+			hooks.unqueued = 0;
+			oldfire = hooks.empty.fire;
+			hooks.empty.fire = function() {
+				if ( !hooks.unqueued ) {
+					oldfire();
+				}
+			};
+		}
+		hooks.unqueued++;
+
+		anim.always( function() {
+
+			// doing this makes sure that the complete handler will be called
+			// before this completes
+			anim.always( function() {
+				hooks.unqueued--;
+				if ( !jQuery.queue( elem, "fx" ).length ) {
+					hooks.empty.fire();
+				}
+			} );
+		} );
+	}
+
+	// height/width overflow pass
+	if ( elem.nodeType === 1 && ( "height" in props || "width" in props ) ) {
+
+		// Make sure that nothing sneaks out
+		// Record all 3 overflow attributes because IE does not
+		// change the overflow attribute when overflowX and
+		// overflowY are set to the same value
+		opts.overflow = [ style.overflow, style.overflowX, style.overflowY ];
+
+		// Set display property to inline-block for height/width
+		// animations on inline elements that are having width/height animated
+		display = jQuery.css( elem, "display" );
+
+		// Test default display if display is currently "none"
+		checkDisplay = display === "none" ?
+			jQuery._data( elem, "olddisplay" ) || defaultDisplay( elem.nodeName ) : display;
+
+		if ( checkDisplay === "inline" && jQuery.css( elem, "float" ) === "none" ) {
+
+			// inline-level elements accept inline-block;
+			// block-level elements need to be inline with layout
+			if ( !support.inlineBlockNeedsLayout || defaultDisplay( elem.nodeName ) === "inline" ) {
+				style.display = "inline-block";
+			} else {
+				style.zoom = 1;
+			}
+		}
+	}
+
+	if ( opts.overflow ) {
+		style.overflow = "hidden";
+		if ( !support.shrinkWrapBlocks() ) {
+			anim.always( function() {
+				style.overflow = opts.overflow[ 0 ];
+				style.overflowX = opts.overflow[ 1 ];
+				style.overflowY = opts.overflow[ 2 ];
+			} );
+		}
+	}
+
+	// show/hide pass
+	for ( prop in props ) {
+		value = props[ prop ];
+		if ( rfxtypes.exec( value ) ) {
+			delete props[ prop ];
+			toggle = toggle || value === "toggle";
+			if ( value === ( hidden ? "hide" : "show" ) ) {
+
+				// If there is dataShow left over from a stopped hide or show
+				// and we are going to proceed with show, we should pretend to be hidden
+				if ( value === "show" && dataShow && dataShow[ prop ] !== undefined ) {
+					hidden = true;
+				} else {
+					continue;
+				}
+			}
+			orig[ prop ] = dataShow && dataShow[ prop ] || jQuery.style( elem, prop );
+
+		// Any non-fx value stops us from restoring the original display value
+		} else {
+			display = undefined;
+		}
+	}
+
+	if ( !jQuery.isEmptyObject( orig ) ) {
+		if ( dataShow ) {
+			if ( "hidden" in dataShow ) {
+				hidden = dataShow.hidden;
+			}
+		} else {
+			dataShow = jQuery._data( elem, "fxshow", {} );
+		}
+
+		// store state if its toggle - enables .stop().toggle() to "reverse"
+		if ( toggle ) {
+			dataShow.hidden = !hidden;
+		}
+		if ( hidden ) {
+			jQuery( elem ).show();
+		} else {
+			anim.done( function() {
+				jQuery( elem ).hide();
+			} );
+		}
+		anim.done( function() {
+			var prop;
+			jQuery._removeData( elem, "fxshow" );
+			for ( prop in orig ) {
+				jQuery.style( elem, prop, orig[ prop ] );
+			}
+		} );
+		for ( prop in orig ) {
+			tween = createTween( hidden ? dataShow[ prop ] : 0, prop, anim );
+
+			if ( !( prop in dataShow ) ) {
+				dataShow[ prop ] = tween.start;
+				if ( hidden ) {
+					tween.end = tween.start;
+					tween.start = prop === "width" || prop === "height" ? 1 : 0;
+				}
+			}
+		}
+
+	// If this is a noop like .hide().hide(), restore an overwritten display value
+	} else if ( ( display === "none" ? defaultDisplay( elem.nodeName ) : display ) === "inline" ) {
+		style.display = display;
+	}
+}
+
+function propFilter( props, specialEasing ) {
+	var index, name, easing, value, hooks;
+
+	// camelCase, specialEasing and expand cssHook pass
+	for ( index in props ) {
+		name = jQuery.camelCase( index );
+		easing = specialEasing[ name ];
+		value = props[ index ];
+		if ( jQuery.isArray( value ) ) {
+			easing = value[ 1 ];
+			value = props[ index ] = value[ 0 ];
+		}
+
+		if ( index !== name ) {
+			props[ name ] = value;
+			delete props[ index ];
+		}
+
+		hooks = jQuery.cssHooks[ name ];
+		if ( hooks && "expand" in hooks ) {
+			value = hooks.expand( value );
+			delete props[ name ];
+
+			// not quite $.extend, this wont overwrite keys already present.
+			// also - reusing 'index' from above because we have the correct "name"
+			for ( index in value ) {
+				if ( !( index in props ) ) {
+					props[ index ] = value[ index ];
+					specialEasing[ index ] = easing;
+				}
+			}
+		} else {
+			specialEasing[ name ] = easing;
+		}
+	}
+}
+
+function Animation( elem, properties, options ) {
+	var result,
+		stopped,
+		index = 0,
+		length = Animation.prefilters.length,
+		deferred = jQuery.Deferred().always( function() {
+
+			// don't match elem in the :animated selector
+			delete tick.elem;
+		} ),
+		tick = function() {
+			if ( stopped ) {
+				return false;
+			}
+			var currentTime = fxNow || createFxNow(),
+				remaining = Math.max( 0, animation.startTime + animation.duration - currentTime ),
+
+				// Support: Android 2.3
+				// Archaic crash bug won't allow us to use `1 - ( 0.5 || 0 )` (#12497)
+				temp = remaining / animation.duration || 0,
+				percent = 1 - temp,
+				index = 0,
+				length = animation.tweens.length;
+
+			for ( ; index < length ; index++ ) {
+				animation.tweens[ index ].run( percent );
+			}
+
+			deferred.notifyWith( elem, [ animation, percent, remaining ] );
+
+			if ( percent < 1 && length ) {
+				return remaining;
+			} else {
+				deferred.resolveWith( elem, [ animation ] );
+				return false;
+			}
+		},
+		animation = deferred.promise( {
+			elem: elem,
+			props: jQuery.extend( {}, properties ),
+			opts: jQuery.extend( true, {
+				specialEasing: {},
+				easing: jQuery.easing._default
+			}, options ),
+			originalProperties: properties,
+			originalOptions: options,
+			startTime: fxNow || createFxNow(),
+			duration: options.duration,
+			tweens: [],
+			createTween: function( prop, end ) {
+				var tween = jQuery.Tween( elem, animation.opts, prop, end,
+						animation.opts.specialEasing[ prop ] || animation.opts.easing );
+				animation.tweens.push( tween );
+				return tween;
+			},
+			stop: function( gotoEnd ) {
+				var index = 0,
+
+					// if we are going to the end, we want to run all the tweens
+					// otherwise we skip this part
+					length = gotoEnd ? animation.tweens.length : 0;
+				if ( stopped ) {
+					return this;
+				}
+				stopped = true;
+				for ( ; index < length ; index++ ) {
+					animation.tweens[ index ].run( 1 );
+				}
+
+				// resolve when we played the last frame
+				// otherwise, reject
+				if ( gotoEnd ) {
+					deferred.notifyWith( elem, [ animation, 1, 0 ] );
+					deferred.resolveWith( elem, [ animation, gotoEnd ] );
+				} else {
+					deferred.rejectWith( elem, [ animation, gotoEnd ] );
+				}
+				return this;
+			}
+		} ),
+		props = animation.props;
+
+	propFilter( props, animation.opts.specialEasing );
+
+	for ( ; index < length ; index++ ) {
+		result = Animation.prefilters[ index ].call( animation, elem, props, animation.opts );
+		if ( result ) {
+			if ( jQuery.isFunction( result.stop ) ) {
+				jQuery._queueHooks( animation.elem, animation.opts.queue ).stop =
+					jQuery.proxy( result.stop, result );
+			}
+			return result;
+		}
+	}
+
+	jQuery.map( props, createTween, animation );
+
+	if ( jQuery.isFunction( animation.opts.start ) ) {
+		animation.opts.start.call( elem, animation );
+	}
+
+	jQuery.fx.timer(
+		jQuery.extend( tick, {
+			elem: elem,
+			anim: animation,
+			queue: animation.opts.queue
+		} )
+	);
+
+	// attach callbacks from options
+	return animation.progress( animation.opts.progress )
+		.done( animation.opts.done, animation.opts.complete )
+		.fail( animation.opts.fail )
+		.always( animation.opts.always );
+}
+
+jQuery.Animation = jQuery.extend( Animation, {
+
+	tweeners: {
+		"*": [ function( prop, value ) {
+			var tween = this.createTween( prop, value );
+			adjustCSS( tween.elem, prop, rcssNum.exec( value ), tween );
+			return tween;
+		} ]
+	},
+
+	tweener: function( props, callback ) {
+		if ( jQuery.isFunction( props ) ) {
+			callback = props;
+			props = [ "*" ];
+		} else {
+			props = props.match( rnotwhite );
+		}
+
+		var prop,
+			index = 0,
+			length = props.length;
+
+		for ( ; index < length ; index++ ) {
+			prop = props[ index ];
+			Animation.tweeners[ prop ] = Animation.tweeners[ prop ] || [];
+			Animation.tweeners[ prop ].unshift( callback );
+		}
+	},
+
+	prefilters: [ defaultPrefilter ],
+
+	prefilter: function( callback, prepend ) {
+		if ( prepend ) {
+			Animation.prefilters.unshift( callback );
+		} else {
+			Animation.prefilters.push( callback );
+		}
+	}
+} );
+
+jQuery.speed = function( speed, easing, fn ) {
+	var opt = speed && typeof speed === "object" ? jQuery.extend( {}, speed ) : {
+		complete: fn || !fn && easing ||
+			jQuery.isFunction( speed ) && speed,
+		duration: speed,
+		easing: fn && easing || easing && !jQuery.isFunction( easing ) && easing
+	};
+
+	opt.duration = jQuery.fx.off ? 0 : typeof opt.duration === "number" ? opt.duration :
+		opt.duration in jQuery.fx.speeds ?
+			jQuery.fx.speeds[ opt.duration ] : jQuery.fx.speeds._default;
+
+	// normalize opt.queue - true/undefined/null -> "fx"
+	if ( opt.queue == null || opt.queue === true ) {
+		opt.queue = "fx";
+	}
+
+	// Queueing
+	opt.old = opt.complete;
+
+	opt.complete = function() {
+		if ( jQuery.isFunction( opt.old ) ) {
+			opt.old.call( this );
+		}
+
+		if ( opt.queue ) {
+			jQuery.dequeue( this, opt.queue );
+		}
+	};
+
+	return opt;
+};
+
+jQuery.fn.extend( {
+	fadeTo: function( speed, to, easing, callback ) {
+
+		// show any hidden elements after setting opacity to 0
+		return this.filter( isHidden ).css( "opacity", 0 ).show()
+
+			// animate to the value specified
+			.end().animate( { opacity: to }, speed, easing, callback );
+	},
+	animate: function( prop, speed, easing, callback ) {
+		var empty = jQuery.isEmptyObject( prop ),
+			optall = jQuery.speed( speed, easing, callback ),
+			doAnimation = function() {
+
+				// Operate on a copy of prop so per-property easing won't be lost
+				var anim = Animation( this, jQuery.extend( {}, prop ), optall );
+
+				// Empty animations, or finishing resolves immediately
+				if ( empty || jQuery._data( this, "finish" ) ) {
+					anim.stop( true );
+				}
+			};
+			doAnimation.finish = doAnimation;
+
+		return empty || optall.queue === false ?
+			this.each( doAnimation ) :
+			this.queue( optall.queue, doAnimation );
+	},
+	stop: function( type, clearQueue, gotoEnd ) {
+		var stopQueue = function( hooks ) {
+			var stop = hooks.stop;
+			delete hooks.stop;
+			stop( gotoEnd );
+		};
+
+		if ( typeof type !== "string" ) {
+			gotoEnd = clearQueue;
+			clearQueue = type;
+			type = undefined;
+		}
+		if ( clearQueue && type !== false ) {
+			this.queue( type || "fx", [] );
+		}
+
+		return this.each( function() {
+			var dequeue = true,
+				index = type != null && type + "queueHooks",
+				timers = jQuery.timers,
+				data = jQuery._data( this );
+
+			if ( index ) {
+				if ( data[ index ] && data[ index ].stop ) {
+					stopQueue( data[ index ] );
+				}
+			} else {
+				for ( index in data ) {
+					if ( data[ index ] && data[ index ].stop && rrun.test( index ) ) {
+						stopQueue( data[ index ] );
+					}
+				}
+			}
+
+			for ( index = timers.length; index--; ) {
+				if ( timers[ index ].elem === this &&
+					( type == null || timers[ index ].queue === type ) ) {
+
+					timers[ index ].anim.stop( gotoEnd );
+					dequeue = false;
+					timers.splice( index, 1 );
+				}
+			}
+
+			// start the next in the queue if the last step wasn't forced
+			// timers currently will call their complete callbacks, which will dequeue
+			// but only if they were gotoEnd
+			if ( dequeue || !gotoEnd ) {
+				jQuery.dequeue( this, type );
+			}
+		} );
+	},
+	finish: function( type ) {
+		if ( type !== false ) {
+			type = type || "fx";
+		}
+		return this.each( function() {
+			var index,
+				data = jQuery._data( this ),
+				queue = data[ type + "queue" ],
+				hooks = data[ type + "queueHooks" ],
+				timers = jQuery.timers,
+				length = queue ? queue.length : 0;
+
+			// enable finishing flag on private data
+			data.finish = true;
+
+			// empty the queue first
+			jQuery.queue( this, type, [] );
+
+			if ( hooks && hooks.stop ) {
+				hooks.stop.call( this, true );
+			}
+
+			// look for any active animations, and finish them
+			for ( index = timers.length; index--; ) {
+				if ( timers[ index ].elem === this && timers[ index ].queue === type ) {
+					timers[ index ].anim.stop( true );
+					timers.splice( index, 1 );
+				}
+			}
+
+			// look for any animations in the old queue and finish them
+			for ( index = 0; index < length; index++ ) {
+				if ( queue[ index ] && queue[ index ].finish ) {
+					queue[ index ].finish.call( this );
+				}
+			}
+
+			// turn off finishing flag
+			delete data.finish;
+		} );
+	}
+} );
+
+jQuery.each( [ "toggle", "show", "hide" ], function( i, name ) {
+	var cssFn = jQuery.fn[ name ];
+	jQuery.fn[ name ] = function( speed, easing, callback ) {
+		return speed == null || typeof speed === "boolean" ?
+			cssFn.apply( this, arguments ) :
+			this.animate( genFx( name, true ), speed, easing, callback );
+	};
+} );
+
+// Generate shortcuts for custom animations
+jQuery.each( {
+	slideDown: genFx( "show" ),
+	slideUp: genFx( "hide" ),
+	slideToggle: genFx( "toggle" ),
+	fadeIn: { opacity: "show" },
+	fadeOut: { opacity: "hide" },
+	fadeToggle: { opacity: "toggle" }
+}, function( name, props ) {
+	jQuery.fn[ name ] = function( speed, easing, callback ) {
+		return this.animate( props, speed, easing, callback );
+	};
+} );
+
+jQuery.timers = [];
+jQuery.fx.tick = function() {
+	var timer,
+		timers = jQuery.timers,
+		i = 0;
+
+	fxNow = jQuery.now();
+
+	for ( ; i < timers.length; i++ ) {
+		timer = timers[ i ];
+
+		// Checks the timer has not already been removed
+		if ( !timer() && timers[ i ] === timer ) {
+			timers.splice( i--, 1 );
+		}
+	}
+
+	if ( !timers.length ) {
+		jQuery.fx.stop();
+	}
+	fxNow = undefined;
+};
+
+jQuery.fx.timer = function( timer ) {
+	jQuery.timers.push( timer );
+	if ( timer() ) {
+		jQuery.fx.start();
+	} else {
+		jQuery.timers.pop();
+	}
+};
+
+jQuery.fx.interval = 13;
+
+jQuery.fx.start = function() {
+	if ( !timerId ) {
+		timerId = window.setInterval( jQuery.fx.tick, jQuery.fx.interval );
+	}
+};
+
+jQuery.fx.stop = function() {
+	window.clearInterval( timerId );
+	timerId = null;
+};
+
+jQuery.fx.speeds = {
+	slow: 600,
+	fast: 200,
+
+	// Default speed
+	_default: 400
+};
+
+
+// Based off of the plugin by Clint Helfers, with permission.
+// http://web.archive.org/web/20100324014747/http://blindsignals.com/index.php/2009/07/jquery-delay/
+jQuery.fn.delay = function( time, type ) {
+	time = jQuery.fx ? jQuery.fx.speeds[ time ] || time : time;
+	type = type || "fx";
+
+	return this.queue( type, function( next, hooks ) {
+		var timeout = window.setTimeout( next, time );
+		hooks.stop = function() {
+			window.clearTimeout( timeout );
+		};
+	} );
+};
+
+
+( function() {
+	var a,
+		input = document.createElement( "input" ),
+		div = document.createElement( "div" ),
+		select = document.createElement( "select" ),
+		opt = select.appendChild( document.createElement( "option" ) );
+
+	// Setup
+	div = document.createElement( "div" );
+	div.setAttribute( "className", "t" );
+	div.innerHTML = "  <link/><table></table><a href='/a'>a</a><input type='checkbox'/>";
+	a = div.getElementsByTagName( "a" )[ 0 ];
+
+	// Support: Windows Web Apps (WWA)
+	// `type` must use .setAttribute for WWA (#14901)
+	input.setAttribute( "type", "checkbox" );
+	div.appendChild( input );
+
+	a = div.getElementsByTagName( "a" )[ 0 ];
+
+	// First batch of tests.
+	a.style.cssText = "top:1px";
+
+	// Test setAttribute on camelCase class.
+	// If it works, we need attrFixes when doing get/setAttribute (ie6/7)
+	support.getSetAttribute = div.className !== "t";
+
+	// Get the style information from getAttribute
+	// (IE uses .cssText instead)
+	support.style = /top/.test( a.getAttribute( "style" ) );
+
+	// Make sure that URLs aren't manipulated
+	// (IE normalizes it by default)
+	support.hrefNormalized = a.getAttribute( "href" ) === "/a";
+
+	// Check the default checkbox/radio value ("" on WebKit; "on" elsewhere)
+	support.checkOn = !!input.value;
+
+	// Make sure that a selected-by-default option has a working selected property.
+	// (WebKit defaults to false instead of true, IE too, if it's in an optgroup)
+	support.optSelected = opt.selected;
+
+	// Tests for enctype support on a form (#6743)
+	support.enctype = !!document.createElement( "form" ).enctype;
+
+	// Make sure that the options inside disabled selects aren't marked as disabled
+	// (WebKit marks them as disabled)
+	select.disabled = true;
+	support.optDisabled = !opt.disabled;
+
+	// Support: IE8 only
+	// Check if we can trust getAttribute("value")
+	input = document.createElement( "input" );
+	input.setAttribute( "value", "" );
+	support.input = input.getAttribute( "value" ) === "";
+
+	// Check if an input maintains its value after becoming a radio
+	input.value = "t";
+	input.setAttribute( "type", "radio" );
+	support.radioValue = input.value === "t";
+} )();
+
+
+var rreturn = /\r/g,
+	rspaces = /[\x20\t\r\n\f]+/g;
+
+jQuery.fn.extend( {
+	val: function( value ) {
+		var hooks, ret, isFunction,
+			elem = this[ 0 ];
+
+		if ( !arguments.length ) {
+			if ( elem ) {
+				hooks = jQuery.valHooks[ elem.type ] ||
+					jQuery.valHooks[ elem.nodeName.toLowerCase() ];
+
+				if (
+					hooks &&
+					"get" in hooks &&
+					( ret = hooks.get( elem, "value" ) ) !== undefined
+				) {
+					return ret;
+				}
+
+				ret = elem.value;
+
+				return typeof ret === "string" ?
+
+					// handle most common string cases
+					ret.replace( rreturn, "" ) :
+
+					// handle cases where value is null/undef or number
+					ret == null ? "" : ret;
+			}
+
+			return;
+		}
+
+		isFunction = jQuery.isFunction( value );
+
+		return this.each( function( i ) {
+			var val;
+
+			if ( this.nodeType !== 1 ) {
+				return;
+			}
+
+			if ( isFunction ) {
+				val = value.call( this, i, jQuery( this ).val() );
+			} else {
+				val = value;
+			}
+
+			// Treat null/undefined as ""; convert numbers to string
+			if ( val == null ) {
+				val = "";
+			} else if ( typeof val === "number" ) {
+				val += "";
+			} else if ( jQuery.isArray( val ) ) {
+				val = jQuery.map( val, function( value ) {
+					return value == null ? "" : value + "";
+				} );
+			}
+
+			hooks = jQuery.valHooks[ this.type ] || jQuery.valHooks[ this.nodeName.toLowerCase() ];
+
+			// If set returns undefined, fall back to normal setting
+			if ( !hooks || !( "set" in hooks ) || hooks.set( this, val, "value" ) === undefined ) {
+				this.value = val;
+			}
+		} );
+	}
+} );
+
+jQuery.extend( {
+	valHooks: {
+		option: {
+			get: function( elem ) {
+				var val = jQuery.find.attr( elem, "value" );
+				return val != null ?
+					val :
+
+					// Support: IE10-11+
+					// option.text throws exceptions (#14686, #14858)
+					// Strip and collapse whitespace
+					// https://html.spec.whatwg.org/#strip-and-collapse-whitespace
+					jQuery.trim( jQuery.text( elem ) ).replace( rspaces, " " );
+			}
+		},
+		select: {
+			get: function( elem ) {
+				var value, option,
+					options = elem.options,
+					index = elem.selectedIndex,
+					one = elem.type === "select-one" || index < 0,
+					values = one ? null : [],
+					max = one ? index + 1 : options.length,
+					i = index < 0 ?
+						max :
+						one ? index : 0;
+
+				// Loop through all the selected options
+				for ( ; i < max; i++ ) {
+					option = options[ i ];
+
+					// oldIE doesn't update selected after form reset (#2551)
+					if ( ( option.selected || i === index ) &&
+
+							// Don't return options that are disabled or in a disabled optgroup
+							( support.optDisabled ?
+								!option.disabled :
+								option.getAttribute( "disabled" ) === null ) &&
+							( !option.parentNode.disabled ||
+								!jQuery.nodeName( option.parentNode, "optgroup" ) ) ) {
+
+						// Get the specific value for the option
+						value = jQuery( option ).val();
+
+						// We don't need an array for one selects
+						if ( one ) {
+							return value;
+						}
+
+						// Multi-Selects return an array
+						values.push( value );
+					}
+				}
+
+				return values;
+			},
+
+			set: function( elem, value ) {
+				var optionSet, option,
+					options = elem.options,
+					values = jQuery.makeArray( value ),
+					i = options.length;
+
+				while ( i-- ) {
+					option = options[ i ];
+
+					if ( jQuery.inArray( jQuery.valHooks.option.get( option ), values ) > -1 ) {
+
+						// Support: IE6
+						// When new option element is added to select box we need to
+						// force reflow of newly added node in order to workaround delay
+						// of initialization properties
+						try {
+							option.selected = optionSet = true;
+
+						} catch ( _ ) {
+
+							// Will be executed only in IE6
+							option.scrollHeight;
+						}
+
+					} else {
+						option.selected = false;
+					}
+				}
+
+				// Force browsers to behave consistently when non-matching value is set
+				if ( !optionSet ) {
+					elem.selectedIndex = -1;
+				}
+
+				return options;
+			}
+		}
+	}
+} );
+
+// Radios and checkboxes getter/setter
+jQuery.each( [ "radio", "checkbox" ], function() {
+	jQuery.valHooks[ this ] = {
+		set: function( elem, value ) {
+			if ( jQuery.isArray( value ) ) {
+				return ( elem.checked = jQuery.inArray( jQuery( elem ).val(), value ) > -1 );
+			}
+		}
+	};
+	if ( !support.checkOn ) {
+		jQuery.valHooks[ this ].get = function( elem ) {
+			return elem.getAttribute( "value" ) === null ? "on" : elem.value;
+		};
+	}
+} );
+
+
+
+
+var nodeHook, boolHook,
+	attrHandle = jQuery.expr.attrHandle,
+	ruseDefault = /^(?:checked|selected)$/i,
+	getSetAttribute = support.getSetAttribute,
+	getSetInput = support.input;
+
+jQuery.fn.extend( {
+	attr: function( name, value ) {
+		return access( this, jQuery.attr, name, value, arguments.length > 1 );
+	},
+
+	removeAttr: function( name ) {
+		return this.each( function() {
+			jQuery.removeAttr( this, name );
+		} );
+	}
+} );
+
+jQuery.extend( {
+	attr: function( elem, name, value ) {
+		var ret, hooks,
+			nType = elem.nodeType;
+
+		// Don't get/set attributes on text, comment and attribute nodes
+		if ( nType === 3 || nType === 8 || nType === 2 ) {
+			return;
+		}
+
+		// Fallback to prop when attributes are not supported
+		if ( typeof elem.getAttribute === "undefined" ) {
+			return jQuery.prop( elem, name, value );
+		}
+
+		// All attributes are lowercase
+		// Grab necessary hook if one is defined
+		if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) {
+			name = name.toLowerCase();
+			hooks = jQuery.attrHooks[ name ] ||
+				( jQuery.expr.match.bool.test( name ) ? boolHook : nodeHook );
+		}
+
+		if ( value !== undefined ) {
+			if ( value === null ) {
+				jQuery.removeAttr( elem, name );
+				return;
+			}
+
+			if ( hooks && "set" in hooks &&
+				( ret = hooks.set( elem, value, name ) ) !== undefined ) {
+				return ret;
+			}
+
+			elem.setAttribute( name, value + "" );
+			return value;
+		}
+
+		if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) {
+			return ret;
+		}
+
+		ret = jQuery.find.attr( elem, name );
+
+		// Non-existent attributes return null, we normalize to undefined
+		return ret == null ? undefined : ret;
+	},
+
+	attrHooks: {
+		type: {
+			set: function( elem, value ) {
+				if ( !support.radioValue && value === "radio" &&
+					jQuery.nodeName( elem, "input" ) ) {
+
+					// Setting the type on a radio button after the value resets the value in IE8-9
+					// Reset value to default in case type is set after value during creation
+					var val = elem.value;
+					elem.setAttribute( "type", value );
+					if ( val ) {
+						elem.value = val;
+					}
+					return value;
+				}
+			}
+		}
+	},
+
+	removeAttr: function( elem, value ) {
+		var name, propName,
+			i = 0,
+			attrNames = value && value.match( rnotwhite );
+
+		if ( attrNames && elem.nodeType === 1 ) {
+			while ( ( name = attrNames[ i++ ] ) ) {
+				propName = jQuery.propFix[ name ] || name;
+
+				// Boolean attributes get special treatment (#10870)
+				if ( jQuery.expr.match.bool.test( name ) ) {
+
+					// Set corresponding property to false
+					if ( getSetInput && getSetAttribute || !ruseDefault.test( name ) ) {
+						elem[ propName ] = false;
+
+					// Support: IE<9
+					// Also clear defaultChecked/defaultSelected (if appropriate)
+					} else {
+						elem[ jQuery.camelCase( "default-" + name ) ] =
+							elem[ propName ] = false;
+					}
+
+				// See #9699 for explanation of this approach (setting first, then removal)
+				} else {
+					jQuery.attr( elem, name, "" );
+				}
+
+				elem.removeAttribute( getSetAttribute ? name : propName );
+			}
+		}
+	}
+} );
+
+// Hooks for boolean attributes
+boolHook = {
+	set: function( elem, value, name ) {
+		if ( value === false ) {
+
+			// Remove boolean attributes when set to false
+			jQuery.removeAttr( elem, name );
+		} else if ( getSetInput && getSetAttribute || !ruseDefault.test( name ) ) {
+
+			// IE<8 needs the *property* name
+			elem.setAttribute( !getSetAttribute && jQuery.propFix[ name ] || name, name );
+
+		} else {
+
+			// Support: IE<9
+			// Use defaultChecked and defaultSelected for oldIE
+			elem[ jQuery.camelCase( "default-" + name ) ] = elem[ name ] = true;
+		}
+		return name;
+	}
+};
+
+jQuery.each( jQuery.expr.match.bool.source.match( /\w+/g ), function( i, name ) {
+	var getter = attrHandle[ name ] || jQuery.find.attr;
+
+	if ( getSetInput && getSetAttribute || !ruseDefault.test( name ) ) {
+		attrHandle[ name ] = function( elem, name, isXML ) {
+			var ret, handle;
+			if ( !isXML ) {
+
+				// Avoid an infinite loop by temporarily removing this function from the getter
+				handle = attrHandle[ name ];
+				attrHandle[ name ] = ret;
+				ret = getter( elem, name, isXML ) != null ?
+					name.toLowerCase() :
+					null;
+				attrHandle[ name ] = handle;
+			}
+			return ret;
+		};
+	} else {
+		attrHandle[ name ] = function( elem, name, isXML ) {
+			if ( !isXML ) {
+				return elem[ jQuery.camelCase( "default-" + name ) ] ?
+					name.toLowerCase() :
+					null;
+			}
+		};
+	}
+} );
+
+// fix oldIE attroperties
+if ( !getSetInput || !getSetAttribute ) {
+	jQuery.attrHooks.value = {
+		set: function( elem, value, name ) {
+			if ( jQuery.nodeName( elem, "input" ) ) {
+
+				// Does not return so that setAttribute is also used
+				elem.defaultValue = value;
+			} else {
+
+				// Use nodeHook if defined (#1954); otherwise setAttribute is fine
+				return nodeHook && nodeHook.set( elem, value, name );
+			}
+		}
+	};
+}
+
+// IE6/7 do not support getting/setting some attributes with get/setAttribute
+if ( !getSetAttribute ) {
+
+	// Use this for any attribute in IE6/7
+	// This fixes almost every IE6/7 issue
+	nodeHook = {
+		set: function( elem, value, name ) {
+
+			// Set the existing or create a new attribute node
+			var ret = elem.getAttributeNode( name );
+			if ( !ret ) {
+				elem.setAttributeNode(
+					( ret = elem.ownerDocument.createAttribute( name ) )
+				);
+			}
+
+			ret.value = value += "";
+
+			// Break association with cloned elements by also using setAttribute (#9646)
+			if ( name === "value" || value === elem.getAttribute( name ) ) {
+				return value;
+			}
+		}
+	};
+
+	// Some attributes are constructed with empty-string values when not defined
+	attrHandle.id = attrHandle.name = attrHandle.coords =
+		function( elem, name, isXML ) {
+			var ret;
+			if ( !isXML ) {
+				return ( ret = elem.getAttributeNode( name ) ) && ret.value !== "" ?
+					ret.value :
+					null;
+			}
+		};
+
+	// Fixing value retrieval on a button requires this module
+	jQuery.valHooks.button = {
+		get: function( elem, name ) {
+			var ret = elem.getAttributeNode( name );
+			if ( ret && ret.specified ) {
+				return ret.value;
+			}
+		},
+		set: nodeHook.set
+	};
+
+	// Set contenteditable to false on removals(#10429)
+	// Setting to empty string throws an error as an invalid value
+	jQuery.attrHooks.contenteditable = {
+		set: function( elem, value, name ) {
+			nodeHook.set( elem, value === "" ? false : value, name );
+		}
+	};
+
+	// Set width and height to auto instead of 0 on empty string( Bug #8150 )
+	// This is for removals
+	jQuery.each( [ "width", "height" ], function( i, name ) {
+		jQuery.attrHooks[ name ] = {
+			set: function( elem, value ) {
+				if ( value === "" ) {
+					elem.setAttribute( name, "auto" );
+					return value;
+				}
+			}
+		};
+	} );
+}
+
+if ( !support.style ) {
+	jQuery.attrHooks.style = {
+		get: function( elem ) {
+
+			// Return undefined in the case of empty string
+			// Note: IE uppercases css property names, but if we were to .toLowerCase()
+			// .cssText, that would destroy case sensitivity in URL's, like in "background"
+			return elem.style.cssText || undefined;
+		},
+		set: function( elem, value ) {
+			return ( elem.style.cssText = value + "" );
+		}
+	};
+}
+
+
+
+
+var rfocusable = /^(?:input|select|textarea|button|object)$/i,
+	rclickable = /^(?:a|area)$/i;
+
+jQuery.fn.extend( {
+	prop: function( name, value ) {
+		return access( this, jQuery.prop, name, value, arguments.length > 1 );
+	},
+
+	removeProp: function( name ) {
+		name = jQuery.propFix[ name ] || name;
+		return this.each( function() {
+
+			// try/catch handles cases where IE balks (such as removing a property on window)
+			try {
+				this[ name ] = undefined;
+				delete this[ name ];
+			} catch ( e ) {}
+		} );
+	}
+} );
+
+jQuery.extend( {
+	prop: function( elem, name, value ) {
+		var ret, hooks,
+			nType = elem.nodeType;
+
+		// Don't get/set properties on text, comment and attribute nodes
+		if ( nType === 3 || nType === 8 || nType === 2 ) {
+			return;
+		}
+
+		if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) {
+
+			// Fix name and attach hooks
+			name = jQuery.propFix[ name ] || name;
+			hooks = jQuery.propHooks[ name ];
+		}
+
+		if ( value !== undefined ) {
+			if ( hooks && "set" in hooks &&
+				( ret = hooks.set( elem, value, name ) ) !== undefined ) {
+				return ret;
+			}
+
+			return ( elem[ name ] = value );
+		}
+
+		if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) {
+			return ret;
+		}
+
+		return elem[ name ];
+	},
+
+	propHooks: {
+		tabIndex: {
+			get: function( elem ) {
+
+				// elem.tabIndex doesn't always return the
+				// correct value when it hasn't been explicitly set
+				// http://fluidproject.org/blog/2008/01/09/getting-setting-and-removing-tabindex-values-with-javascript/
+				// Use proper attribute retrieval(#12072)
+				var tabindex = jQuery.find.attr( elem, "tabindex" );
+
+				return tabindex ?
+					parseInt( tabindex, 10 ) :
+					rfocusable.test( elem.nodeName ) ||
+						rclickable.test( elem.nodeName ) && elem.href ?
+							0 :
+							-1;
+			}
+		}
+	},
+
+	propFix: {
+		"for": "htmlFor",
+		"class": "className"
+	}
+} );
+
+// Some attributes require a special call on IE
+// http://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx
+if ( !support.hrefNormalized ) {
+
+	// href/src property should get the full normalized URL (#10299/#12915)
+	jQuery.each( [ "href", "src" ], function( i, name ) {
+		jQuery.propHooks[ name ] = {
+			get: function( elem ) {
+				return elem.getAttribute( name, 4 );
+			}
+		};
+	} );
+}
+
+// Support: Safari, IE9+
+// Accessing the selectedIndex property
+// forces the browser to respect setting selected
+// on the option
+// The getter ensures a default option is selected
+// when in an optgroup
+if ( !support.optSelected ) {
+	jQuery.propHooks.selected = {
+		get: function( elem ) {
+			var parent = elem.parentNode;
+
+			if ( parent ) {
+				parent.selectedIndex;
+
+				// Make sure that it also works with optgroups, see #5701
+				if ( parent.parentNode ) {
+					parent.parentNode.selectedIndex;
+				}
+			}
+			return null;
+		},
+		set: function( elem ) {
+			var parent = elem.parentNode;
+			if ( parent ) {
+				parent.selectedIndex;
+
+				if ( parent.parentNode ) {
+					parent.parentNode.selectedIndex;
+				}
+			}
+		}
+	};
+}
+
+jQuery.each( [
+	"tabIndex",
+	"readOnly",
+	"maxLength",
+	"cellSpacing",
+	"cellPadding",
+	"rowSpan",
+	"colSpan",
+	"useMap",
+	"frameBorder",
+	"contentEditable"
+], function() {
+	jQuery.propFix[ this.toLowerCase() ] = this;
+} );
+
+// IE6/7 call enctype encoding
+if ( !support.enctype ) {
+	jQuery.propFix.enctype = "encoding";
+}
+
+
+
+
+var rclass = /[\t\r\n\f]/g;
+
+function getClass( elem ) {
+	return jQuery.attr( elem, "class" ) || "";
+}
+
+jQuery.fn.extend( {
+	addClass: function( value ) {
+		var classes, elem, cur, curValue, clazz, j, finalValue,
+			i = 0;
+
+		if ( jQuery.isFunction( value ) ) {
+			return this.each( function( j ) {
+				jQuery( this ).addClass( value.call( this, j, getClass( this ) ) );
+			} );
+		}
+
+		if ( typeof value === "string" && value ) {
+			classes = value.match( rnotwhite ) || [];
+
+			while ( ( elem = this[ i++ ] ) ) {
+				curValue = getClass( elem );
+				cur = elem.nodeType === 1 &&
+					( " " + curValue + " " ).replace( rclass, " " );
+
+				if ( cur ) {
+					j = 0;
+					while ( ( clazz = classes[ j++ ] ) ) {
+						if ( cur.indexOf( " " + clazz + " " ) < 0 ) {
+							cur += clazz + " ";
+						}
+					}
+
+					// only assign if different to avoid unneeded rendering.
+					finalValue = jQuery.trim( cur );
+					if ( curValue !== finalValue ) {
+						jQuery.attr( elem, "class", finalValue );
+					}
+				}
+			}
+		}
+
+		return this;
+	},
+
+	removeClass: function( value ) {
+		var classes, elem, cur, curValue, clazz, j, finalValue,
+			i = 0;
+
+		if ( jQuery.isFunction( value ) ) {
+			return this.each( function( j ) {
+				jQuery( this ).removeClass( value.call( this, j, getClass( this ) ) );
+			} );
+		}
+
+		if ( !arguments.length ) {
+			return this.attr( "class", "" );
+		}
+
+		if ( typeof value === "string" && value ) {
+			classes = value.match( rnotwhite ) || [];
+
+			while ( ( elem = this[ i++ ] ) ) {
+				curValue = getClass( elem );
+
+				// This expression is here for better compressibility (see addClass)
+				cur = elem.nodeType === 1 &&
+					( " " + curValue + " " ).replace( rclass, " " );
+
+				if ( cur ) {
+					j = 0;
+					while ( ( clazz = classes[ j++ ] ) ) {
+
+						// Remove *all* instances
+						while ( cur.indexOf( " " + clazz + " " ) > -1 ) {
+							cur = cur.replace( " " + clazz + " ", " " );
+						}
+					}
+
+					// Only assign if different to avoid unneeded rendering.
+					finalValue = jQuery.trim( cur );
+					if ( curValue !== finalValue ) {
+						jQuery.attr( elem, "class", finalValue );
+					}
+				}
+			}
+		}
+
+		return this;
+	},
+
+	toggleClass: function( value, stateVal ) {
+		var type = typeof value;
+
+		if ( typeof stateVal === "boolean" && type === "string" ) {
+			return stateVal ? this.addClass( value ) : this.removeClass( value );
+		}
+
+		if ( jQuery.isFunction( value ) ) {
+			return this.each( function( i ) {
+				jQuery( this ).toggleClass(
+					value.call( this, i, getClass( this ), stateVal ),
+					stateVal
+				);
+			} );
+		}
+
+		return this.each( function() {
+			var className, i, self, classNames;
+
+			if ( type === "string" ) {
+
+				// Toggle individual class names
+				i = 0;
+				self = jQuery( this );
+				classNames = value.match( rnotwhite ) || [];
+
+				while ( ( className = classNames[ i++ ] ) ) {
+
+					// Check each className given, space separated list
+					if ( self.hasClass( className ) ) {
+						self.removeClass( className );
+					} else {
+						self.addClass( className );
+					}
+				}
+
+			// Toggle whole class name
+			} else if ( value === undefined || type === "boolean" ) {
+				className = getClass( this );
+				if ( className ) {
+
+					// store className if set
+					jQuery._data( this, "__className__", className );
+				}
+
+				// If the element has a class name or if we're passed "false",
+				// then remove the whole classname (if there was one, the above saved it).
+				// Otherwise bring back whatever was previously saved (if anything),
+				// falling back to the empty string if nothing was stored.
+				jQuery.attr( this, "class",
+					className || value === false ?
+					"" :
+					jQuery._data( this, "__className__" ) || ""
+				);
+			}
+		} );
+	},
+
+	hasClass: function( selector ) {
+		var className, elem,
+			i = 0;
+
+		className = " " + selector + " ";
+		while ( ( elem = this[ i++ ] ) ) {
+			if ( elem.nodeType === 1 &&
+				( " " + getClass( elem ) + " " ).replace( rclass, " " )
+					.indexOf( className ) > -1
+			) {
+				return true;
+			}
+		}
+
+		return false;
+	}
+} );
+
+
+
+
+// Return jQuery for attributes-only inclusion
+
+
+jQuery.each( ( "blur focus focusin focusout load resize scroll unload click dblclick " +
+	"mousedown mouseup mousemove mouseover mouseout mouseenter mouseleave " +
+	"change select submit keydown keypress keyup error contextmenu" ).split( " " ),
+	function( i, name ) {
+
+	// Handle event binding
+	jQuery.fn[ name ] = function( data, fn ) {
+		return arguments.length > 0 ?
+			this.on( name, null, data, fn ) :
+			this.trigger( name );
+	};
+} );
+
+jQuery.fn.extend( {
+	hover: function( fnOver, fnOut ) {
+		return this.mouseenter( fnOver ).mouseleave( fnOut || fnOver );
+	}
+} );
+
+
+var location = window.location;
+
+var nonce = jQuery.now();
+
+var rquery = ( /\?/ );
+
+
+
+var rvalidtokens = /(,)|(\[|{)|(}|])|"(?:[^"\\\r\n]|\\["\\\/bfnrt]|\\u[\da-fA-F]{4})*"\s*:?|true|false|null|-?(?!0\d)\d+(?:\.\d+|)(?:[eE][+-]?\d+|)/g;
+
+jQuery.parseJSON = function( data ) {
+
+	// Attempt to parse using the native JSON parser first
+	if ( window.JSON && window.JSON.parse ) {
+
+		// Support: Android 2.3
+		// Workaround failure to string-cast null input
+		return window.JSON.parse( data + "" );
+	}
+
+	var requireNonComma,
+		depth = null,
+		str = jQuery.trim( data + "" );
+
+	// Guard against invalid (and possibly dangerous) input by ensuring that nothing remains
+	// after removing valid tokens
+	return str && !jQuery.trim( str.replace( rvalidtokens, function( token, comma, open, close ) {
+
+		// Force termination if we see a misplaced comma
+		if ( requireNonComma && comma ) {
+			depth = 0;
+		}
+
+		// Perform no more replacements after returning to outermost depth
+		if ( depth === 0 ) {
+			return token;
+		}
+
+		// Commas must not follow "[", "{", or ","
+		requireNonComma = open || comma;
+
+		// Determine new depth
+		// array/object open ("[" or "{"): depth += true - false (increment)
+		// array/object close ("]" or "}"): depth += false - true (decrement)
+		// other cases ("," or primitive): depth += true - true (numeric cast)
+		depth += !close - !open;
+
+		// Remove this token
+		return "";
+	} ) ) ?
+		( Function( "return " + str ) )() :
+		jQuery.error( "Invalid JSON: " + data );
+};
+
+
+// Cross-browser xml parsing
+jQuery.parseXML = function( data ) {
+	var xml, tmp;
+	if ( !data || typeof data !== "string" ) {
+		return null;
+	}
+	try {
+		if ( window.DOMParser ) { // Standard
+			tmp = new window.DOMParser();
+			xml = tmp.parseFromString( data, "text/xml" );
+		} else { // IE
+			xml = new window.ActiveXObject( "Microsoft.XMLDOM" );
+			xml.async = "false";
+			xml.loadXML( data );
+		}
+	} catch ( e ) {
+		xml = undefined;
+	}
+	if ( !xml || !xml.documentElement || xml.getElementsByTagName( "parsererror" ).length ) {
+		jQuery.error( "Invalid XML: " + data );
+	}
+	return xml;
+};
+
+
+var
+	rhash = /#.*$/,
+	rts = /([?&])_=[^&]*/,
+
+	// IE leaves an \r character at EOL
+	rheaders = /^(.*?):[ \t]*([^\r\n]*)\r?$/mg,
+
+	// #7653, #8125, #8152: local protocol detection
+	rlocalProtocol = /^(?:about|app|app-storage|.+-extension|file|res|widget):$/,
+	rnoContent = /^(?:GET|HEAD)$/,
+	rprotocol = /^\/\//,
+	rurl = /^([\w.+-]+:)(?:\/\/(?:[^\/?#]*@|)([^\/?#:]*)(?::(\d+)|)|)/,
+
+	/* Prefilters
+	 * 1) They are useful to introduce custom dataTypes (see ajax/jsonp.js for an example)
+	 * 2) These are called:
+	 *    - BEFORE asking for a transport
+	 *    - AFTER param serialization (s.data is a string if s.processData is true)
+	 * 3) key is the dataType
+	 * 4) the catchall symbol "*" can be used
+	 * 5) execution will start with transport dataType and THEN continue down to "*" if needed
+	 */
+	prefilters = {},
+
+	/* Transports bindings
+	 * 1) key is the dataType
+	 * 2) the catchall symbol "*" can be used
+	 * 3) selection will start with transport dataType and THEN go to "*" if needed
+	 */
+	transports = {},
+
+	// Avoid comment-prolog char sequence (#10098); must appease lint and evade compression
+	allTypes = "*/".concat( "*" ),
+
+	// Document location
+	ajaxLocation = location.href,
+
+	// Segment location into parts
+	ajaxLocParts = rurl.exec( ajaxLocation.toLowerCase() ) || [];
+
+// Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport
+function addToPrefiltersOrTransports( structure ) {
+
+	// dataTypeExpression is optional and defaults to "*"
+	return function( dataTypeExpression, func ) {
+
+		if ( typeof dataTypeExpression !== "string" ) {
+			func = dataTypeExpression;
+			dataTypeExpression = "*";
+		}
+
+		var dataType,
+			i = 0,
+			dataTypes = dataTypeExpression.toLowerCase().match( rnotwhite ) || [];
+
+		if ( jQuery.isFunction( func ) ) {
+
+			// For each dataType in the dataTypeExpression
+			while ( ( dataType = dataTypes[ i++ ] ) ) {
+
+				// Prepend if requested
+				if ( dataType.charAt( 0 ) === "+" ) {
+					dataType = dataType.slice( 1 ) || "*";
+					( structure[ dataType ] = structure[ dataType ] || [] ).unshift( func );
+
+				// Otherwise append
+				} else {
+					( structure[ dataType ] = structure[ dataType ] || [] ).push( func );
+				}
+			}
+		}
+	};
+}
+
+// Base inspection function for prefilters and transports
+function inspectPrefiltersOrTransports( structure, options, originalOptions, jqXHR ) {
+
+	var inspected = {},
+		seekingTransport = ( structure === transports );
+
+	function inspect( dataType ) {
+		var selected;
+		inspected[ dataType ] = true;
+		jQuery.each( structure[ dataType ] || [], function( _, prefilterOrFactory ) {
+			var dataTypeOrTransport = prefilterOrFactory( options, originalOptions, jqXHR );
+			if ( typeof dataTypeOrTransport === "string" &&
+				!seekingTransport && !inspected[ dataTypeOrTransport ] ) {
+
+				options.dataTypes.unshift( dataTypeOrTransport );
+				inspect( dataTypeOrTransport );
+				return false;
+			} else if ( seekingTransport ) {
+				return !( selected = dataTypeOrTransport );
+			}
+		} );
+		return selected;
+	}
+
+	return inspect( options.dataTypes[ 0 ] ) || !inspected[ "*" ] && inspect( "*" );
+}
+
+// A special extend for ajax options
+// that takes "flat" options (not to be deep extended)
+// Fixes #9887
+function ajaxExtend( target, src ) {
+	var deep, key,
+		flatOptions = jQuery.ajaxSettings.flatOptions || {};
+
+	for ( key in src ) {
+		if ( src[ key ] !== undefined ) {
+			( flatOptions[ key ] ? target : ( deep || ( deep = {} ) ) )[ key ] = src[ key ];
+		}
+	}
+	if ( deep ) {
+		jQuery.extend( true, target, deep );
+	}
+
+	return target;
+}
+
+/* Handles responses to an ajax request:
+ * - finds the right dataType (mediates between content-type and expected dataType)
+ * - returns the corresponding response
+ */
+function ajaxHandleResponses( s, jqXHR, responses ) {
+	var firstDataType, ct, finalDataType, type,
+		contents = s.contents,
+		dataTypes = s.dataTypes;
+
+	// Remove auto dataType and get content-type in the process
+	while ( dataTypes[ 0 ] === "*" ) {
+		dataTypes.shift();
+		if ( ct === undefined ) {
+			ct = s.mimeType || jqXHR.getResponseHeader( "Content-Type" );
+		}
+	}
+
+	// Check if we're dealing with a known content-type
+	if ( ct ) {
+		for ( type in contents ) {
+			if ( contents[ type ] && contents[ type ].test( ct ) ) {
+				dataTypes.unshift( type );
+				break;
+			}
+		}
+	}
+
+	// Check to see if we have a response for the expected dataType
+	if ( dataTypes[ 0 ] in responses ) {
+		finalDataType = dataTypes[ 0 ];
+	} else {
+
+		// Try convertible dataTypes
+		for ( type in responses ) {
+			if ( !dataTypes[ 0 ] || s.converters[ type + " " + dataTypes[ 0 ] ] ) {
+				finalDataType = type;
+				break;
+			}
+			if ( !firstDataType ) {
+				firstDataType = type;
+			}
+		}
+
+		// Or just use first one
+		finalDataType = finalDataType || firstDataType;
+	}
+
+	// If we found a dataType
+	// We add the dataType to the list if needed
+	// and return the corresponding response
+	if ( finalDataType ) {
+		if ( finalDataType !== dataTypes[ 0 ] ) {
+			dataTypes.unshift( finalDataType );
+		}
+		return responses[ finalDataType ];
+	}
+}
+
+/* Chain conversions given the request and the original response
+ * Also sets the responseXXX fields on the jqXHR instance
+ */
+function ajaxConvert( s, response, jqXHR, isSuccess ) {
+	var conv2, current, conv, tmp, prev,
+		converters = {},
+
+		// Work with a copy of dataTypes in case we need to modify it for conversion
+		dataTypes = s.dataTypes.slice();
+
+	// Create converters map with lowercased keys
+	if ( dataTypes[ 1 ] ) {
+		for ( conv in s.converters ) {
+			converters[ conv.toLowerCase() ] = s.converters[ conv ];
+		}
+	}
+
+	current = dataTypes.shift();
+
+	// Convert to each sequential dataType
+	while ( current ) {
+
+		if ( s.responseFields[ current ] ) {
+			jqXHR[ s.responseFields[ current ] ] = response;
+		}
+
+		// Apply the dataFilter if provided
+		if ( !prev && isSuccess && s.dataFilter ) {
+			response = s.dataFilter( response, s.dataType );
+		}
+
+		prev = current;
+		current = dataTypes.shift();
+
+		if ( current ) {
+
+			// There's only work to do if current dataType is non-auto
+			if ( current === "*" ) {
+
+				current = prev;
+
+			// Convert response if prev dataType is non-auto and differs from current
+			} else if ( prev !== "*" && prev !== current ) {
+
+				// Seek a direct converter
+				conv = converters[ prev + " " + current ] || converters[ "* " + current ];
+
+				// If none found, seek a pair
+				if ( !conv ) {
+					for ( conv2 in converters ) {
+
+						// If conv2 outputs current
+						tmp = conv2.split( " " );
+						if ( tmp[ 1 ] === current ) {
+
+							// If prev can be converted to accepted input
+							conv = converters[ prev + " " + tmp[ 0 ] ] ||
+								converters[ "* " + tmp[ 0 ] ];
+							if ( conv ) {
+
+								// Condense equivalence converters
+								if ( conv === true ) {
+									conv = converters[ conv2 ];
+
+								// Otherwise, insert the intermediate dataType
+								} else if ( converters[ conv2 ] !== true ) {
+									current = tmp[ 0 ];
+									dataTypes.unshift( tmp[ 1 ] );
+								}
+								break;
+							}
+						}
+					}
+				}
+
+				// Apply converter (if not an equivalence)
+				if ( conv !== true ) {
+
+					// Unless errors are allowed to bubble, catch and return them
+					if ( conv && s[ "throws" ] ) { // jscs:ignore requireDotNotation
+						response = conv( response );
+					} else {
+						try {
+							response = conv( response );
+						} catch ( e ) {
+							return {
+								state: "parsererror",
+								error: conv ? e : "No conversion from " + prev + " to " + current
+							};
+						}
+					}
+				}
+			}
+		}
+	}
+
+	return { state: "success", data: response };
+}
+
+jQuery.extend( {
+
+	// Counter for holding the number of active queries
+	active: 0,
+
+	// Last-Modified header cache for next request
+	lastModified: {},
+	etag: {},
+
+	ajaxSettings: {
+		url: ajaxLocation,
+		type: "GET",
+		isLocal: rlocalProtocol.test( ajaxLocParts[ 1 ] ),
+		global: true,
+		processData: true,
+		async: true,
+		contentType: "application/x-www-form-urlencoded; charset=UTF-8",
+		/*
+		timeout: 0,
+		data: null,
+		dataType: null,
+		username: null,
+		password: null,
+		cache: null,
+		throws: false,
+		traditional: false,
+		headers: {},
+		*/
+
+		accepts: {
+			"*": allTypes,
+			text: "text/plain",
+			html: "text/html",
+			xml: "application/xml, text/xml",
+			json: "application/json, text/javascript"
+		},
+
+		contents: {
+			xml: /\bxml\b/,
+			html: /\bhtml/,
+			json: /\bjson\b/
+		},
+
+		responseFields: {
+			xml: "responseXML",
+			text: "responseText",
+			json: "responseJSON"
+		},
+
+		// Data converters
+		// Keys separate source (or catchall "*") and destination types with a single space
+		converters: {
+
+			// Convert anything to text
+			"* text": String,
+
+			// Text to html (true = no transformation)
+			"text html": true,
+
+			// Evaluate text as a json expression
+			"text json": jQuery.parseJSON,
+
+			// Parse text as xml
+			"text xml": jQuery.parseXML
+		},
+
+		// For options that shouldn't be deep extended:
+		// you can add your own custom options here if
+		// and when you create one that shouldn't be
+		// deep extended (see ajaxExtend)
+		flatOptions: {
+			url: true,
+			context: true
+		}
+	},
+
+	// Creates a full fledged settings object into target
+	// with both ajaxSettings and settings fields.
+	// If target is omitted, writes into ajaxSettings.
+	ajaxSetup: function( target, settings ) {
+		return settings ?
+
+			// Building a settings object
+			ajaxExtend( ajaxExtend( target, jQuery.ajaxSettings ), settings ) :
+
+			// Extending ajaxSettings
+			ajaxExtend( jQuery.ajaxSettings, target );
+	},
+
+	ajaxPrefilter: addToPrefiltersOrTransports( prefilters ),
+	ajaxTransport: addToPrefiltersOrTransports( transports ),
+
+	// Main method
+	ajax: function( url, options ) {
+
+		// If url is an object, simulate pre-1.5 signature
+		if ( typeof url === "object" ) {
+			options = url;
+			url = undefined;
+		}
+
+		// Force options to be an object
+		options = options || {};
+
+		var
+
+			// Cross-domain detection vars
+			parts,
+
+			// Loop variable
+			i,
+
+			// URL without anti-cache param
+			cacheURL,
+
+			// Response headers as string
+			responseHeadersString,
+
+			// timeout handle
+			timeoutTimer,
+
+			// To know if global events are to be dispatched
+			fireGlobals,
+
+			transport,
+
+			// Response headers
+			responseHeaders,
+
+			// Create the final options object
+			s = jQuery.ajaxSetup( {}, options ),
+
+			// Callbacks context
+			callbackContext = s.context || s,
+
+			// Context for global events is callbackContext if it is a DOM node or jQuery collection
+			globalEventContext = s.context &&
+				( callbackContext.nodeType || callbackContext.jquery ) ?
+					jQuery( callbackContext ) :
+					jQuery.event,
+
+			// Deferreds
+			deferred = jQuery.Deferred(),
+			completeDeferred = jQuery.Callbacks( "once memory" ),
+
+			// Status-dependent callbacks
+			statusCode = s.statusCode || {},
+
+			// Headers (they are sent all at once)
+			requestHeaders = {},
+			requestHeadersNames = {},
+
+			// The jqXHR state
+			state = 0,
+
+			// Default abort message
+			strAbort = "canceled",
+
+			// Fake xhr
+			jqXHR = {
+				readyState: 0,
+
+				// Builds headers hashtable if needed
+				getResponseHeader: function( key ) {
+					var match;
+					if ( state === 2 ) {
+						if ( !responseHeaders ) {
+							responseHeaders = {};
+							while ( ( match = rheaders.exec( responseHeadersString ) ) ) {
+								responseHeaders[ match[ 1 ].toLowerCase() ] = match[ 2 ];
+							}
+						}
+						match = responseHeaders[ key.toLowerCase() ];
+					}
+					return match == null ? null : match;
+				},
+
+				// Raw string
+				getAllResponseHeaders: function() {
+					return state === 2 ? responseHeadersString : null;
+				},
+
+				// Caches the header
+				setRequestHeader: function( name, value ) {
+					var lname = name.toLowerCase();
+					if ( !state ) {
+						name = requestHeadersNames[ lname ] = requestHeadersNames[ lname ] || name;
+						requestHeaders[ name ] = value;
+					}
+					return this;
+				},
+
+				// Overrides response content-type header
+				overrideMimeType: function( type ) {
+					if ( !state ) {
+						s.mimeType = type;
+					}
+					return this;
+				},
+
+				// Status-dependent callbacks
+				statusCode: function( map ) {
+					var code;
+					if ( map ) {
+						if ( state < 2 ) {
+							for ( code in map ) {
+
+								// Lazy-add the new callback in a way that preserves old ones
+								statusCode[ code ] = [ statusCode[ code ], map[ code ] ];
+							}
+						} else {
+
+							// Execute the appropriate callbacks
+							jqXHR.always( map[ jqXHR.status ] );
+						}
+					}
+					return this;
+				},
+
+				// Cancel the request
+				abort: function( statusText ) {
+					var finalText = statusText || strAbort;
+					if ( transport ) {
+						transport.abort( finalText );
+					}
+					done( 0, finalText );
+					return this;
+				}
+			};
+
+		// Attach deferreds
+		deferred.promise( jqXHR ).complete = completeDeferred.add;
+		jqXHR.success = jqXHR.done;
+		jqXHR.error = jqXHR.fail;
+
+		// Remove hash character (#7531: and string promotion)
+		// Add protocol if not provided (#5866: IE7 issue with protocol-less urls)
+		// Handle falsy url in the settings object (#10093: consistency with old signature)
+		// We also use the url parameter if available
+		s.url = ( ( url || s.url || ajaxLocation ) + "" )
+			.replace( rhash, "" )
+			.replace( rprotocol, ajaxLocParts[ 1 ] + "//" );
+
+		// Alias method option to type as per ticket #12004
+		s.type = options.method || options.type || s.method || s.type;
+
+		// Extract dataTypes list
+		s.dataTypes = jQuery.trim( s.dataType || "*" ).toLowerCase().match( rnotwhite ) || [ "" ];
+
+		// A cross-domain request is in order when we have a protocol:host:port mismatch
+		if ( s.crossDomain == null ) {
+			parts = rurl.exec( s.url.toLowerCase() );
+			s.crossDomain = !!( parts &&
+				( parts[ 1 ] !== ajaxLocParts[ 1 ] || parts[ 2 ] !== ajaxLocParts[ 2 ] ||
+					( parts[ 3 ] || ( parts[ 1 ] === "http:" ? "80" : "443" ) ) !==
+						( ajaxLocParts[ 3 ] || ( ajaxLocParts[ 1 ] === "http:" ? "80" : "443" ) ) )
+			);
+		}
+
+		// Convert data if not already a string
+		if ( s.data && s.processData && typeof s.data !== "string" ) {
+			s.data = jQuery.param( s.data, s.traditional );
+		}
+
+		// Apply prefilters
+		inspectPrefiltersOrTransports( prefilters, s, options, jqXHR );
+
+		// If request was aborted inside a prefilter, stop there
+		if ( state === 2 ) {
+			return jqXHR;
+		}
+
+		// We can fire global events as of now if asked to
+		// Don't fire events if jQuery.event is undefined in an AMD-usage scenario (#15118)
+		fireGlobals = jQuery.event && s.global;
+
+		// Watch for a new set of requests
+		if ( fireGlobals && jQuery.active++ === 0 ) {
+			jQuery.event.trigger( "ajaxStart" );
+		}
+
+		// Uppercase the type
+		s.type = s.type.toUpperCase();
+
+		// Determine if request has content
+		s.hasContent = !rnoContent.test( s.type );
+
+		// Save the URL in case we're toying with the If-Modified-Since
+		// and/or If-None-Match header later on
+		cacheURL = s.url;
+
+		// More options handling for requests with no content
+		if ( !s.hasContent ) {
+
+			// If data is available, append data to url
+			if ( s.data ) {
+				cacheURL = ( s.url += ( rquery.test( cacheURL ) ? "&" : "?" ) + s.data );
+
+				// #9682: remove data so that it's not used in an eventual retry
+				delete s.data;
+			}
+
+			// Add anti-cache in url if needed
+			if ( s.cache === false ) {
+				s.url = rts.test( cacheURL ) ?
+
+					// If there is already a '_' parameter, set its value
+					cacheURL.replace( rts, "$1_=" + nonce++ ) :
+
+					// Otherwise add one to the end
+					cacheURL + ( rquery.test( cacheURL ) ? "&" : "?" ) + "_=" + nonce++;
+			}
+		}
+
+		// Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode.
+		if ( s.ifModified ) {
+			if ( jQuery.lastModified[ cacheURL ] ) {
+				jqXHR.setRequestHeader( "If-Modified-Since", jQuery.lastModified[ cacheURL ] );
+			}
+			if ( jQuery.etag[ cacheURL ] ) {
+				jqXHR.setRequestHeader( "If-None-Match", jQuery.etag[ cacheURL ] );
+			}
+		}
+
+		// Set the correct header, if data is being sent
+		if ( s.data && s.hasContent && s.contentType !== false || options.contentType ) {
+			jqXHR.setRequestHeader( "Content-Type", s.contentType );
+		}
+
+		// Set the Accepts header for the server, depending on the dataType
+		jqXHR.setRequestHeader(
+			"Accept",
+			s.dataTypes[ 0 ] && s.accepts[ s.dataTypes[ 0 ] ] ?
+				s.accepts[ s.dataTypes[ 0 ] ] +
+					( s.dataTypes[ 0 ] !== "*" ? ", " + allTypes + "; q=0.01" : "" ) :
+				s.accepts[ "*" ]
+		);
+
+		// Check for headers option
+		for ( i in s.headers ) {
+			jqXHR.setRequestHeader( i, s.headers[ i ] );
+		}
+
+		// Allow custom headers/mimetypes and early abort
+		if ( s.beforeSend &&
+			( s.beforeSend.call( callbackContext, jqXHR, s ) === false || state === 2 ) ) {
+
+			// Abort if not done already and return
+			return jqXHR.abort();
+		}
+
+		// aborting is no longer a cancellation
+		strAbort = "abort";
+
+		// Install callbacks on deferreds
+		for ( i in { success: 1, error: 1, complete: 1 } ) {
+			jqXHR[ i ]( s[ i ] );
+		}
+
+		// Get transport
+		transport = inspectPrefiltersOrTransports( transports, s, options, jqXHR );
+
+		// If no transport, we auto-abort
+		if ( !transport ) {
+			done( -1, "No Transport" );
+		} else {
+			jqXHR.readyState = 1;
+
+			// Send global event
+			if ( fireGlobals ) {
+				globalEventContext.trigger( "ajaxSend", [ jqXHR, s ] );
+			}
+
+			// If request was aborted inside ajaxSend, stop there
+			if ( state === 2 ) {
+				return jqXHR;
+			}
+
+			// Timeout
+			if ( s.async && s.timeout > 0 ) {
+				timeoutTimer = window.setTimeout( function() {
+					jqXHR.abort( "timeout" );
+				}, s.timeout );
+			}
+
+			try {
+				state = 1;
+				transport.send( requestHeaders, done );
+			} catch ( e ) {
+
+				// Propagate exception as error if not done
+				if ( state < 2 ) {
+					done( -1, e );
+
+				// Simply rethrow otherwise
+				} else {
+					throw e;
+				}
+			}
+		}
+
+		// Callback for when everything is done
+		function done( status, nativeStatusText, responses, headers ) {
+			var isSuccess, success, error, response, modified,
+				statusText = nativeStatusText;
+
+			// Called once
+			if ( state === 2 ) {
+				return;
+			}
+
+			// State is "done" now
+			state = 2;
+
+			// Clear timeout if it exists
+			if ( timeoutTimer ) {
+				window.clearTimeout( timeoutTimer );
+			}
+
+			// Dereference transport for early garbage collection
+			// (no matter how long the jqXHR object will be used)
+			transport = undefined;
+
+			// Cache response headers
+			responseHeadersString = headers || "";
+
+			// Set readyState
+			jqXHR.readyState = status > 0 ? 4 : 0;
+
+			// Determine if successful
+			isSuccess = status >= 200 && status < 300 || status === 304;
+
+			// Get response data
+			if ( responses ) {
+				response = ajaxHandleResponses( s, jqXHR, responses );
+			}
+
+			// Convert no matter what (that way responseXXX fields are always set)
+			response = ajaxConvert( s, response, jqXHR, isSuccess );
+
+			// If successful, handle type chaining
+			if ( isSuccess ) {
+
+				// Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode.
+				if ( s.ifModified ) {
+					modified = jqXHR.getResponseHeader( "Last-Modified" );
+					if ( modified ) {
+						jQuery.lastModified[ cacheURL ] = modified;
+					}
+					modified = jqXHR.getResponseHeader( "etag" );
+					if ( modified ) {
+						jQuery.etag[ cacheURL ] = modified;
+					}
+				}
+
+				// if no content
+				if ( status === 204 || s.type === "HEAD" ) {
+					statusText = "nocontent";
+
+				// if not modified
+				} else if ( status === 304 ) {
+					statusText = "notmodified";
+
+				// If we have data, let's convert it
+				} else {
+					statusText = response.state;
+					success = response.data;
+					error = response.error;
+					isSuccess = !error;
+				}
+			} else {
+
+				// We extract error from statusText
+				// then normalize statusText and status for non-aborts
+				error = statusText;
+				if ( status || !statusText ) {
+					statusText = "error";
+					if ( status < 0 ) {
+						status = 0;
+					}
+				}
+			}
+
+			// Set data for the fake xhr object
+			jqXHR.status = status;
+			jqXHR.statusText = ( nativeStatusText || statusText ) + "";
+
+			// Success/Error
+			if ( isSuccess ) {
+				deferred.resolveWith( callbackContext, [ success, statusText, jqXHR ] );
+			} else {
+				deferred.rejectWith( callbackContext, [ jqXHR, statusText, error ] );
+			}
+
+			// Status-dependent callbacks
+			jqXHR.statusCode( statusCode );
+			statusCode = undefined;
+
+			if ( fireGlobals ) {
+				globalEventContext.trigger( isSuccess ? "ajaxSuccess" : "ajaxError",
+					[ jqXHR, s, isSuccess ? success : error ] );
+			}
+
+			// Complete
+			completeDeferred.fireWith( callbackContext, [ jqXHR, statusText ] );
+
+			if ( fireGlobals ) {
+				globalEventContext.trigger( "ajaxComplete", [ jqXHR, s ] );
+
+				// Handle the global AJAX counter
+				if ( !( --jQuery.active ) ) {
+					jQuery.event.trigger( "ajaxStop" );
+				}
+			}
+		}
+
+		return jqXHR;
+	},
+
+	getJSON: function( url, data, callback ) {
+		return jQuery.get( url, data, callback, "json" );
+	},
+
+	getScript: function( url, callback ) {
+		return jQuery.get( url, undefined, callback, "script" );
+	}
+} );
+
+jQuery.each( [ "get", "post" ], function( i, method ) {
+	jQuery[ method ] = function( url, data, callback, type ) {
+
+		// shift arguments if data argument was omitted
+		if ( jQuery.isFunction( data ) ) {
+			type = type || callback;
+			callback = data;
+			data = undefined;
+		}
+
+		// The url can be an options object (which then must have .url)
+		return jQuery.ajax( jQuery.extend( {
+			url: url,
+			type: method,
+			dataType: type,
+			data: data,
+			success: callback
+		}, jQuery.isPlainObject( url ) && url ) );
+	};
+} );
+
+
+jQuery._evalUrl = function( url ) {
+	return jQuery.ajax( {
+		url: url,
+
+		// Make this explicit, since user can override this through ajaxSetup (#11264)
+		type: "GET",
+		dataType: "script",
+		cache: true,
+		async: false,
+		global: false,
+		"throws": true
+	} );
+};
+
+
+jQuery.fn.extend( {
+	wrapAll: function( html ) {
+		if ( jQuery.isFunction( html ) ) {
+			return this.each( function( i ) {
+				jQuery( this ).wrapAll( html.call( this, i ) );
+			} );
+		}
+
+		if ( this[ 0 ] ) {
+
+			// The elements to wrap the target around
+			var wrap = jQuery( html, this[ 0 ].ownerDocument ).eq( 0 ).clone( true );
+
+			if ( this[ 0 ].parentNode ) {
+				wrap.insertBefore( this[ 0 ] );
+			}
+
+			wrap.map( function() {
+				var elem = this;
+
+				while ( elem.firstChild && elem.firstChild.nodeType === 1 ) {
+					elem = elem.firstChild;
+				}
+
+				return elem;
+			} ).append( this );
+		}
+
+		return this;
+	},
+
+	wrapInner: function( html ) {
+		if ( jQuery.isFunction( html ) ) {
+			return this.each( function( i ) {
+				jQuery( this ).wrapInner( html.call( this, i ) );
+			} );
+		}
+
+		return this.each( function() {
+			var self = jQuery( this ),
+				contents = self.contents();
+
+			if ( contents.length ) {
+				contents.wrapAll( html );
+
+			} else {
+				self.append( html );
+			}
+		} );
+	},
+
+	wrap: function( html ) {
+		var isFunction = jQuery.isFunction( html );
+
+		return this.each( function( i ) {
+			jQuery( this ).wrapAll( isFunction ? html.call( this, i ) : html );
+		} );
+	},
+
+	unwrap: function() {
+		return this.parent().each( function() {
+			if ( !jQuery.nodeName( this, "body" ) ) {
+				jQuery( this ).replaceWith( this.childNodes );
+			}
+		} ).end();
+	}
+} );
+
+
+function getDisplay( elem ) {
+	return elem.style && elem.style.display || jQuery.css( elem, "display" );
+}
+
+function filterHidden( elem ) {
+
+	// Disconnected elements are considered hidden
+	if ( !jQuery.contains( elem.ownerDocument || document, elem ) ) {
+		return true;
+	}
+	while ( elem && elem.nodeType === 1 ) {
+		if ( getDisplay( elem ) === "none" || elem.type === "hidden" ) {
+			return true;
+		}
+		elem = elem.parentNode;
+	}
+	return false;
+}
+
+jQuery.expr.filters.hidden = function( elem ) {
+
+	// Support: Opera <= 12.12
+	// Opera reports offsetWidths and offsetHeights less than zero on some elements
+	return support.reliableHiddenOffsets() ?
+		( elem.offsetWidth <= 0 && elem.offsetHeight <= 0 &&
+			!elem.getClientRects().length ) :
+			filterHidden( elem );
+};
+
+jQuery.expr.filters.visible = function( elem ) {
+	return !jQuery.expr.filters.hidden( elem );
+};
+
+
+
+
+var r20 = /%20/g,
+	rbracket = /\[\]$/,
+	rCRLF = /\r?\n/g,
+	rsubmitterTypes = /^(?:submit|button|image|reset|file)$/i,
+	rsubmittable = /^(?:input|select|textarea|keygen)/i;
+
+function buildParams( prefix, obj, traditional, add ) {
+	var name;
+
+	if ( jQuery.isArray( obj ) ) {
+
+		// Serialize array item.
+		jQuery.each( obj, function( i, v ) {
+			if ( traditional || rbracket.test( prefix ) ) {
+
+				// Treat each array item as a scalar.
+				add( prefix, v );
+
+			} else {
+
+				// Item is non-scalar (array or object), encode its numeric index.
+				buildParams(
+					prefix + "[" + ( typeof v === "object" && v != null ? i : "" ) + "]",
+					v,
+					traditional,
+					add
+				);
+			}
+		} );
+
+	} else if ( !traditional && jQuery.type( obj ) === "object" ) {
+
+		// Serialize object item.
+		for ( name in obj ) {
+			buildParams( prefix + "[" + name + "]", obj[ name ], traditional, add );
+		}
+
+	} else {
+
+		// Serialize scalar item.
+		add( prefix, obj );
+	}
+}
+
+// Serialize an array of form elements or a set of
+// key/values into a query string
+jQuery.param = function( a, traditional ) {
+	var prefix,
+		s = [],
+		add = function( key, value ) {
+
+			// If value is a function, invoke it and return its value
+			value = jQuery.isFunction( value ) ? value() : ( value == null ? "" : value );
+			s[ s.length ] = encodeURIComponent( key ) + "=" + encodeURIComponent( value );
+		};
+
+	// Set traditional to true for jQuery <= 1.3.2 behavior.
+	if ( traditional === undefined ) {
+		traditional = jQuery.ajaxSettings && jQuery.ajaxSettings.traditional;
+	}
+
+	// If an array was passed in, assume that it is an array of form elements.
+	if ( jQuery.isArray( a ) || ( a.jquery && !jQuery.isPlainObject( a ) ) ) {
+
+		// Serialize the form elements
+		jQuery.each( a, function() {
+			add( this.name, this.value );
+		} );
+
+	} else {
+
+		// If traditional, encode the "old" way (the way 1.3.2 or older
+		// did it), otherwise encode params recursively.
+		for ( prefix in a ) {
+			buildParams( prefix, a[ prefix ], traditional, add );
+		}
+	}
+
+	// Return the resulting serialization
+	return s.join( "&" ).replace( r20, "+" );
+};
+
+jQuery.fn.extend( {
+	serialize: function() {
+		return jQuery.param( this.serializeArray() );
+	},
+	serializeArray: function() {
+		return this.map( function() {
+
+			// Can add propHook for "elements" to filter or add form elements
+			var elements = jQuery.prop( this, "elements" );
+			return elements ? jQuery.makeArray( elements ) : this;
+		} )
+		.filter( function() {
+			var type = this.type;
+
+			// Use .is(":disabled") so that fieldset[disabled] works
+			return this.name && !jQuery( this ).is( ":disabled" ) &&
+				rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) &&
+				( this.checked || !rcheckableType.test( type ) );
+		} )
+		.map( function( i, elem ) {
+			var val = jQuery( this ).val();
+
+			return val == null ?
+				null :
+				jQuery.isArray( val ) ?
+					jQuery.map( val, function( val ) {
+						return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) };
+					} ) :
+					{ name: elem.name, value: val.replace( rCRLF, "\r\n" ) };
+		} ).get();
+	}
+} );
+
+
+// Create the request object
+// (This is still attached to ajaxSettings for backward compatibility)
+jQuery.ajaxSettings.xhr = window.ActiveXObject !== undefined ?
+
+	// Support: IE6-IE8
+	function() {
+
+		// XHR cannot access local files, always use ActiveX for that case
+		if ( this.isLocal ) {
+			return createActiveXHR();
+		}
+
+		// Support: IE 9-11
+		// IE seems to error on cross-domain PATCH requests when ActiveX XHR
+		// is used. In IE 9+ always use the native XHR.
+		// Note: this condition won't catch Edge as it doesn't define
+		// document.documentMode but it also doesn't support ActiveX so it won't
+		// reach this code.
+		if ( document.documentMode > 8 ) {
+			return createStandardXHR();
+		}
+
+		// Support: IE<9
+		// oldIE XHR does not support non-RFC2616 methods (#13240)
+		// See http://msdn.microsoft.com/en-us/library/ie/ms536648(v=vs.85).aspx
+		// and http://www.w3.org/Protocols/rfc2616/rfc2616-sec9.html#sec9
+		// Although this check for six methods instead of eight
+		// since IE also does not support "trace" and "connect"
+		return /^(get|post|head|put|delete|options)$/i.test( this.type ) &&
+			createStandardXHR() || createActiveXHR();
+	} :
+
+	// For all other browsers, use the standard XMLHttpRequest object
+	createStandardXHR;
+
+var xhrId = 0,
+	xhrCallbacks = {},
+	xhrSupported = jQuery.ajaxSettings.xhr();
+
+// Support: IE<10
+// Open requests must be manually aborted on unload (#5280)
+// See https://support.microsoft.com/kb/2856746 for more info
+if ( window.attachEvent ) {
+	window.attachEvent( "onunload", function() {
+		for ( var key in xhrCallbacks ) {
+			xhrCallbacks[ key ]( undefined, true );
+		}
+	} );
+}
+
+// Determine support properties
+support.cors = !!xhrSupported && ( "withCredentials" in xhrSupported );
+xhrSupported = support.ajax = !!xhrSupported;
+
+// Create transport if the browser can provide an xhr
+if ( xhrSupported ) {
+
+	jQuery.ajaxTransport( function( options ) {
+
+		// Cross domain only allowed if supported through XMLHttpRequest
+		if ( !options.crossDomain || support.cors ) {
+
+			var callback;
+
+			return {
+				send: function( headers, complete ) {
+					var i,
+						xhr = options.xhr(),
+						id = ++xhrId;
+
+					// Open the socket
+					xhr.open(
+						options.type,
+						options.url,
+						options.async,
+						options.username,
+						options.password
+					);
+
+					// Apply custom fields if provided
+					if ( options.xhrFields ) {
+						for ( i in options.xhrFields ) {
+							xhr[ i ] = options.xhrFields[ i ];
+						}
+					}
+
+					// Override mime type if needed
+					if ( options.mimeType && xhr.overrideMimeType ) {
+						xhr.overrideMimeType( options.mimeType );
+					}
+
+					// X-Requested-With header
+					// For cross-domain requests, seeing as conditions for a preflight are
+					// akin to a jigsaw puzzle, we simply never set it to be sure.
+					// (it can always be set on a per-request basis or even using ajaxSetup)
+					// For same-domain requests, won't change header if already provided.
+					if ( !options.crossDomain && !headers[ "X-Requested-With" ] ) {
+						headers[ "X-Requested-With" ] = "XMLHttpRequest";
+					}
+
+					// Set headers
+					for ( i in headers ) {
+
+						// Support: IE<9
+						// IE's ActiveXObject throws a 'Type Mismatch' exception when setting
+						// request header to a null-value.
+						//
+						// To keep consistent with other XHR implementations, cast the value
+						// to string and ignore `undefined`.
+						if ( headers[ i ] !== undefined ) {
+							xhr.setRequestHeader( i, headers[ i ] + "" );
+						}
+					}
+
+					// Do send the request
+					// This may raise an exception which is actually
+					// handled in jQuery.ajax (so no try/catch here)
+					xhr.send( ( options.hasContent && options.data ) || null );
+
+					// Listener
+					callback = function( _, isAbort ) {
+						var status, statusText, responses;
+
+						// Was never called and is aborted or complete
+						if ( callback && ( isAbort || xhr.readyState === 4 ) ) {
+
+							// Clean up
+							delete xhrCallbacks[ id ];
+							callback = undefined;
+							xhr.onreadystatechange = jQuery.noop;
+
+							// Abort manually if needed
+							if ( isAbort ) {
+								if ( xhr.readyState !== 4 ) {
+									xhr.abort();
+								}
+							} else {
+								responses = {};
+								status = xhr.status;
+
+								// Support: IE<10
+								// Accessing binary-data responseText throws an exception
+								// (#11426)
+								if ( typeof xhr.responseText === "string" ) {
+									responses.text = xhr.responseText;
+								}
+
+								// Firefox throws an exception when accessing
+								// statusText for faulty cross-domain requests
+								try {
+									statusText = xhr.statusText;
+								} catch ( e ) {
+
+									// We normalize with Webkit giving an empty statusText
+									statusText = "";
+								}
+
+								// Filter status for non standard behaviors
+
+								// If the request is local and we have data: assume a success
+								// (success with no data won't get notified, that's the best we
+								// can do given current implementations)
+								if ( !status && options.isLocal && !options.crossDomain ) {
+									status = responses.text ? 200 : 404;
+
+								// IE - #1450: sometimes returns 1223 when it should be 204
+								} else if ( status === 1223 ) {
+									status = 204;
+								}
+							}
+						}
+
+						// Call complete if needed
+						if ( responses ) {
+							complete( status, statusText, responses, xhr.getAllResponseHeaders() );
+						}
+					};
+
+					// Do send the request
+					// `xhr.send` may raise an exception, but it will be
+					// handled in jQuery.ajax (so no try/catch here)
+					if ( !options.async ) {
+
+						// If we're in sync mode we fire the callback
+						callback();
+					} else if ( xhr.readyState === 4 ) {
+
+						// (IE6 & IE7) if it's in cache and has been
+						// retrieved directly we need to fire the callback
+						window.setTimeout( callback );
+					} else {
+
+						// Register the callback, but delay it in case `xhr.send` throws
+						// Add to the list of active xhr callbacks
+						xhr.onreadystatechange = xhrCallbacks[ id ] = callback;
+					}
+				},
+
+				abort: function() {
+					if ( callback ) {
+						callback( undefined, true );
+					}
+				}
+			};
+		}
+	} );
+}
+
+// Functions to create xhrs
+function createStandardXHR() {
+	try {
+		return new window.XMLHttpRequest();
+	} catch ( e ) {}
+}
+
+function createActiveXHR() {
+	try {
+		return new window.ActiveXObject( "Microsoft.XMLHTTP" );
+	} catch ( e ) {}
+}
+
+
+
+
+// Install script dataType
+jQuery.ajaxSetup( {
+	accepts: {
+		script: "text/javascript, application/javascript, " +
+			"application/ecmascript, application/x-ecmascript"
+	},
+	contents: {
+		script: /\b(?:java|ecma)script\b/
+	},
+	converters: {
+		"text script": function( text ) {
+			jQuery.globalEval( text );
+			return text;
+		}
+	}
+} );
+
+// Handle cache's special case and global
+jQuery.ajaxPrefilter( "script", function( s ) {
+	if ( s.cache === undefined ) {
+		s.cache = false;
+	}
+	if ( s.crossDomain ) {
+		s.type = "GET";
+		s.global = false;
+	}
+} );
+
+// Bind script tag hack transport
+jQuery.ajaxTransport( "script", function( s ) {
+
+	// This transport only deals with cross domain requests
+	if ( s.crossDomain ) {
+
+		var script,
+			head = document.head || jQuery( "head" )[ 0 ] || document.documentElement;
+
+		return {
+
+			send: function( _, callback ) {
+
+				script = document.createElement( "script" );
+
+				script.async = true;
+
+				if ( s.scriptCharset ) {
+					script.charset = s.scriptCharset;
+				}
+
+				script.src = s.url;
+
+				// Attach handlers for all browsers
+				script.onload = script.onreadystatechange = function( _, isAbort ) {
+
+					if ( isAbort || !script.readyState || /loaded|complete/.test( script.readyState ) ) {
+
+						// Handle memory leak in IE
+						script.onload = script.onreadystatechange = null;
+
+						// Remove the script
+						if ( script.parentNode ) {
+							script.parentNode.removeChild( script );
+						}
+
+						// Dereference the script
+						script = null;
+
+						// Callback if not abort
+						if ( !isAbort ) {
+							callback( 200, "success" );
+						}
+					}
+				};
+
+				// Circumvent IE6 bugs with base elements (#2709 and #4378) by prepending
+				// Use native DOM manipulation to avoid our domManip AJAX trickery
+				head.insertBefore( script, head.firstChild );
+			},
+
+			abort: function() {
+				if ( script ) {
+					script.onload( undefined, true );
+				}
+			}
+		};
+	}
+} );
+
+
+
+
+var oldCallbacks = [],
+	rjsonp = /(=)\?(?=&|$)|\?\?/;
+
+// Default jsonp settings
+jQuery.ajaxSetup( {
+	jsonp: "callback",
+	jsonpCallback: function() {
+		var callback = oldCallbacks.pop() || ( jQuery.expando + "_" + ( nonce++ ) );
+		this[ callback ] = true;
+		return callback;
+	}
+} );
+
+// Detect, normalize options and install callbacks for jsonp requests
+jQuery.ajaxPrefilter( "json jsonp", function( s, originalSettings, jqXHR ) {
+
+	var callbackName, overwritten, responseContainer,
+		jsonProp = s.jsonp !== false && ( rjsonp.test( s.url ) ?
+			"url" :
+			typeof s.data === "string" &&
+				( s.contentType || "" )
+					.indexOf( "application/x-www-form-urlencoded" ) === 0 &&
+				rjsonp.test( s.data ) && "data"
+		);
+
+	// Handle iff the expected data type is "jsonp" or we have a parameter to set
+	if ( jsonProp || s.dataTypes[ 0 ] === "jsonp" ) {
+
+		// Get callback name, remembering preexisting value associated with it
+		callbackName = s.jsonpCallback = jQuery.isFunction( s.jsonpCallback ) ?
+			s.jsonpCallback() :
+			s.jsonpCallback;
+
+		// Insert callback into url or form data
+		if ( jsonProp ) {
+			s[ jsonProp ] = s[ jsonProp ].replace( rjsonp, "$1" + callbackName );
+		} else if ( s.jsonp !== false ) {
+			s.url += ( rquery.test( s.url ) ? "&" : "?" ) + s.jsonp + "=" + callbackName;
+		}
+
+		// Use data converter to retrieve json after script execution
+		s.converters[ "script json" ] = function() {
+			if ( !responseContainer ) {
+				jQuery.error( callbackName + " was not called" );
+			}
+			return responseContainer[ 0 ];
+		};
+
+		// force json dataType
+		s.dataTypes[ 0 ] = "json";
+
+		// Install callback
+		overwritten = window[ callbackName ];
+		window[ callbackName ] = function() {
+			responseContainer = arguments;
+		};
+
+		// Clean-up function (fires after converters)
+		jqXHR.always( function() {
+
+			// If previous value didn't exist - remove it
+			if ( overwritten === undefined ) {
+				jQuery( window ).removeProp( callbackName );
+
+			// Otherwise restore preexisting value
+			} else {
+				window[ callbackName ] = overwritten;
+			}
+
+			// Save back as free
+			if ( s[ callbackName ] ) {
+
+				// make sure that re-using the options doesn't screw things around
+				s.jsonpCallback = originalSettings.jsonpCallback;
+
+				// save the callback name for future use
+				oldCallbacks.push( callbackName );
+			}
+
+			// Call if it was a function and we have a response
+			if ( responseContainer && jQuery.isFunction( overwritten ) ) {
+				overwritten( responseContainer[ 0 ] );
+			}
+
+			responseContainer = overwritten = undefined;
+		} );
+
+		// Delegate to script
+		return "script";
+	}
+} );
+
+
+
+
+// data: string of html
+// context (optional): If specified, the fragment will be created in this context,
+// defaults to document
+// keepScripts (optional): If true, will include scripts passed in the html string
+jQuery.parseHTML = function( data, context, keepScripts ) {
+	if ( !data || typeof data !== "string" ) {
+		return null;
+	}
+	if ( typeof context === "boolean" ) {
+		keepScripts = context;
+		context = false;
+	}
+	context = context || document;
+
+	var parsed = rsingleTag.exec( data ),
+		scripts = !keepScripts && [];
+
+	// Single tag
+	if ( parsed ) {
+		return [ context.createElement( parsed[ 1 ] ) ];
+	}
+
+	parsed = buildFragment( [ data ], context, scripts );
+
+	if ( scripts && scripts.length ) {
+		jQuery( scripts ).remove();
+	}
+
+	return jQuery.merge( [], parsed.childNodes );
+};
+
+
+// Keep a copy of the old load method
+var _load = jQuery.fn.load;
+
+/**
+ * Load a url into a page
+ */
+jQuery.fn.load = function( url, params, callback ) {
+	if ( typeof url !== "string" && _load ) {
+		return _load.apply( this, arguments );
+	}
+
+	var selector, type, response,
+		self = this,
+		off = url.indexOf( " " );
+
+	if ( off > -1 ) {
+		selector = jQuery.trim( url.slice( off, url.length ) );
+		url = url.slice( 0, off );
+	}
+
+	// If it's a function
+	if ( jQuery.isFunction( params ) ) {
+
+		// We assume that it's the callback
+		callback = params;
+		params = undefined;
+
+	// Otherwise, build a param string
+	} else if ( params && typeof params === "object" ) {
+		type = "POST";
+	}
+
+	// If we have elements to modify, make the request
+	if ( self.length > 0 ) {
+		jQuery.ajax( {
+			url: url,
+
+			// If "type" variable is undefined, then "GET" method will be used.
+			// Make value of this field explicit since
+			// user can override it through ajaxSetup method
+			type: type || "GET",
+			dataType: "html",
+			data: params
+		} ).done( function( responseText ) {
+
+			// Save response for use in complete callback
+			response = arguments;
+
+			self.html( selector ?
+
+				// If a selector was specified, locate the right elements in a dummy div
+				// Exclude scripts to avoid IE 'Permission Denied' errors
+				jQuery( "<div>" ).append( jQuery.parseHTML( responseText ) ).find( selector ) :
+
+				// Otherwise use the full result
+				responseText );
+
+		// If the request succeeds, this function gets "data", "status", "jqXHR"
+		// but they are ignored because response was set above.
+		// If it fails, this function gets "jqXHR", "status", "error"
+		} ).always( callback && function( jqXHR, status ) {
+			self.each( function() {
+				callback.apply( this, response || [ jqXHR.responseText, status, jqXHR ] );
+			} );
+		} );
+	}
+
+	return this;
+};
+
+
+
+
+// Attach a bunch of functions for handling common AJAX events
+jQuery.each( [
+	"ajaxStart",
+	"ajaxStop",
+	"ajaxComplete",
+	"ajaxError",
+	"ajaxSuccess",
+	"ajaxSend"
+], function( i, type ) {
+	jQuery.fn[ type ] = function( fn ) {
+		return this.on( type, fn );
+	};
+} );
+
+
+
+
+jQuery.expr.filters.animated = function( elem ) {
+	return jQuery.grep( jQuery.timers, function( fn ) {
+		return elem === fn.elem;
+	} ).length;
+};
+
+
+
+
+
+/**
+ * Gets a window from an element
+ */
+function getWindow( elem ) {
+	return jQuery.isWindow( elem ) ?
+		elem :
+		elem.nodeType === 9 ?
+			elem.defaultView || elem.parentWindow :
+			false;
+}
+
+jQuery.offset = {
+	setOffset: function( elem, options, i ) {
+		var curPosition, curLeft, curCSSTop, curTop, curOffset, curCSSLeft, calculatePosition,
+			position = jQuery.css( elem, "position" ),
+			curElem = jQuery( elem ),
+			props = {};
+
+		// set position first, in-case top/left are set even on static elem
+		if ( position === "static" ) {
+			elem.style.position = "relative";
+		}
+
+		curOffset = curElem.offset();
+		curCSSTop = jQuery.css( elem, "top" );
+		curCSSLeft = jQuery.css( elem, "left" );
+		calculatePosition = ( position === "absolute" || position === "fixed" ) &&
+			jQuery.inArray( "auto", [ curCSSTop, curCSSLeft ] ) > -1;
+
+		// need to be able to calculate position if either top or left
+		// is auto and position is either absolute or fixed
+		if ( calculatePosition ) {
+			curPosition = curElem.position();
+			curTop = curPosition.top;
+			curLeft = curPosition.left;
+		} else {
+			curTop = parseFloat( curCSSTop ) || 0;
+			curLeft = parseFloat( curCSSLeft ) || 0;
+		}
+
+		if ( jQuery.isFunction( options ) ) {
+
+			// Use jQuery.extend here to allow modification of coordinates argument (gh-1848)
+			options = options.call( elem, i, jQuery.extend( {}, curOffset ) );
+		}
+
+		if ( options.top != null ) {
+			props.top = ( options.top - curOffset.top ) + curTop;
+		}
+		if ( options.left != null ) {
+			props.left = ( options.left - curOffset.left ) + curLeft;
+		}
+
+		if ( "using" in options ) {
+			options.using.call( elem, props );
+		} else {
+			curElem.css( props );
+		}
+	}
+};
+
+jQuery.fn.extend( {
+	offset: function( options ) {
+		if ( arguments.length ) {
+			return options === undefined ?
+				this :
+				this.each( function( i ) {
+					jQuery.offset.setOffset( this, options, i );
+				} );
+		}
+
+		var docElem, win,
+			box = { top: 0, left: 0 },
+			elem = this[ 0 ],
+			doc = elem && elem.ownerDocument;
+
+		if ( !doc ) {
+			return;
+		}
+
+		docElem = doc.documentElement;
+
+		// Make sure it's not a disconnected DOM node
+		if ( !jQuery.contains( docElem, elem ) ) {
+			return box;
+		}
+
+		// If we don't have gBCR, just use 0,0 rather than error
+		// BlackBerry 5, iOS 3 (original iPhone)
+		if ( typeof elem.getBoundingClientRect !== "undefined" ) {
+			box = elem.getBoundingClientRect();
+		}
+		win = getWindow( doc );
+		return {
+			top: box.top  + ( win.pageYOffset || docElem.scrollTop )  - ( docElem.clientTop  || 0 ),
+			left: box.left + ( win.pageXOffset || docElem.scrollLeft ) - ( docElem.clientLeft || 0 )
+		};
+	},
+
+	position: function() {
+		if ( !this[ 0 ] ) {
+			return;
+		}
+
+		var offsetParent, offset,
+			parentOffset = { top: 0, left: 0 },
+			elem = this[ 0 ];
+
+		// Fixed elements are offset from window (parentOffset = {top:0, left: 0},
+		// because it is its only offset parent
+		if ( jQuery.css( elem, "position" ) === "fixed" ) {
+
+			// we assume that getBoundingClientRect is available when computed position is fixed
+			offset = elem.getBoundingClientRect();
+		} else {
+
+			// Get *real* offsetParent
+			offsetParent = this.offsetParent();
+
+			// Get correct offsets
+			offset = this.offset();
+			if ( !jQuery.nodeName( offsetParent[ 0 ], "html" ) ) {
+				parentOffset = offsetParent.offset();
+			}
+
+			// Add offsetParent borders
+			parentOffset.top  += jQuery.css( offsetParent[ 0 ], "borderTopWidth", true );
+			parentOffset.left += jQuery.css( offsetParent[ 0 ], "borderLeftWidth", true );
+		}
+
+		// Subtract parent offsets and element margins
+		// note: when an element has margin: auto the offsetLeft and marginLeft
+		// are the same in Safari causing offset.left to incorrectly be 0
+		return {
+			top:  offset.top  - parentOffset.top - jQuery.css( elem, "marginTop", true ),
+			left: offset.left - parentOffset.left - jQuery.css( elem, "marginLeft", true )
+		};
+	},
+
+	offsetParent: function() {
+		return this.map( function() {
+			var offsetParent = this.offsetParent;
+
+			while ( offsetParent && ( !jQuery.nodeName( offsetParent, "html" ) &&
+				jQuery.css( offsetParent, "position" ) === "static" ) ) {
+				offsetParent = offsetParent.offsetParent;
+			}
+			return offsetParent || documentElement;
+		} );
+	}
+} );
+
+// Create scrollLeft and scrollTop methods
+jQuery.each( { scrollLeft: "pageXOffset", scrollTop: "pageYOffset" }, function( method, prop ) {
+	var top = /Y/.test( prop );
+
+	jQuery.fn[ method ] = function( val ) {
+		return access( this, function( elem, method, val ) {
+			var win = getWindow( elem );
+
+			if ( val === undefined ) {
+				return win ? ( prop in win ) ? win[ prop ] :
+					win.document.documentElement[ method ] :
+					elem[ method ];
+			}
+
+			if ( win ) {
+				win.scrollTo(
+					!top ? val : jQuery( win ).scrollLeft(),
+					top ? val : jQuery( win ).scrollTop()
+				);
+
+			} else {
+				elem[ method ] = val;
+			}
+		}, method, val, arguments.length, null );
+	};
+} );
+
+// Support: Safari<7-8+, Chrome<37-44+
+// Add the top/left cssHooks using jQuery.fn.position
+// Webkit bug: https://bugs.webkit.org/show_bug.cgi?id=29084
+// getComputedStyle returns percent when specified for top/left/bottom/right
+// rather than make the css module depend on the offset module, we just check for it here
+jQuery.each( [ "top", "left" ], function( i, prop ) {
+	jQuery.cssHooks[ prop ] = addGetHookIf( support.pixelPosition,
+		function( elem, computed ) {
+			if ( computed ) {
+				computed = curCSS( elem, prop );
+
+				// if curCSS returns percentage, fallback to offset
+				return rnumnonpx.test( computed ) ?
+					jQuery( elem ).position()[ prop ] + "px" :
+					computed;
+			}
+		}
+	);
+} );
+
+
+// Create innerHeight, innerWidth, height, width, outerHeight and outerWidth methods
+jQuery.each( { Height: "height", Width: "width" }, function( name, type ) {
+	jQuery.each( { padding: "inner" + name, content: type, "": "outer" + name },
+	function( defaultExtra, funcName ) {
+
+		// margin is only for outerHeight, outerWidth
+		jQuery.fn[ funcName ] = function( margin, value ) {
+			var chainable = arguments.length && ( defaultExtra || typeof margin !== "boolean" ),
+				extra = defaultExtra || ( margin === true || value === true ? "margin" : "border" );
+
+			return access( this, function( elem, type, value ) {
+				var doc;
+
+				if ( jQuery.isWindow( elem ) ) {
+
+					// As of 5/8/2012 this will yield incorrect results for Mobile Safari, but there
+					// isn't a whole lot we can do. See pull request at this URL for discussion:
+					// https://github.com/jquery/jquery/pull/764
+					return elem.document.documentElement[ "client" + name ];
+				}
+
+				// Get document width or height
+				if ( elem.nodeType === 9 ) {
+					doc = elem.documentElement;
+
+					// Either scroll[Width/Height] or offset[Width/Height] or client[Width/Height],
+					// whichever is greatest
+					// unfortunately, this causes bug #3838 in IE6/8 only,
+					// but there is currently no good, small way to fix it.
+					return Math.max(
+						elem.body[ "scroll" + name ], doc[ "scroll" + name ],
+						elem.body[ "offset" + name ], doc[ "offset" + name ],
+						doc[ "client" + name ]
+					);
+				}
+
+				return value === undefined ?
+
+					// Get width or height on the element, requesting but not forcing parseFloat
+					jQuery.css( elem, type, extra ) :
+
+					// Set width or height on the element
+					jQuery.style( elem, type, value, extra );
+			}, type, chainable ? margin : undefined, chainable, null );
+		};
+	} );
+} );
+
+
+jQuery.fn.extend( {
+
+	bind: function( types, data, fn ) {
+		return this.on( types, null, data, fn );
+	},
+	unbind: function( types, fn ) {
+		return this.off( types, null, fn );
+	},
+
+	delegate: function( selector, types, data, fn ) {
+		return this.on( types, selector, data, fn );
+	},
+	undelegate: function( selector, types, fn ) {
+
+		// ( namespace ) or ( selector, types [, fn] )
+		return arguments.length === 1 ?
+			this.off( selector, "**" ) :
+			this.off( types, selector || "**", fn );
+	}
+} );
+
+// The number of elements contained in the matched element set
+jQuery.fn.size = function() {
+	return this.length;
+};
+
+jQuery.fn.andSelf = jQuery.fn.addBack;
+
+
+
+
+// Register as a named AMD module, since jQuery can be concatenated with other
+// files that may use define, but not via a proper concatenation script that
+// understands anonymous AMD modules. A named AMD is safest and most robust
+// way to register. Lowercase jquery is used because AMD module names are
+// derived from file names, and jQuery is normally delivered in a lowercase
+// file name. Do this after creating the global so that if an AMD module wants
+// to call noConflict to hide this version of jQuery, it will work.
+
+// Note that for maximum portability, libraries that are not jQuery should
+// declare themselves as anonymous modules, and avoid setting a global if an
+// AMD loader is present. jQuery is a special case. For more information, see
+// https://github.com/jrburke/requirejs/wiki/Updating-existing-libraries#wiki-anon
+
+if ( typeof define === "function" && define.amd ) {
+	define( "jquery", [], function() {
+		return jQuery;
+	} );
+}
+
+
+
+var
+
+	// Map over jQuery in case of overwrite
+	_jQuery = window.jQuery,
+
+	// Map over the $ in case of overwrite
+	_$ = window.$;
+
+jQuery.noConflict = function( deep ) {
+	if ( window.$ === jQuery ) {
+		window.$ = _$;
+	}
+
+	if ( deep && window.jQuery === jQuery ) {
+		window.jQuery = _jQuery;
+	}
+
+	return jQuery;
+};
+
+// Expose jQuery and $ identifiers, even in
+// AMD (#7102#comment:10, https://github.com/jquery/jquery/pull/557)
+// and CommonJS for browser emulators (#13566)
+if ( !noGlobal ) {
+	window.jQuery = window.$ = jQuery;
+}
+
+return jQuery;
+}));
diff --git a/docs/pt/katex/README.md b/docs/pt/katex/README.md
new file mode 100644
index 0000000..307fdbf
--- /dev/null
+++ b/docs/pt/katex/README.md
@@ -0,0 +1,91 @@
+# [<img src="https://katex.org/img/katex-logo-black.svg" width="130" alt="KaTeX">](https://katex.org/)
+[![npm](https://img.shields.io/npm/v/katex.svg)](https://www.npmjs.com/package/katex)
+[![CircleCI](https://circleci.com/gh/KaTeX/KaTeX.svg?style=shield)](https://circleci.com/gh/KaTeX/KaTeX)
+[![codecov](https://codecov.io/gh/KaTeX/KaTeX/branch/master/graph/badge.svg)](https://codecov.io/gh/KaTeX/KaTeX)
+[![Join the chat at https://gitter.im/KaTeX/KaTeX](https://badges.gitter.im/Join%20Chat.svg)](https://gitter.im/KaTeX/KaTeX?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge)
+[![Dependabot Status](https://api.dependabot.com/badges/status?host=github&repo=KaTeX/KaTeX)](https://dependabot.com)
+[![jsDelivr](https://data.jsdelivr.com/v1/package/npm/katex/badge?style=rounded)](https://www.jsdelivr.com/package/npm/katex)
+![](https://img.badgesize.io/KaTeX/KaTeX/v0.12.0/dist/katex.min.js?compression=gzip)
+
+KaTeX is a fast, easy-to-use JavaScript library for TeX math rendering on the web.
+
+ * **Fast:** KaTeX renders its math synchronously and doesn't need to reflow the page. See how it compares to a competitor in [this speed test](http://www.intmath.com/cg5/katex-mathjax-comparison.php).
+ * **Print quality:** KaTeX's layout is based on Donald Knuth's TeX, the gold standard for math typesetting.
+ * **Self contained:** KaTeX has no dependencies and can easily be bundled with your website resources.
+ * **Server side rendering:** KaTeX produces the same output regardless of browser or environment, so you can pre-render expressions using Node.js and send them as plain HTML.
+
+KaTeX is compatible with all major browsers, including Chrome, Safari, Firefox, Opera, Edge, and IE 11.
+
+KaTeX supports much (but not all) of LaTeX and many LaTeX packages. See the [list of supported functions](https://katex.org/docs/supported.html).
+
+Try out KaTeX [on the demo page](https://katex.org/#demo)!
+
+## Getting started
+
+### Starter template
+
+```html
+<!DOCTYPE html>
+<!-- KaTeX requires the use of the HTML5 doctype. Without it, KaTeX may not render properly -->
+<html>
+  <head>
+    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+
+    <!-- The loading of KaTeX is deferred to speed up page rendering -->
+    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.js" integrity="sha384-g7c+Jr9ZivxKLnZTDUhnkOnsh30B4H0rpLUpJ4jAIKs4fnJI+sEnkvrMWph2EDg4" crossorigin="anonymous"></script>
+
+    <!-- To automatically render math in text elements, include the auto-render extension: -->
+    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/contrib/auto-render.min.js" integrity="sha384-mll67QQFJfxn0IYznZYonOWZ644AWYC+Pt2cHqMaRhXVrursRwvLnLaebdGIlYNa" crossorigin="anonymous"
+        onload="renderMathInElement(document.body);"></script>
+  </head>
+  ...
+</html>
+```
+
+You can also [download KaTeX](https://github.com/KaTeX/KaTeX/releases) and host it yourself.
+
+For details on how to configure auto-render extension, refer to [the documentation](https://katex.org/docs/autorender.html).
+
+### API
+
+Call `katex.render` to render a TeX expression directly into a DOM element.
+For example:
+
+```js
+katex.render("c = \\pm\\sqrt{a^2 + b^2}", element, {
+    throwOnError: false
+});
+```
+
+Call `katex.renderToString` to generate an HTML string of the rendered math,
+e.g., for server-side rendering.  For example:
+
+```js
+var html = katex.renderToString("c = \\pm\\sqrt{a^2 + b^2}", {
+    throwOnError: false
+});
+// '<span class="katex">...</span>'
+```
+
+Make sure to include the CSS and font files in both cases.
+If you are doing all rendering on the server, there is no need to include the
+JavaScript on the client.
+
+The examples above use the `throwOnError: false` option, which renders invalid
+inputs as the TeX source code in red (by default), with the error message as
+hover text.  For other available options, see the
+[API documentation](https://katex.org/docs/api.html),
+[options documentation](https://katex.org/docs/options.html), and
+[handling errors documentation](https://katex.org/docs/error.html).
+
+## Demo and Documentation
+
+Learn more about using KaTeX [on the website](https://katex.org)!
+
+## Contributing
+
+See [CONTRIBUTING.md](CONTRIBUTING.md)
+
+## License
+
+KaTeX is licensed under the [MIT License](http://opensource.org/licenses/MIT).
diff --git a/docs/pt/katex/contrib/auto-render.js b/docs/pt/katex/contrib/auto-render.js
new file mode 100644
index 0000000..d594434
--- /dev/null
+++ b/docs/pt/katex/contrib/auto-render.js
@@ -0,0 +1,350 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory(require("katex"));
+	else if(typeof define === 'function' && define.amd)
+		define(["katex"], factory);
+	else if(typeof exports === 'object')
+		exports["renderMathInElement"] = factory(require("katex"));
+	else
+		root["renderMathInElement"] = factory(root["katex"]);
+})((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__0__) {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
+/******/ 	function __webpack_require__(moduleId) {
+/******/
+/******/ 		// Check if module is in cache
+/******/ 		if(installedModules[moduleId]) {
+/******/ 			return installedModules[moduleId].exports;
+/******/ 		}
+/******/ 		// Create a new module (and put it into the cache)
+/******/ 		var module = installedModules[moduleId] = {
+/******/ 			i: moduleId,
+/******/ 			l: false,
+/******/ 			exports: {}
+/******/ 		};
+/******/
+/******/ 		// Execute the module function
+/******/ 		modules[moduleId].call(module.exports, module, module.exports, __webpack_require__);
+/******/
+/******/ 		// Flag the module as loaded
+/******/ 		module.l = true;
+/******/
+/******/ 		// Return the exports of the module
+/******/ 		return module.exports;
+/******/ 	}
+/******/
+/******/
+/******/ 	// expose the modules object (__webpack_modules__)
+/******/ 	__webpack_require__.m = modules;
+/******/
+/******/ 	// expose the module cache
+/******/ 	__webpack_require__.c = installedModules;
+/******/
+/******/ 	// define getter function for harmony exports
+/******/ 	__webpack_require__.d = function(exports, name, getter) {
+/******/ 		if(!__webpack_require__.o(exports, name)) {
+/******/ 			Object.defineProperty(exports, name, { enumerable: true, get: getter });
+/******/ 		}
+/******/ 	};
+/******/
+/******/ 	// define __esModule on exports
+/******/ 	__webpack_require__.r = function(exports) {
+/******/ 		if(typeof Symbol !== 'undefined' && Symbol.toStringTag) {
+/******/ 			Object.defineProperty(exports, Symbol.toStringTag, { value: 'Module' });
+/******/ 		}
+/******/ 		Object.defineProperty(exports, '__esModule', { value: true });
+/******/ 	};
+/******/
+/******/ 	// create a fake namespace object
+/******/ 	// mode & 1: value is a module id, require it
+/******/ 	// mode & 2: merge all properties of value into the ns
+/******/ 	// mode & 4: return value when already ns object
+/******/ 	// mode & 8|1: behave like require
+/******/ 	__webpack_require__.t = function(value, mode) {
+/******/ 		if(mode & 1) value = __webpack_require__(value);
+/******/ 		if(mode & 8) return value;
+/******/ 		if((mode & 4) && typeof value === 'object' && value && value.__esModule) return value;
+/******/ 		var ns = Object.create(null);
+/******/ 		__webpack_require__.r(ns);
+/******/ 		Object.defineProperty(ns, 'default', { enumerable: true, value: value });
+/******/ 		if(mode & 2 && typeof value != 'string') for(var key in value) __webpack_require__.d(ns, key, function(key) { return value[key]; }.bind(null, key));
+/******/ 		return ns;
+/******/ 	};
+/******/
+/******/ 	// getDefaultExport function for compatibility with non-harmony modules
+/******/ 	__webpack_require__.n = function(module) {
+/******/ 		var getter = module && module.__esModule ?
+/******/ 			function getDefault() { return module['default']; } :
+/******/ 			function getModuleExports() { return module; };
+/******/ 		__webpack_require__.d(getter, 'a', getter);
+/******/ 		return getter;
+/******/ 	};
+/******/
+/******/ 	// Object.prototype.hasOwnProperty.call
+/******/ 	__webpack_require__.o = function(object, property) { return Object.prototype.hasOwnProperty.call(object, property); };
+/******/
+/******/ 	// __webpack_public_path__
+/******/ 	__webpack_require__.p = "";
+/******/
+/******/
+/******/ 	// Load entry module and return exports
+/******/ 	return __webpack_require__(__webpack_require__.s = 1);
+/******/ })
+/************************************************************************/
+/******/ ([
+/* 0 */
+/***/ (function(module, exports) {
+
+module.exports = __WEBPACK_EXTERNAL_MODULE__0__;
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+
+// EXTERNAL MODULE: external "katex"
+var external_katex_ = __webpack_require__(0);
+var external_katex_default = /*#__PURE__*/__webpack_require__.n(external_katex_);
+
+// CONCATENATED MODULE: ./contrib/auto-render/splitAtDelimiters.js
+/* eslint no-constant-condition:0 */
+var findEndOfMath = function findEndOfMath(delimiter, text, startIndex) {
+  // Adapted from
+  // https://github.com/Khan/perseus/blob/master/src/perseus-markdown.jsx
+  var index = startIndex;
+  var braceLevel = 0;
+  var delimLength = delimiter.length;
+
+  while (index < text.length) {
+    var character = text[index];
+
+    if (braceLevel <= 0 && text.slice(index, index + delimLength) === delimiter) {
+      return index;
+    } else if (character === "\\") {
+      index++;
+    } else if (character === "{") {
+      braceLevel++;
+    } else if (character === "}") {
+      braceLevel--;
+    }
+
+    index++;
+  }
+
+  return -1;
+};
+
+var splitAtDelimiters = function splitAtDelimiters(startData, leftDelim, rightDelim, display) {
+  var finalData = [];
+
+  for (var i = 0; i < startData.length; i++) {
+    if (startData[i].type === "text") {
+      var text = startData[i].data;
+      var lookingForLeft = true;
+      var currIndex = 0;
+      var nextIndex = void 0;
+      nextIndex = text.indexOf(leftDelim);
+
+      if (nextIndex !== -1) {
+        currIndex = nextIndex;
+        finalData.push({
+          type: "text",
+          data: text.slice(0, currIndex)
+        });
+        lookingForLeft = false;
+      }
+
+      while (true) {
+        if (lookingForLeft) {
+          nextIndex = text.indexOf(leftDelim, currIndex);
+
+          if (nextIndex === -1) {
+            break;
+          }
+
+          finalData.push({
+            type: "text",
+            data: text.slice(currIndex, nextIndex)
+          });
+          currIndex = nextIndex;
+        } else {
+          nextIndex = findEndOfMath(rightDelim, text, currIndex + leftDelim.length);
+
+          if (nextIndex === -1) {
+            break;
+          }
+
+          finalData.push({
+            type: "math",
+            data: text.slice(currIndex + leftDelim.length, nextIndex),
+            rawData: text.slice(currIndex, nextIndex + rightDelim.length),
+            display: display
+          });
+          currIndex = nextIndex + rightDelim.length;
+        }
+
+        lookingForLeft = !lookingForLeft;
+      }
+
+      finalData.push({
+        type: "text",
+        data: text.slice(currIndex)
+      });
+    } else {
+      finalData.push(startData[i]);
+    }
+  }
+
+  return finalData;
+};
+
+/* harmony default export */ var auto_render_splitAtDelimiters = (splitAtDelimiters);
+// CONCATENATED MODULE: ./contrib/auto-render/auto-render.js
+/* eslint no-console:0 */
+
+
+
+var auto_render_splitWithDelimiters = function splitWithDelimiters(text, delimiters) {
+  var data = [{
+    type: "text",
+    data: text
+  }];
+
+  for (var i = 0; i < delimiters.length; i++) {
+    var delimiter = delimiters[i];
+    data = auto_render_splitAtDelimiters(data, delimiter.left, delimiter.right, delimiter.display || false);
+  }
+
+  return data;
+};
+/* Note: optionsCopy is mutated by this method. If it is ever exposed in the
+ * API, we should copy it before mutating.
+ */
+
+
+var auto_render_renderMathInText = function renderMathInText(text, optionsCopy) {
+  var data = auto_render_splitWithDelimiters(text, optionsCopy.delimiters);
+
+  if (data.length === 1 && data[0].type === 'text') {
+    // There is no formula in the text.
+    // Let's return null which means there is no need to replace
+    // the current text node with a new one.
+    return null;
+  }
+
+  var fragment = document.createDocumentFragment();
+
+  for (var i = 0; i < data.length; i++) {
+    if (data[i].type === "text") {
+      fragment.appendChild(document.createTextNode(data[i].data));
+    } else {
+      var span = document.createElement("span");
+      var math = data[i].data; // Override any display mode defined in the settings with that
+      // defined by the text itself
+
+      optionsCopy.displayMode = data[i].display;
+
+      try {
+        if (optionsCopy.preProcess) {
+          math = optionsCopy.preProcess(math);
+        }
+
+        external_katex_default.a.render(math, span, optionsCopy);
+      } catch (e) {
+        if (!(e instanceof external_katex_default.a.ParseError)) {
+          throw e;
+        }
+
+        optionsCopy.errorCallback("KaTeX auto-render: Failed to parse `" + data[i].data + "` with ", e);
+        fragment.appendChild(document.createTextNode(data[i].rawData));
+        continue;
+      }
+
+      fragment.appendChild(span);
+    }
+  }
+
+  return fragment;
+};
+
+var renderElem = function renderElem(elem, optionsCopy) {
+  for (var i = 0; i < elem.childNodes.length; i++) {
+    var childNode = elem.childNodes[i];
+
+    if (childNode.nodeType === 3) {
+      // Text node
+      var frag = auto_render_renderMathInText(childNode.textContent, optionsCopy);
+
+      if (frag) {
+        i += frag.childNodes.length - 1;
+        elem.replaceChild(frag, childNode);
+      }
+    } else if (childNode.nodeType === 1) {
+      (function () {
+        // Element node
+        var className = ' ' + childNode.className + ' ';
+        var shouldRender = optionsCopy.ignoredTags.indexOf(childNode.nodeName.toLowerCase()) === -1 && optionsCopy.ignoredClasses.every(function (x) {
+          return className.indexOf(' ' + x + ' ') === -1;
+        });
+
+        if (shouldRender) {
+          renderElem(childNode, optionsCopy);
+        }
+      })();
+    } // Otherwise, it's something else, and ignore it.
+
+  }
+};
+
+var renderMathInElement = function renderMathInElement(elem, options) {
+  if (!elem) {
+    throw new Error("No element provided to render");
+  }
+
+  var optionsCopy = {}; // Object.assign(optionsCopy, option)
+
+  for (var option in options) {
+    if (options.hasOwnProperty(option)) {
+      optionsCopy[option] = options[option];
+    }
+  } // default options
+
+
+  optionsCopy.delimiters = optionsCopy.delimiters || [{
+    left: "$$",
+    right: "$$",
+    display: true
+  }, {
+    left: "\\(",
+    right: "\\)",
+    display: false
+  }, // LaTeX uses $…$, but it ruins the display of normal `$` in text:
+  // {left: "$", right: "$", display: false},
+  //  \[…\] must come last in this array. Otherwise, renderMathInElement
+  //  will search for \[ before it searches for $$ or  \(
+  // That makes it susceptible to finding a \\[0.3em] row delimiter and
+  // treating it as if it were the start of a KaTeX math zone.
+  {
+    left: "\\[",
+    right: "\\]",
+    display: true
+  }];
+  optionsCopy.ignoredTags = optionsCopy.ignoredTags || ["script", "noscript", "style", "textarea", "pre", "code", "option"];
+  optionsCopy.ignoredClasses = optionsCopy.ignoredClasses || [];
+  optionsCopy.errorCallback = optionsCopy.errorCallback || console.error; // Enable sharing of global macros defined via `\gdef` between different
+  // math elements within a single call to `renderMathInElement`.
+
+  optionsCopy.macros = optionsCopy.macros || {};
+  renderElem(elem, optionsCopy);
+};
+
+/* harmony default export */ var auto_render = __webpack_exports__["default"] = (renderMathInElement);
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/docs/pt/katex/contrib/auto-render.min.js b/docs/pt/katex/contrib/auto-render.min.js
new file mode 100644
index 0000000..840e74b
--- /dev/null
+++ b/docs/pt/katex/contrib/auto-render.min.js
@@ -0,0 +1 @@
+!function(e,t){"object"==typeof exports&&"object"==typeof module?module.exports=t(require("katex")):"function"==typeof define&&define.amd?define(["katex"],t):"object"==typeof exports?exports.renderMathInElement=t(require("katex")):e.renderMathInElement=t(e.katex)}("undefined"!=typeof self?self:this,function(e){return function(e){var t={};function r(n){if(t[n])return t[n].exports;var o=t[n]={i:n,l:!1,exports:{}};return e[n].call(o.exports,o,o.exports,r),o.l=!0,o.exports}return r.m=e,r.c=t,r.d=function(e,t,n){r.o(e,t)||Object.defineProperty(e,t,{enumerable:!0,get:n})},r.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},r.t=function(e,t){if(1&t&&(e=r(e)),8&t)return e;if(4&t&&"object"==typeof e&&e&&e.__esModule)return e;var n=Object.create(null);if(r.r(n),Object.defineProperty(n,"default",{enumerable:!0,value:e}),2&t&&"string"!=typeof e)for(var o in e)r.d(n,o,function(t){return e[t]}.bind(null,o));return n},r.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return r.d(t,"a",t),t},r.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)},r.p="",r(r.s=1)}([function(t,r){t.exports=e},function(e,t,r){"use strict";r.r(t);var n=r(0),o=r.n(n),a=function(e,t,r){for(var n=r,o=0,a=e.length;n<t.length;){var i=t[n];if(o<=0&&t.slice(n,n+a)===e)return n;"\\"===i?n++:"{"===i?o++:"}"===i&&o--,n++}return-1},i=function(e,t,r,n){for(var o=[],i=0;i<e.length;i++)if("text"===e[i].type){var l=e[i].data,d=!0,s=0,f=void 0;for(-1!==(f=l.indexOf(t))&&(s=f,o.push({type:"text",data:l.slice(0,s)}),d=!1);;){if(d){if(-1===(f=l.indexOf(t,s)))break;o.push({type:"text",data:l.slice(s,f)}),s=f}else{if(-1===(f=a(r,l,s+t.length)))break;o.push({type:"math",data:l.slice(s+t.length,f),rawData:l.slice(s,f+r.length),display:n}),s=f+r.length}d=!d}o.push({type:"text",data:l.slice(s)})}else o.push(e[i]);return o},l=function(e,t){var r=function(e,t){for(var r=[{type:"text",data:e}],n=0;n<t.length;n++){var o=t[n];r=i(r,o.left,o.right,o.display||!1)}return r}(e,t.delimiters);if(1===r.length&&"text"===r[0].type)return null;for(var n=document.createDocumentFragment(),a=0;a<r.length;a++)if("text"===r[a].type)n.appendChild(document.createTextNode(r[a].data));else{var l=document.createElement("span"),d=r[a].data;t.displayMode=r[a].display;try{t.preProcess&&(d=t.preProcess(d)),o.a.render(d,l,t)}catch(e){if(!(e instanceof o.a.ParseError))throw e;t.errorCallback("KaTeX auto-render: Failed to parse `"+r[a].data+"` with ",e),n.appendChild(document.createTextNode(r[a].rawData));continue}n.appendChild(l)}return n};t.default=function(e,t){if(!e)throw new Error("No element provided to render");var r={};for(var n in t)t.hasOwnProperty(n)&&(r[n]=t[n]);r.delimiters=r.delimiters||[{left:"$$",right:"$$",display:!0},{left:"\\(",right:"\\)",display:!1},{left:"\\[",right:"\\]",display:!0}],r.ignoredTags=r.ignoredTags||["script","noscript","style","textarea","pre","code","option"],r.ignoredClasses=r.ignoredClasses||[],r.errorCallback=r.errorCallback||console.error,r.macros=r.macros||{},function e(t,r){for(var n=0;n<t.childNodes.length;n++){var o=t.childNodes[n];if(3===o.nodeType){var a=l(o.textContent,r);a&&(n+=a.childNodes.length-1,t.replaceChild(a,o))}else 1===o.nodeType&&function(){var t=" "+o.className+" ";-1===r.ignoredTags.indexOf(o.nodeName.toLowerCase())&&r.ignoredClasses.every(function(e){return-1===t.indexOf(" "+e+" ")})&&e(o,r)}()}}(e,r)}}]).default});
\ No newline at end of file
diff --git a/docs/pt/katex/contrib/auto-render.mjs b/docs/pt/katex/contrib/auto-render.mjs
new file mode 100644
index 0000000..bc9a71f
--- /dev/null
+++ b/docs/pt/katex/contrib/auto-render.mjs
@@ -0,0 +1,226 @@
+import katex from '../katex.mjs';
+
+/* eslint no-constant-condition:0 */
+const findEndOfMath = function findEndOfMath(delimiter, text, startIndex) {
+  // Adapted from
+  // https://github.com/Khan/perseus/blob/master/src/perseus-markdown.jsx
+  let index = startIndex;
+  let braceLevel = 0;
+  const delimLength = delimiter.length;
+
+  while (index < text.length) {
+    const character = text[index];
+
+    if (braceLevel <= 0 && text.slice(index, index + delimLength) === delimiter) {
+      return index;
+    } else if (character === "\\") {
+      index++;
+    } else if (character === "{") {
+      braceLevel++;
+    } else if (character === "}") {
+      braceLevel--;
+    }
+
+    index++;
+  }
+
+  return -1;
+};
+
+const splitAtDelimiters = function splitAtDelimiters(startData, leftDelim, rightDelim, display) {
+  const finalData = [];
+
+  for (let i = 0; i < startData.length; i++) {
+    if (startData[i].type === "text") {
+      const text = startData[i].data;
+      let lookingForLeft = true;
+      let currIndex = 0;
+      let nextIndex;
+      nextIndex = text.indexOf(leftDelim);
+
+      if (nextIndex !== -1) {
+        currIndex = nextIndex;
+        finalData.push({
+          type: "text",
+          data: text.slice(0, currIndex)
+        });
+        lookingForLeft = false;
+      }
+
+      while (true) {
+        if (lookingForLeft) {
+          nextIndex = text.indexOf(leftDelim, currIndex);
+
+          if (nextIndex === -1) {
+            break;
+          }
+
+          finalData.push({
+            type: "text",
+            data: text.slice(currIndex, nextIndex)
+          });
+          currIndex = nextIndex;
+        } else {
+          nextIndex = findEndOfMath(rightDelim, text, currIndex + leftDelim.length);
+
+          if (nextIndex === -1) {
+            break;
+          }
+
+          finalData.push({
+            type: "math",
+            data: text.slice(currIndex + leftDelim.length, nextIndex),
+            rawData: text.slice(currIndex, nextIndex + rightDelim.length),
+            display: display
+          });
+          currIndex = nextIndex + rightDelim.length;
+        }
+
+        lookingForLeft = !lookingForLeft;
+      }
+
+      finalData.push({
+        type: "text",
+        data: text.slice(currIndex)
+      });
+    } else {
+      finalData.push(startData[i]);
+    }
+  }
+
+  return finalData;
+};
+
+/* eslint no-console:0 */
+
+const splitWithDelimiters = function splitWithDelimiters(text, delimiters) {
+  let data = [{
+    type: "text",
+    data: text
+  }];
+
+  for (let i = 0; i < delimiters.length; i++) {
+    const delimiter = delimiters[i];
+    data = splitAtDelimiters(data, delimiter.left, delimiter.right, delimiter.display || false);
+  }
+
+  return data;
+};
+/* Note: optionsCopy is mutated by this method. If it is ever exposed in the
+ * API, we should copy it before mutating.
+ */
+
+
+const renderMathInText = function renderMathInText(text, optionsCopy) {
+  const data = splitWithDelimiters(text, optionsCopy.delimiters);
+
+  if (data.length === 1 && data[0].type === 'text') {
+    // There is no formula in the text.
+    // Let's return null which means there is no need to replace
+    // the current text node with a new one.
+    return null;
+  }
+
+  const fragment = document.createDocumentFragment();
+
+  for (let i = 0; i < data.length; i++) {
+    if (data[i].type === "text") {
+      fragment.appendChild(document.createTextNode(data[i].data));
+    } else {
+      const span = document.createElement("span");
+      let math = data[i].data; // Override any display mode defined in the settings with that
+      // defined by the text itself
+
+      optionsCopy.displayMode = data[i].display;
+
+      try {
+        if (optionsCopy.preProcess) {
+          math = optionsCopy.preProcess(math);
+        }
+
+        katex.render(math, span, optionsCopy);
+      } catch (e) {
+        if (!(e instanceof katex.ParseError)) {
+          throw e;
+        }
+
+        optionsCopy.errorCallback("KaTeX auto-render: Failed to parse `" + data[i].data + "` with ", e);
+        fragment.appendChild(document.createTextNode(data[i].rawData));
+        continue;
+      }
+
+      fragment.appendChild(span);
+    }
+  }
+
+  return fragment;
+};
+
+const renderElem = function renderElem(elem, optionsCopy) {
+  for (let i = 0; i < elem.childNodes.length; i++) {
+    const childNode = elem.childNodes[i];
+
+    if (childNode.nodeType === 3) {
+      // Text node
+      const frag = renderMathInText(childNode.textContent, optionsCopy);
+
+      if (frag) {
+        i += frag.childNodes.length - 1;
+        elem.replaceChild(frag, childNode);
+      }
+    } else if (childNode.nodeType === 1) {
+      // Element node
+      const className = ' ' + childNode.className + ' ';
+      const shouldRender = optionsCopy.ignoredTags.indexOf(childNode.nodeName.toLowerCase()) === -1 && optionsCopy.ignoredClasses.every(x => className.indexOf(' ' + x + ' ') === -1);
+
+      if (shouldRender) {
+        renderElem(childNode, optionsCopy);
+      }
+    } // Otherwise, it's something else, and ignore it.
+
+  }
+};
+
+const renderMathInElement = function renderMathInElement(elem, options) {
+  if (!elem) {
+    throw new Error("No element provided to render");
+  }
+
+  const optionsCopy = {}; // Object.assign(optionsCopy, option)
+
+  for (const option in options) {
+    if (options.hasOwnProperty(option)) {
+      optionsCopy[option] = options[option];
+    }
+  } // default options
+
+
+  optionsCopy.delimiters = optionsCopy.delimiters || [{
+    left: "$$",
+    right: "$$",
+    display: true
+  }, {
+    left: "\\(",
+    right: "\\)",
+    display: false
+  }, // LaTeX uses $…$, but it ruins the display of normal `$` in text:
+  // {left: "$", right: "$", display: false},
+  //  \[…\] must come last in this array. Otherwise, renderMathInElement
+  //  will search for \[ before it searches for $$ or  \(
+  // That makes it susceptible to finding a \\[0.3em] row delimiter and
+  // treating it as if it were the start of a KaTeX math zone.
+  {
+    left: "\\[",
+    right: "\\]",
+    display: true
+  }];
+  optionsCopy.ignoredTags = optionsCopy.ignoredTags || ["script", "noscript", "style", "textarea", "pre", "code", "option"];
+  optionsCopy.ignoredClasses = optionsCopy.ignoredClasses || [];
+  optionsCopy.errorCallback = optionsCopy.errorCallback || console.error; // Enable sharing of global macros defined via `\gdef` between different
+  // math elements within a single call to `renderMathInElement`.
+
+  optionsCopy.macros = optionsCopy.macros || {};
+  renderElem(elem, optionsCopy);
+};
+
+export default renderMathInElement;
diff --git a/docs/pt/katex/contrib/copy-tex.css b/docs/pt/katex/contrib/copy-tex.css
new file mode 100644
index 0000000..90801d8
--- /dev/null
+++ b/docs/pt/katex/contrib/copy-tex.css
@@ -0,0 +1,14 @@
+/* Force selection of entire .katex/.katex-display blocks, so that we can
+ * copy/paste the entire source code.  If you omit this CSS, partial
+ * selections of a formula will work, but will copy the ugly HTML
+ * representation instead of the LaTeX source code.  (Full selections will
+ * still produce the LaTeX source code.)
+ */
+.katex,
+.katex-display {
+    user-select: all;
+    -moz-user-select: all;
+    -webkit-user-select: all;
+    -ms-user-select: all;
+}
+
diff --git a/docs/pt/katex/contrib/copy-tex.js b/docs/pt/katex/contrib/copy-tex.js
new file mode 100644
index 0000000..4649f8f
--- /dev/null
+++ b/docs/pt/katex/contrib/copy-tex.js
@@ -0,0 +1,213 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory();
+	else if(typeof define === 'function' && define.amd)
+		define([], factory);
+	else {
+		var a = factory();
+		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
+	}
+})((typeof self !== 'undefined' ? self : this), function() {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
+/******/ 	function __webpack_require__(moduleId) {
+/******/
+/******/ 		// Check if module is in cache
+/******/ 		if(installedModules[moduleId]) {
+/******/ 			return installedModules[moduleId].exports;
+/******/ 		}
+/******/ 		// Create a new module (and put it into the cache)
+/******/ 		var module = installedModules[moduleId] = {
+/******/ 			i: moduleId,
+/******/ 			l: false,
+/******/ 			exports: {}
+/******/ 		};
+/******/
+/******/ 		// Execute the module function
+/******/ 		modules[moduleId].call(module.exports, module, module.exports, __webpack_require__);
+/******/
+/******/ 		// Flag the module as loaded
+/******/ 		module.l = true;
+/******/
+/******/ 		// Return the exports of the module
+/******/ 		return module.exports;
+/******/ 	}
+/******/
+/******/
+/******/ 	// expose the modules object (__webpack_modules__)
+/******/ 	__webpack_require__.m = modules;
+/******/
+/******/ 	// expose the module cache
+/******/ 	__webpack_require__.c = installedModules;
+/******/
+/******/ 	// define getter function for harmony exports
+/******/ 	__webpack_require__.d = function(exports, name, getter) {
+/******/ 		if(!__webpack_require__.o(exports, name)) {
+/******/ 			Object.defineProperty(exports, name, { enumerable: true, get: getter });
+/******/ 		}
+/******/ 	};
+/******/
+/******/ 	// define __esModule on exports
+/******/ 	__webpack_require__.r = function(exports) {
+/******/ 		if(typeof Symbol !== 'undefined' && Symbol.toStringTag) {
+/******/ 			Object.defineProperty(exports, Symbol.toStringTag, { value: 'Module' });
+/******/ 		}
+/******/ 		Object.defineProperty(exports, '__esModule', { value: true });
+/******/ 	};
+/******/
+/******/ 	// create a fake namespace object
+/******/ 	// mode & 1: value is a module id, require it
+/******/ 	// mode & 2: merge all properties of value into the ns
+/******/ 	// mode & 4: return value when already ns object
+/******/ 	// mode & 8|1: behave like require
+/******/ 	__webpack_require__.t = function(value, mode) {
+/******/ 		if(mode & 1) value = __webpack_require__(value);
+/******/ 		if(mode & 8) return value;
+/******/ 		if((mode & 4) && typeof value === 'object' && value && value.__esModule) return value;
+/******/ 		var ns = Object.create(null);
+/******/ 		__webpack_require__.r(ns);
+/******/ 		Object.defineProperty(ns, 'default', { enumerable: true, value: value });
+/******/ 		if(mode & 2 && typeof value != 'string') for(var key in value) __webpack_require__.d(ns, key, function(key) { return value[key]; }.bind(null, key));
+/******/ 		return ns;
+/******/ 	};
+/******/
+/******/ 	// getDefaultExport function for compatibility with non-harmony modules
+/******/ 	__webpack_require__.n = function(module) {
+/******/ 		var getter = module && module.__esModule ?
+/******/ 			function getDefault() { return module['default']; } :
+/******/ 			function getModuleExports() { return module; };
+/******/ 		__webpack_require__.d(getter, 'a', getter);
+/******/ 		return getter;
+/******/ 	};
+/******/
+/******/ 	// Object.prototype.hasOwnProperty.call
+/******/ 	__webpack_require__.o = function(object, property) { return Object.prototype.hasOwnProperty.call(object, property); };
+/******/
+/******/ 	// __webpack_public_path__
+/******/ 	__webpack_require__.p = "";
+/******/
+/******/
+/******/ 	// Load entry module and return exports
+/******/ 	return __webpack_require__(__webpack_require__.s = 1);
+/******/ })
+/************************************************************************/
+/******/ ([
+/* 0 */
+/***/ (function(module, exports, __webpack_require__) {
+
+// extracted by mini-css-extract-plugin
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+
+// EXTERNAL MODULE: ./contrib/copy-tex/copy-tex.css
+var copy_tex = __webpack_require__(0);
+
+// CONCATENATED MODULE: ./contrib/copy-tex/katex2tex.js
+// Set these to how you want inline and display math to be delimited.
+var defaultCopyDelimiters = {
+  inline: ['$', '$'],
+  // alternative: ['\(', '\)']
+  display: ['$$', '$$'] // alternative: ['\[', '\]']
+
+}; // Replace .katex elements with their TeX source (<annotation> element).
+// Modifies fragment in-place.  Useful for writing your own 'copy' handler,
+// as in copy-tex.js.
+
+var katexReplaceWithTex = function katexReplaceWithTex(fragment, copyDelimiters) {
+  if (copyDelimiters === void 0) {
+    copyDelimiters = defaultCopyDelimiters;
+  }
+
+  // Remove .katex-html blocks that are preceded by .katex-mathml blocks
+  // (which will get replaced below).
+  var katexHtml = fragment.querySelectorAll('.katex-mathml + .katex-html');
+
+  for (var i = 0; i < katexHtml.length; i++) {
+    var element = katexHtml[i];
+
+    if (element.remove) {
+      element.remove(null);
+    } else {
+      element.parentNode.removeChild(element);
+    }
+  } // Replace .katex-mathml elements with their annotation (TeX source)
+  // descendant, with inline delimiters.
+
+
+  var katexMathml = fragment.querySelectorAll('.katex-mathml');
+
+  for (var _i = 0; _i < katexMathml.length; _i++) {
+    var _element = katexMathml[_i];
+
+    var texSource = _element.querySelector('annotation');
+
+    if (texSource) {
+      if (_element.replaceWith) {
+        _element.replaceWith(texSource);
+      } else {
+        _element.parentNode.replaceChild(texSource, _element);
+      }
+
+      texSource.innerHTML = copyDelimiters.inline[0] + texSource.innerHTML + copyDelimiters.inline[1];
+    }
+  } // Switch display math to display delimiters.
+
+
+  var displays = fragment.querySelectorAll('.katex-display annotation');
+
+  for (var _i2 = 0; _i2 < displays.length; _i2++) {
+    var _element2 = displays[_i2];
+    _element2.innerHTML = copyDelimiters.display[0] + _element2.innerHTML.substr(copyDelimiters.inline[0].length, _element2.innerHTML.length - copyDelimiters.inline[0].length - copyDelimiters.inline[1].length) + copyDelimiters.display[1];
+  }
+
+  return fragment;
+};
+/* harmony default export */ var katex2tex = (katexReplaceWithTex);
+// CONCATENATED MODULE: ./contrib/copy-tex/copy-tex.js
+ // Global copy handler to modify behavior on .katex elements.
+
+document.addEventListener('copy', function (event) {
+  var selection = window.getSelection();
+
+  if (selection.isCollapsed) {
+    return; // default action OK if selection is empty
+  }
+
+  var fragment = selection.getRangeAt(0).cloneContents();
+
+  if (!fragment.querySelector('.katex-mathml')) {
+    return; // default action OK if no .katex-mathml elements
+  } // Preserve usual HTML copy/paste behavior.
+
+
+  var html = [];
+
+  for (var i = 0; i < fragment.childNodes.length; i++) {
+    html.push(fragment.childNodes[i].outerHTML);
+  }
+
+  event.clipboardData.setData('text/html', html.join('')); // Rewrite plain-text version.
+
+  event.clipboardData.setData('text/plain', katex2tex(fragment).textContent); // Prevent normal copy handling.
+
+  event.preventDefault();
+});
+// CONCATENATED MODULE: ./contrib/copy-tex/copy-tex.webpack.js
+/**
+ * This is the webpack entry point for KaTeX. As ECMAScript doesn't support
+ * CSS modules natively, a separate entry point is used.
+ */
+
+
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/docs/pt/katex/contrib/copy-tex.min.css b/docs/pt/katex/contrib/copy-tex.min.css
new file mode 100644
index 0000000..555ed11
--- /dev/null
+++ b/docs/pt/katex/contrib/copy-tex.min.css
@@ -0,0 +1 @@
+.katex,.katex-display{user-select:all;-moz-user-select:all;-webkit-user-select:all;-ms-user-select:all}
diff --git a/docs/pt/katex/contrib/copy-tex.min.js b/docs/pt/katex/contrib/copy-tex.min.js
new file mode 100644
index 0000000..e0354d6
--- /dev/null
+++ b/docs/pt/katex/contrib/copy-tex.min.js
@@ -0,0 +1 @@
+!function(e,t){if("object"==typeof exports&&"object"==typeof module)module.exports=t();else if("function"==typeof define&&define.amd)define([],t);else{var n=t();for(var r in n)("object"==typeof exports?exports:e)[r]=n[r]}}("undefined"!=typeof self?self:this,function(){return function(e){var t={};function n(r){if(t[r])return t[r].exports;var o=t[r]={i:r,l:!1,exports:{}};return e[r].call(o.exports,o,o.exports,n),o.l=!0,o.exports}return n.m=e,n.c=t,n.d=function(e,t,r){n.o(e,t)||Object.defineProperty(e,t,{enumerable:!0,get:r})},n.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},n.t=function(e,t){if(1&t&&(e=n(e)),8&t)return e;if(4&t&&"object"==typeof e&&e&&e.__esModule)return e;var r=Object.create(null);if(n.r(r),Object.defineProperty(r,"default",{enumerable:!0,value:e}),2&t&&"string"!=typeof e)for(var o in e)n.d(r,o,function(t){return e[t]}.bind(null,o));return r},n.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return n.d(t,"a",t),t},n.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)},n.p="",n(n.s=1)}([function(e,t,n){},function(e,t,n){"use strict";n.r(t);n(0);var r={inline:["$","$"],display:["$$","$$"]},o=function(e,t){void 0===t&&(t=r);for(var n=e.querySelectorAll(".katex-mathml + .katex-html"),o=0;o<n.length;o++){var l=n[o];l.remove?l.remove(null):l.parentNode.removeChild(l)}for(var i=e.querySelectorAll(".katex-mathml"),a=0;a<i.length;a++){var u=i[a],f=u.querySelector("annotation");f&&(u.replaceWith?u.replaceWith(f):u.parentNode.replaceChild(f,u),f.innerHTML=t.inline[0]+f.innerHTML+t.inline[1])}for(var c=e.querySelectorAll(".katex-display annotation"),d=0;d<c.length;d++){var p=c[d];p.innerHTML=t.display[0]+p.innerHTML.substr(t.inline[0].length,p.innerHTML.length-t.inline[0].length-t.inline[1].length)+t.display[1]}return e};document.addEventListener("copy",function(e){var t=window.getSelection();if(!t.isCollapsed){var n=t.getRangeAt(0).cloneContents();if(n.querySelector(".katex-mathml")){for(var r=[],l=0;l<n.childNodes.length;l++)r.push(n.childNodes[l].outerHTML);e.clipboardData.setData("text/html",r.join("")),e.clipboardData.setData("text/plain",o(n).textContent),e.preventDefault()}}})}]).default});
\ No newline at end of file
diff --git a/docs/pt/katex/contrib/copy-tex.mjs b/docs/pt/katex/contrib/copy-tex.mjs
new file mode 100644
index 0000000..b6ed1ee
--- /dev/null
+++ b/docs/pt/katex/contrib/copy-tex.mjs
@@ -0,0 +1,85 @@
+// Set these to how you want inline and display math to be delimited.
+const defaultCopyDelimiters = {
+  inline: ['$', '$'],
+  // alternative: ['\(', '\)']
+  display: ['$$', '$$'] // alternative: ['\[', '\]']
+
+}; // Replace .katex elements with their TeX source (<annotation> element).
+// Modifies fragment in-place.  Useful for writing your own 'copy' handler,
+// as in copy-tex.js.
+
+const katexReplaceWithTex = function katexReplaceWithTex(fragment, copyDelimiters) {
+  if (copyDelimiters === void 0) {
+    copyDelimiters = defaultCopyDelimiters;
+  }
+
+  // Remove .katex-html blocks that are preceded by .katex-mathml blocks
+  // (which will get replaced below).
+  const katexHtml = fragment.querySelectorAll('.katex-mathml + .katex-html');
+
+  for (let i = 0; i < katexHtml.length; i++) {
+    const element = katexHtml[i];
+
+    if (element.remove) {
+      element.remove(null);
+    } else {
+      element.parentNode.removeChild(element);
+    }
+  } // Replace .katex-mathml elements with their annotation (TeX source)
+  // descendant, with inline delimiters.
+
+
+  const katexMathml = fragment.querySelectorAll('.katex-mathml');
+
+  for (let i = 0; i < katexMathml.length; i++) {
+    const element = katexMathml[i];
+    const texSource = element.querySelector('annotation');
+
+    if (texSource) {
+      if (element.replaceWith) {
+        element.replaceWith(texSource);
+      } else {
+        element.parentNode.replaceChild(texSource, element);
+      }
+
+      texSource.innerHTML = copyDelimiters.inline[0] + texSource.innerHTML + copyDelimiters.inline[1];
+    }
+  } // Switch display math to display delimiters.
+
+
+  const displays = fragment.querySelectorAll('.katex-display annotation');
+
+  for (let i = 0; i < displays.length; i++) {
+    const element = displays[i];
+    element.innerHTML = copyDelimiters.display[0] + element.innerHTML.substr(copyDelimiters.inline[0].length, element.innerHTML.length - copyDelimiters.inline[0].length - copyDelimiters.inline[1].length) + copyDelimiters.display[1];
+  }
+
+  return fragment;
+};
+
+document.addEventListener('copy', function (event) {
+  const selection = window.getSelection();
+
+  if (selection.isCollapsed) {
+    return; // default action OK if selection is empty
+  }
+
+  const fragment = selection.getRangeAt(0).cloneContents();
+
+  if (!fragment.querySelector('.katex-mathml')) {
+    return; // default action OK if no .katex-mathml elements
+  } // Preserve usual HTML copy/paste behavior.
+
+
+  const html = [];
+
+  for (let i = 0; i < fragment.childNodes.length; i++) {
+    html.push(fragment.childNodes[i].outerHTML);
+  }
+
+  event.clipboardData.setData('text/html', html.join('')); // Rewrite plain-text version.
+
+  event.clipboardData.setData('text/plain', katexReplaceWithTex(fragment).textContent); // Prevent normal copy handling.
+
+  event.preventDefault();
+});
diff --git a/docs/pt/katex/contrib/mathtex-script-type.js b/docs/pt/katex/contrib/mathtex-script-type.js
new file mode 100644
index 0000000..daa01a6
--- /dev/null
+++ b/docs/pt/katex/contrib/mathtex-script-type.js
@@ -0,0 +1,137 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory(require("katex"));
+	else if(typeof define === 'function' && define.amd)
+		define(["katex"], factory);
+	else {
+		var a = typeof exports === 'object' ? factory(require("katex")) : factory(root["katex"]);
+		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
+	}
+})((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__0__) {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
+/******/ 	function __webpack_require__(moduleId) {
+/******/
+/******/ 		// Check if module is in cache
+/******/ 		if(installedModules[moduleId]) {
+/******/ 			return installedModules[moduleId].exports;
+/******/ 		}
+/******/ 		// Create a new module (and put it into the cache)
+/******/ 		var module = installedModules[moduleId] = {
+/******/ 			i: moduleId,
+/******/ 			l: false,
+/******/ 			exports: {}
+/******/ 		};
+/******/
+/******/ 		// Execute the module function
+/******/ 		modules[moduleId].call(module.exports, module, module.exports, __webpack_require__);
+/******/
+/******/ 		// Flag the module as loaded
+/******/ 		module.l = true;
+/******/
+/******/ 		// Return the exports of the module
+/******/ 		return module.exports;
+/******/ 	}
+/******/
+/******/
+/******/ 	// expose the modules object (__webpack_modules__)
+/******/ 	__webpack_require__.m = modules;
+/******/
+/******/ 	// expose the module cache
+/******/ 	__webpack_require__.c = installedModules;
+/******/
+/******/ 	// define getter function for harmony exports
+/******/ 	__webpack_require__.d = function(exports, name, getter) {
+/******/ 		if(!__webpack_require__.o(exports, name)) {
+/******/ 			Object.defineProperty(exports, name, { enumerable: true, get: getter });
+/******/ 		}
+/******/ 	};
+/******/
+/******/ 	// define __esModule on exports
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+
+module.exports = __WEBPACK_EXTERNAL_MODULE__0__;
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0__ = __webpack_require__(0);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0___default = /*#__PURE__*/__webpack_require__.n(katex__WEBPACK_IMPORTED_MODULE_0__);
+
+var scripts = document.body.getElementsByTagName("script");
+scripts = Array.prototype.slice.call(scripts);
+scripts.forEach(function (script) {
+  if (!script.type || !script.type.match(/math\/tex/i)) {
+    return -1;
+  }
+
+  var display = script.type.match(/mode\s*=\s*display(;|\s|\n|$)/) != null;
+  var katexElement = document.createElement(display ? "div" : "span");
+  katexElement.setAttribute("class", display ? "equation" : "inline-equation");
+
+  try {
+    katex__WEBPACK_IMPORTED_MODULE_0___default.a.render(script.text, katexElement, {
+      displayMode: display
+    });
+  } catch (err) {
+    //console.error(err); linter doesn't like this
+    katexElement.textContent = script.text;
+  }
+
+  script.parentNode.replaceChild(katexElement, script);
+});
+
+/***/ })
+/******/ ])["default"];
+});
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diff --git a/docs/pt/katex/contrib/mathtex-script-type.min.js b/docs/pt/katex/contrib/mathtex-script-type.min.js
new file mode 100644
index 0000000..ae9f528
--- /dev/null
+++ b/docs/pt/katex/contrib/mathtex-script-type.min.js
@@ -0,0 +1 @@
+!function(e,t){if("object"==typeof exports&&"object"==typeof module)module.exports=t(require("katex"));else if("function"==typeof define&&define.amd)define(["katex"],t);else{var r="object"==typeof exports?t(require("katex")):t(e.katex);for(var n in r)("object"==typeof exports?exports:e)[n]=r[n]}}("undefined"!=typeof self?self:this,function(e){return function(e){var t={};function r(n){if(t[n])return t[n].exports;var o=t[n]={i:n,l:!1,exports:{}};return e[n].call(o.exports,o,o.exports,r),o.l=!0,o.exports}return r.m=e,r.c=t,r.d=function(e,t,n){r.o(e,t)||Object.defineProperty(e,t,{enumerable:!0,get:n})},r.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},r.t=function(e,t){if(1&t&&(e=r(e)),8&t)return e;if(4&t&&"object"==typeof e&&e&&e.__esModule)return e;var n=Object.create(null);if(r.r(n),Object.defineProperty(n,"default",{enumerable:!0,value:e}),2&t&&"string"!=typeof e)for(var o in e)r.d(n,o,function(t){return e[t]}.bind(null,o));return n},r.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return r.d(t,"a",t),t},r.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)},r.p="",r(r.s=1)}([function(t,r){t.exports=e},function(e,t,r){"use strict";r.r(t);var n=r(0),o=r.n(n),u=document.body.getElementsByTagName("script");(u=Array.prototype.slice.call(u)).forEach(function(e){if(!e.type||!e.type.match(/math\/tex/i))return-1;var t=null!=e.type.match(/mode\s*=\s*display(;|\s|\n|$)/),r=document.createElement(t?"div":"span");r.setAttribute("class",t?"equation":"inline-equation");try{o.a.render(e.text,r,{displayMode:t})}catch(t){r.textContent=e.text}e.parentNode.replaceChild(r,e)})}]).default});
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diff --git a/docs/pt/katex/contrib/mathtex-script-type.mjs b/docs/pt/katex/contrib/mathtex-script-type.mjs
new file mode 100644
index 0000000..7cfb90e
--- /dev/null
+++ b/docs/pt/katex/contrib/mathtex-script-type.mjs
@@ -0,0 +1,24 @@
+import katex from '../katex.mjs';
+
+let scripts = document.body.getElementsByTagName("script");
+scripts = Array.prototype.slice.call(scripts);
+scripts.forEach(function (script) {
+  if (!script.type || !script.type.match(/math\/tex/i)) {
+    return -1;
+  }
+
+  const display = script.type.match(/mode\s*=\s*display(;|\s|\n|$)/) != null;
+  const katexElement = document.createElement(display ? "div" : "span");
+  katexElement.setAttribute("class", display ? "equation" : "inline-equation");
+
+  try {
+    katex.render(script.text, katexElement, {
+      displayMode: display
+    });
+  } catch (err) {
+    //console.error(err); linter doesn't like this
+    katexElement.textContent = script.text;
+  }
+
+  script.parentNode.replaceChild(katexElement, script);
+});
diff --git a/docs/pt/katex/contrib/mhchem.js b/docs/pt/katex/contrib/mhchem.js
new file mode 100644
index 0000000..f84566e
--- /dev/null
+++ b/docs/pt/katex/contrib/mhchem.js
@@ -0,0 +1,3241 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory(require("katex"));
+	else if(typeof define === 'function' && define.amd)
+		define(["katex"], factory);
+	else {
+		var a = typeof exports === 'object' ? factory(require("katex")) : factory(root["katex"]);
+		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
+	}
+})((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__0__) {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
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+/******/
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+/******/ 		module.l = true;
+/******/
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+/******/ 		return module.exports;
+/******/ 	}
+/******/
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+/******/ 	__webpack_require__.m = modules;
+/******/
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+/******/ })
+/************************************************************************/
+/******/ ([
+/* 0 */
+/***/ (function(module, exports) {
+
+module.exports = __WEBPACK_EXTERNAL_MODULE__0__;
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0__ = __webpack_require__(0);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0___default = /*#__PURE__*/__webpack_require__.n(katex__WEBPACK_IMPORTED_MODULE_0__);
+/* eslint-disable */
+
+/* -*- Mode: Javascript; indent-tabs-mode:nil; js-indent-level: 2 -*- */
+
+/* vim: set ts=2 et sw=2 tw=80: */
+
+/*************************************************************
+ *
+ *  KaTeX mhchem.js
+ *
+ *  This file implements a KaTeX version of mhchem version 3.3.0.
+ *  It is adapted from MathJax/extensions/TeX/mhchem.js
+ *  It differs from the MathJax version as follows:
+ *    1. The interface is changed so that it can be called from KaTeX, not MathJax.
+ *    2. \rlap and \llap are replaced with \mathrlap and \mathllap.
+ *    3. Four lines of code are edited in order to use \raisebox instead of \raise.
+ *    4. The reaction arrow code is simplified. All reaction arrows are rendered
+ *       using KaTeX extensible arrows instead of building non-extensible arrows.
+ *    5. \tripledash vertical alignment is slightly adjusted.
+ *
+ *    This code, as other KaTeX code, is released under the MIT license.
+ * 
+ * /*************************************************************
+ *
+ *  MathJax/extensions/TeX/mhchem.js
+ *
+ *  Implements the \ce command for handling chemical formulas
+ *  from the mhchem LaTeX package.
+ *
+ *  ---------------------------------------------------------------------
+ *
+ *  Copyright (c) 2011-2015 The MathJax Consortium
+ *  Copyright (c) 2015-2018 Martin Hensel
+ *
+ *  Licensed under the Apache License, Version 2.0 (the "License");
+ *  you may not use this file except in compliance with the License.
+ *  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ *  Unless required by applicable law or agreed to in writing, software
+ *  distributed under the License is distributed on an "AS IS" BASIS,
+ *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ *  See the License for the specific language governing permissions and
+ *  limitations under the License.
+ */
+//
+// Coding Style
+//   - use '' for identifiers that can by minified/uglified
+//   - use "" for strings that need to stay untouched
+// version: "3.3.0" for MathJax and KaTeX
+// Add \ce, \pu, and \tripledash to the KaTeX macros.
+katex__WEBPACK_IMPORTED_MODULE_0___default.a.__defineMacro("\\ce", function (context) {
+  return chemParse(context.consumeArgs(1)[0], "ce");
+});
+
+katex__WEBPACK_IMPORTED_MODULE_0___default.a.__defineMacro("\\pu", function (context) {
+  return chemParse(context.consumeArgs(1)[0], "pu");
+}); //  Needed for \bond for the ~ forms
+//  Raise by 2.56mu, not 2mu. We're raising a hyphen-minus, U+002D, not 
+//  a mathematical minus, U+2212. So we need that extra 0.56.
+
+
+katex__WEBPACK_IMPORTED_MODULE_0___default.a.__defineMacro("\\tripledash", "{\\vphantom{-}\\raisebox{2.56mu}{$\\mkern2mu" + "\\tiny\\text{-}\\mkern1mu\\text{-}\\mkern1mu\\text{-}\\mkern2mu$}}");
+
+ //
+//  This is the main function for handing the \ce and \pu commands.
+//  It takes the argument to \ce or \pu and returns the corresponding TeX string.
+//
+
+var chemParse = function chemParse(tokens, stateMachine) {
+  // Recreate the argument string from KaTeX's array of tokens.
+  var str = "";
+  var expectedLoc = tokens[tokens.length - 1].loc.start;
+
+  for (var i = tokens.length - 1; i >= 0; i--) {
+    if (tokens[i].loc.start > expectedLoc) {
+      // context.consumeArgs has eaten a space.
+      str += " ";
+      expectedLoc = tokens[i].loc.start;
+    }
+
+    str += tokens[i].text;
+    expectedLoc += tokens[i].text.length;
+  }
+
+  var tex = texify.go(mhchemParser.go(str, stateMachine));
+  return tex;
+}; //
+// Core parser for mhchem syntax  (recursive)
+//
+
+/** @type {MhchemParser} */
+
+
+var mhchemParser = {
+  //
+  // Parses mchem \ce syntax
+  //
+  // Call like
+  //   go("H2O");
+  //
+  go: function go(input, stateMachine) {
+    if (!input) {
+      return [];
+    }
+
+    if (stateMachine === undefined) {
+      stateMachine = 'ce';
+    }
+
+    var state = '0'; //
+    // String buffers for parsing:
+    //
+    // buffer.a == amount
+    // buffer.o == element
+    // buffer.b == left-side superscript
+    // buffer.p == left-side subscript
+    // buffer.q == right-side subscript
+    // buffer.d == right-side superscript
+    //
+    // buffer.r == arrow
+    // buffer.rdt == arrow, script above, type
+    // buffer.rd == arrow, script above, content
+    // buffer.rqt == arrow, script below, type
+    // buffer.rq == arrow, script below, content
+    //
+    // buffer.text_
+    // buffer.rm
+    // etc.
+    //
+    // buffer.parenthesisLevel == int, starting at 0
+    // buffer.sb == bool, space before
+    // buffer.beginsWithBond == bool
+    //
+    // These letters are also used as state names.
+    //
+    // Other states:
+    // 0 == begin of main part (arrow/operator unlikely)
+    // 1 == next entity
+    // 2 == next entity (arrow/operator unlikely)
+    // 3 == next atom
+    // c == macro
+    //
+
+    /** @type {Buffer} */
+
+    var buffer = {};
+    buffer['parenthesisLevel'] = 0;
+    input = input.replace(/\n/g, " ");
+    input = input.replace(/[\u2212\u2013\u2014\u2010]/g, "-");
+    input = input.replace(/[\u2026]/g, "..."); //
+    // Looks through mhchemParser.transitions, to execute a matching action
+    // (recursive)
+    //
+
+    var lastInput;
+    var watchdog = 10;
+    /** @type {ParserOutput[]} */
+
+    var output = [];
+
+    while (true) {
+      if (lastInput !== input) {
+        watchdog = 10;
+        lastInput = input;
+      } else {
+        watchdog--;
+      } //
+      // Find actions in transition table
+      //
+
+
+      var machine = mhchemParser.stateMachines[stateMachine];
+      var t = machine.transitions[state] || machine.transitions['*'];
+
+      iterateTransitions: for (var i = 0; i < t.length; i++) {
+        var matches = mhchemParser.patterns.match_(t[i].pattern, input);
+
+        if (matches) {
+          //
+          // Execute actions
+          //
+          var task = t[i].task;
+
+          for (var iA = 0; iA < task.action_.length; iA++) {
+            var o; //
+            // Find and execute action
+            //
+
+            if (machine.actions[task.action_[iA].type_]) {
+              o = machine.actions[task.action_[iA].type_](buffer, matches.match_, task.action_[iA].option);
+            } else if (mhchemParser.actions[task.action_[iA].type_]) {
+              o = mhchemParser.actions[task.action_[iA].type_](buffer, matches.match_, task.action_[iA].option);
+            } else {
+              throw ["MhchemBugA", "mhchem bug A. Please report. (" + task.action_[iA].type_ + ")"]; // Trying to use non-existing action
+            } //
+            // Add output
+            //
+
+
+            mhchemParser.concatArray(output, o);
+          } //
+          // Set next state,
+          // Shorten input,
+          // Continue with next character
+          //   (= apply only one transition per position)
+          //
+
+
+          state = task.nextState || state;
+
+          if (input.length > 0) {
+            if (!task.revisit) {
+              input = matches.remainder;
+            }
+
+            if (!task.toContinue) {
+              break iterateTransitions;
+            }
+          } else {
+            return output;
+          }
+        }
+      } //
+      // Prevent infinite loop
+      //
+
+
+      if (watchdog <= 0) {
+        throw ["MhchemBugU", "mhchem bug U. Please report."]; // Unexpected character
+      }
+    }
+  },
+  concatArray: function concatArray(a, b) {
+    if (b) {
+      if (Array.isArray(b)) {
+        for (var iB = 0; iB < b.length; iB++) {
+          a.push(b[iB]);
+        }
+      } else {
+        a.push(b);
+      }
+    }
+  },
+  patterns: {
+    //
+    // Matching patterns
+    // either regexps or function that return null or {match_:"a", remainder:"bc"}
+    //
+    patterns: {
+      // property names must not look like integers ("2") for correct property traversal order, later on
+      'empty': /^$/,
+      'else': /^./,
+      'else2': /^./,
+      'space': /^\s/,
+      'space A': /^\s(?=[A-Z\\$])/,
+      'space$': /^\s$/,
+      'a-z': /^[a-z]/,
+      'x': /^x/,
+      'x$': /^x$/,
+      'i$': /^i$/,
+      'letters': /^(?:[a-zA-Z\u03B1-\u03C9\u0391-\u03A9?@]|(?:\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega|Gamma|Delta|Theta|Lambda|Xi|Pi|Sigma|Upsilon|Phi|Psi|Omega)(?:\s+|\{\}|(?![a-zA-Z]))))+/,
+      '\\greek': /^\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega|Gamma|Delta|Theta|Lambda|Xi|Pi|Sigma|Upsilon|Phi|Psi|Omega)(?:\s+|\{\}|(?![a-zA-Z]))/,
+      'one lowercase latin letter $': /^(?:([a-z])(?:$|[^a-zA-Z]))$/,
+      '$one lowercase latin letter$ $': /^\$(?:([a-z])(?:$|[^a-zA-Z]))\$$/,
+      'one lowercase greek letter $': /^(?:\$?[\u03B1-\u03C9]\$?|\$?\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega)\s*\$?)(?:\s+|\{\}|(?![a-zA-Z]))$/,
+      'digits': /^[0-9]+/,
+      '-9.,9': /^[+\-]?(?:[0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))/,
+      '-9.,9 no missing 0': /^[+\-]?[0-9]+(?:[.,][0-9]+)?/,
+      '(-)(9.,9)(e)(99)': function e99(input) {
+        var m = input.match(/^(\+\-|\+\/\-|\+|\-|\\pm\s?)?([0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))?(\((?:[0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))\))?(?:([eE]|\s*(\*|x|\\times|\u00D7)\s*10\^)([+\-]?[0-9]+|\{[+\-]?[0-9]+\}))?/);
+
+        if (m && m[0]) {
+          return {
+            match_: m.splice(1),
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      '(-)(9)^(-9)': function _(input) {
+        var m = input.match(/^(\+\-|\+\/\-|\+|\-|\\pm\s?)?([0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+)?)\^([+\-]?[0-9]+|\{[+\-]?[0-9]+\})/);
+
+        if (m && m[0]) {
+          return {
+            match_: m.splice(1),
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      'state of aggregation $': function stateOfAggregation$(input) {
+        // ... or crystal system
+        var a = mhchemParser.patterns.findObserveGroups(input, "", /^\([a-z]{1,3}(?=[\),])/, ")", ""); // (aq), (aq,$\infty$), (aq, sat)
+
+        if (a && a.remainder.match(/^($|[\s,;\)\]\}])/)) {
+          return a;
+        } //  AND end of 'phrase'
+
+
+        var m = input.match(/^(?:\((?:\\ca\s?)?\$[amothc]\$\))/); // OR crystal system ($o$) (\ca$c$)
+
+        if (m) {
+          return {
+            match_: m[0],
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      '_{(state of aggregation)}$': /^_\{(\([a-z]{1,3}\))\}/,
+      '{[(': /^(?:\\\{|\[|\()/,
+      ')]}': /^(?:\)|\]|\\\})/,
+      ', ': /^[,;]\s*/,
+      ',': /^[,;]/,
+      '.': /^[.]/,
+      '. ': /^([.\u22C5\u00B7\u2022])\s*/,
+      '...': /^\.\.\.(?=$|[^.])/,
+      '* ': /^([*])\s*/,
+      '^{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^{", "", "", "}");
+      },
+      '^($...$)': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", "$", "$", "");
+      },
+      '^a': /^\^([0-9]+|[^\\_])/,
+      '^\\x{}{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '^\\x{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '^\\x': /^\^(\\[a-zA-Z]+)\s*/,
+      '^(-1)': /^\^(-?\d+)/,
+      '\'': /^'/,
+      '_{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_{", "", "", "}");
+      },
+      '_($...$)': function _$$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", "$", "$", "");
+      },
+      '_9': /^_([+\-]?[0-9]+|[^\\])/,
+      '_\\x{}{}': function _X(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '_\\x{}': function _X(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '_\\x': /^_(\\[a-zA-Z]+)\s*/,
+      '^_': /^(?:\^(?=_)|\_(?=\^)|[\^_]$)/,
+      '{}': /^\{\}/,
+      '{...}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", "{", "}", "");
+      },
+      '{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "{", "", "", "}");
+      },
+      '$...$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", "$", "$", "");
+      },
+      '${(...)}$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "${", "", "", "}$");
+      },
+      '$(...)$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "$", "", "", "$");
+      },
+      '=<>': /^[=<>]/,
+      '#': /^[#\u2261]/,
+      '+': /^\+/,
+      '-$': /^-(?=[\s_},;\]/]|$|\([a-z]+\))/,
+      // -space -, -; -] -/ -$ -state-of-aggregation
+      '-9': /^-(?=[0-9])/,
+      '- orbital overlap': /^-(?=(?:[spd]|sp)(?:$|[\s,;\)\]\}]))/,
+      '-': /^-/,
+      'pm-operator': /^(?:\\pm|\$\\pm\$|\+-|\+\/-)/,
+      'operator': /^(?:\+|(?:[\-=<>]|<<|>>|\\approx|\$\\approx\$)(?=\s|$|-?[0-9]))/,
+      'arrowUpDown': /^(?:v|\(v\)|\^|\(\^\))(?=$|[\s,;\)\]\}])/,
+      '\\bond{(...)}': function bond(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\bond{", "", "", "}");
+      },
+      '->': /^(?:<->|<-->|->|<-|<=>>|<<=>|<=>|[\u2192\u27F6\u21CC])/,
+      'CMT': /^[CMT](?=\[)/,
+      '[(...)]': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "[", "", "", "]");
+      },
+      '1st-level escape': /^(&|\\\\|\\hline)\s*/,
+      '\\,': /^(?:\\[,\ ;:])/,
+      // \\x - but output no space before
+      '\\x{}{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '\\x{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '\\ca': /^\\ca(?:\s+|(?![a-zA-Z]))/,
+      '\\x': /^(?:\\[a-zA-Z]+\s*|\\[_&{}%])/,
+      'orbital': /^(?:[0-9]{1,2}[spdfgh]|[0-9]{0,2}sp)(?=$|[^a-zA-Z])/,
+      // only those with numbers in front, because the others will be formatted correctly anyway
+      'others': /^[\/~|]/,
+      '\\frac{(...)}': function frac(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\frac{", "", "", "}", "{", "", "", "}");
+      },
+      '\\overset{(...)}': function overset(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\overset{", "", "", "}", "{", "", "", "}");
+      },
+      "\\underset{(...)}": function underset(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\underset{", "", "", "}", "{", "", "", "}");
+      },
+      "\\underbrace{(...)}": function underbrace(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\underbrace{", "", "", "}_", "{", "", "", "}");
+      },
+      '\\color{(...)}0': function color0(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color{", "", "", "}");
+      },
+      '\\color{(...)}{(...)}1': function color1(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color{", "", "", "}", "{", "", "", "}");
+      },
+      '\\color(...){(...)}2': function color2(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color", "\\", "", /^(?=\{)/, "{", "", "", "}");
+      },
+      '\\ce{(...)}': function ce(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\ce{", "", "", "}");
+      },
+      'oxidation$': /^(?:[+-][IVX]+|\\pm\s*0|\$\\pm\$\s*0)$/,
+      'd-oxidation$': /^(?:[+-]?\s?[IVX]+|\\pm\s*0|\$\\pm\$\s*0)$/,
+      // 0 could be oxidation or charge
+      'roman numeral': /^[IVX]+/,
+      '1/2$': /^[+\-]?(?:[0-9]+|\$[a-z]\$|[a-z])\/[0-9]+(?:\$[a-z]\$|[a-z])?$/,
+      'amount': function amount(input) {
+        var match; // e.g. 2, 0.5, 1/2, -2, n/2, +;  $a$ could be added later in parsing
+
+        match = input.match(/^(?:(?:(?:\([+\-]?[0-9]+\/[0-9]+\)|[+\-]?(?:[0-9]+|\$[a-z]\$|[a-z])\/[0-9]+|[+\-]?[0-9]+[.,][0-9]+|[+\-]?\.[0-9]+|[+\-]?[0-9]+)(?:[a-z](?=\s*[A-Z]))?)|[+\-]?[a-z](?=\s*[A-Z])|\+(?!\s))/);
+
+        if (match) {
+          return {
+            match_: match[0],
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        var a = mhchemParser.patterns.findObserveGroups(input, "", "$", "$", "");
+
+        if (a) {
+          // e.g. $2n-1$, $-$
+          match = a.match_.match(/^\$(?:\(?[+\-]?(?:[0-9]*[a-z]?[+\-])?[0-9]*[a-z](?:[+\-][0-9]*[a-z]?)?\)?|\+|-)\$$/);
+
+          if (match) {
+            return {
+              match_: match[0],
+              remainder: input.substr(match[0].length)
+            };
+          }
+        }
+
+        return null;
+      },
+      'amount2': function amount2(input) {
+        return this['amount'](input);
+      },
+      '(KV letters),': /^(?:[A-Z][a-z]{0,2}|i)(?=,)/,
+      'formula$': function formula$(input) {
+        if (input.match(/^\([a-z]+\)$/)) {
+          return null;
+        } // state of aggregation = no formula
+
+
+        var match = input.match(/^(?:[a-z]|(?:[0-9\ \+\-\,\.\(\)]+[a-z])+[0-9\ \+\-\,\.\(\)]*|(?:[a-z][0-9\ \+\-\,\.\(\)]+)+[a-z]?)$/);
+
+        if (match) {
+          return {
+            match_: match[0],
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        return null;
+      },
+      'uprightEntities': /^(?:pH|pOH|pC|pK|iPr|iBu)(?=$|[^a-zA-Z])/,
+      '/': /^\s*(\/)\s*/,
+      '//': /^\s*(\/\/)\s*/,
+      '*': /^\s*[*.]\s*/
+    },
+    findObserveGroups: function findObserveGroups(input, begExcl, begIncl, endIncl, endExcl, beg2Excl, beg2Incl, end2Incl, end2Excl, combine) {
+      /** @type {{(input: string, pattern: string | RegExp): string | string[] | null;}} */
+      var _match = function _match(input, pattern) {
+        if (typeof pattern === "string") {
+          if (input.indexOf(pattern) !== 0) {
+            return null;
+          }
+
+          return pattern;
+        } else {
+          var match = input.match(pattern);
+
+          if (!match) {
+            return null;
+          }
+
+          return match[0];
+        }
+      };
+      /** @type {{(input: string, i: number, endChars: string | RegExp): {endMatchBegin: number, endMatchEnd: number} | null;}} */
+
+
+      var _findObserveGroups = function _findObserveGroups(input, i, endChars) {
+        var braces = 0;
+
+        while (i < input.length) {
+          var a = input.charAt(i);
+
+          var match = _match(input.substr(i), endChars);
+
+          if (match !== null && braces === 0) {
+            return {
+              endMatchBegin: i,
+              endMatchEnd: i + match.length
+            };
+          } else if (a === "{") {
+            braces++;
+          } else if (a === "}") {
+            if (braces === 0) {
+              throw ["ExtraCloseMissingOpen", "Extra close brace or missing open brace"];
+            } else {
+              braces--;
+            }
+          }
+
+          i++;
+        }
+
+        if (braces > 0) {
+          return null;
+        }
+
+        return null;
+      };
+
+      var match = _match(input, begExcl);
+
+      if (match === null) {
+        return null;
+      }
+
+      input = input.substr(match.length);
+      match = _match(input, begIncl);
+
+      if (match === null) {
+        return null;
+      }
+
+      var e = _findObserveGroups(input, match.length, endIncl || endExcl);
+
+      if (e === null) {
+        return null;
+      }
+
+      var match1 = input.substring(0, endIncl ? e.endMatchEnd : e.endMatchBegin);
+
+      if (!(beg2Excl || beg2Incl)) {
+        return {
+          match_: match1,
+          remainder: input.substr(e.endMatchEnd)
+        };
+      } else {
+        var group2 = this.findObserveGroups(input.substr(e.endMatchEnd), beg2Excl, beg2Incl, end2Incl, end2Excl);
+
+        if (group2 === null) {
+          return null;
+        }
+        /** @type {string[]} */
+
+
+        var matchRet = [match1, group2.match_];
+        return {
+          match_: combine ? matchRet.join("") : matchRet,
+          remainder: group2.remainder
+        };
+      }
+    },
+    //
+    // Matching function
+    // e.g. match("a", input) will look for the regexp called "a" and see if it matches
+    // returns null or {match_:"a", remainder:"bc"}
+    //
+    match_: function match_(m, input) {
+      var pattern = mhchemParser.patterns.patterns[m];
+
+      if (pattern === undefined) {
+        throw ["MhchemBugP", "mhchem bug P. Please report. (" + m + ")"]; // Trying to use non-existing pattern
+      } else if (typeof pattern === "function") {
+        return mhchemParser.patterns.patterns[m](input); // cannot use cached var pattern here, because some pattern functions need this===mhchemParser
+      } else {
+        // RegExp
+        var match = input.match(pattern);
+
+        if (match) {
+          var mm;
+
+          if (match[2]) {
+            mm = [match[1], match[2]];
+          } else if (match[1]) {
+            mm = match[1];
+          } else {
+            mm = match[0];
+          }
+
+          return {
+            match_: mm,
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        return null;
+      }
+    }
+  },
+  //
+  // Generic state machine actions
+  //
+  actions: {
+    'a=': function a(buffer, m) {
+      buffer.a = (buffer.a || "") + m;
+    },
+    'b=': function b(buffer, m) {
+      buffer.b = (buffer.b || "") + m;
+    },
+    'p=': function p(buffer, m) {
+      buffer.p = (buffer.p || "") + m;
+    },
+    'o=': function o(buffer, m) {
+      buffer.o = (buffer.o || "") + m;
+    },
+    'q=': function q(buffer, m) {
+      buffer.q = (buffer.q || "") + m;
+    },
+    'd=': function d(buffer, m) {
+      buffer.d = (buffer.d || "") + m;
+    },
+    'rm=': function rm(buffer, m) {
+      buffer.rm = (buffer.rm || "") + m;
+    },
+    'text=': function text(buffer, m) {
+      buffer.text_ = (buffer.text_ || "") + m;
+    },
+    'insert': function insert(buffer, m, a) {
+      return {
+        type_: a
+      };
+    },
+    'insert+p1': function insertP1(buffer, m, a) {
+      return {
+        type_: a,
+        p1: m
+      };
+    },
+    'insert+p1+p2': function insertP1P2(buffer, m, a) {
+      return {
+        type_: a,
+        p1: m[0],
+        p2: m[1]
+      };
+    },
+    'copy': function copy(buffer, m) {
+      return m;
+    },
+    'rm': function rm(buffer, m) {
+      return {
+        type_: 'rm',
+        p1: m || ""
+      };
+    },
+    'text': function text(buffer, m) {
+      return mhchemParser.go(m, 'text');
+    },
+    '{text}': function text(buffer, m) {
+      var ret = ["{"];
+      mhchemParser.concatArray(ret, mhchemParser.go(m, 'text'));
+      ret.push("}");
+      return ret;
+    },
+    'tex-math': function texMath(buffer, m) {
+      return mhchemParser.go(m, 'tex-math');
+    },
+    'tex-math tight': function texMathTight(buffer, m) {
+      return mhchemParser.go(m, 'tex-math tight');
+    },
+    'bond': function bond(buffer, m, k) {
+      return {
+        type_: 'bond',
+        kind_: k || m
+      };
+    },
+    'color0-output': function color0Output(buffer, m) {
+      return {
+        type_: 'color0',
+        color: m[0]
+      };
+    },
+    'ce': function ce(buffer, m) {
+      return mhchemParser.go(m);
+    },
+    '1/2': function _(buffer, m) {
+      /** @type {ParserOutput[]} */
+      var ret = [];
+
+      if (m.match(/^[+\-]/)) {
+        ret.push(m.substr(0, 1));
+        m = m.substr(1);
+      }
+
+      var n = m.match(/^([0-9]+|\$[a-z]\$|[a-z])\/([0-9]+)(\$[a-z]\$|[a-z])?$/);
+      n[1] = n[1].replace(/\$/g, "");
+      ret.push({
+        type_: 'frac',
+        p1: n[1],
+        p2: n[2]
+      });
+
+      if (n[3]) {
+        n[3] = n[3].replace(/\$/g, "");
+        ret.push({
+          type_: 'tex-math',
+          p1: n[3]
+        });
+      }
+
+      return ret;
+    },
+    '9,9': function _(buffer, m) {
+      return mhchemParser.go(m, '9,9');
+    }
+  },
+  //
+  // createTransitions
+  // convert  { 'letter': { 'state': { action_: 'output' } } }  to  { 'state' => [ { pattern: 'letter', task: { action_: [{type_: 'output'}] } } ] }
+  // with expansion of 'a|b' to 'a' and 'b' (at 2 places)
+  //
+  createTransitions: function createTransitions(o) {
+    var pattern, state;
+    /** @type {string[]} */
+
+    var stateArray;
+    var i; //
+    // 1. Collect all states
+    //
+
+    /** @type {Transitions} */
+
+    var transitions = {};
+
+    for (pattern in o) {
+      for (state in o[pattern]) {
+        stateArray = state.split("|");
+        o[pattern][state].stateArray = stateArray;
+
+        for (i = 0; i < stateArray.length; i++) {
+          transitions[stateArray[i]] = [];
+        }
+      }
+    } //
+    // 2. Fill states
+    //
+
+
+    for (pattern in o) {
+      for (state in o[pattern]) {
+        stateArray = o[pattern][state].stateArray || [];
+
+        for (i = 0; i < stateArray.length; i++) {
+          //
+          // 2a. Normalize actions into array:  'text=' ==> [{type_:'text='}]
+          // (Note to myself: Resolving the function here would be problematic. It would need .bind (for *this*) and currying (for *option*).)
+          //
+
+          /** @type {any} */
+          var p = o[pattern][state];
+
+          if (p.action_) {
+            p.action_ = [].concat(p.action_);
+
+            for (var k = 0; k < p.action_.length; k++) {
+              if (typeof p.action_[k] === "string") {
+                p.action_[k] = {
+                  type_: p.action_[k]
+                };
+              }
+            }
+          } else {
+            p.action_ = [];
+          } //
+          // 2.b Multi-insert
+          //
+
+
+          var patternArray = pattern.split("|");
+
+          for (var j = 0; j < patternArray.length; j++) {
+            if (stateArray[i] === '*') {
+              // insert into all
+              for (var t in transitions) {
+                transitions[t].push({
+                  pattern: patternArray[j],
+                  task: p
+                });
+              }
+            } else {
+              transitions[stateArray[i]].push({
+                pattern: patternArray[j],
+                task: p
+              });
+            }
+          }
+        }
+      }
+    }
+
+    return transitions;
+  },
+  stateMachines: {}
+}; //
+// Definition of state machines
+//
+
+mhchemParser.stateMachines = {
+  //
+  // \ce state machines
+  //
+  //#region ce
+  'ce': {
+    // main parser
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      'else': {
+        '0|1|2': {
+          action_: 'beginsWithBond=false',
+          revisit: true,
+          toContinue: true
+        }
+      },
+      'oxidation$': {
+        '0': {
+          action_: 'oxidation-output'
+        }
+      },
+      'CMT': {
+        'r': {
+          action_: 'rdt=',
+          nextState: 'rt'
+        },
+        'rd': {
+          action_: 'rqt=',
+          nextState: 'rdt'
+        }
+      },
+      'arrowUpDown': {
+        '0|1|2|as': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '1'
+        }
+      },
+      'uprightEntities': {
+        '0|1|2': {
+          action_: ['o=', 'output'],
+          nextState: '1'
+        }
+      },
+      'orbital': {
+        '0|1|2|3': {
+          action_: 'o=',
+          nextState: 'o'
+        }
+      },
+      '->': {
+        '0|1|2|3': {
+          action_: 'r=',
+          nextState: 'r'
+        },
+        'a|as': {
+          action_: ['output', 'r='],
+          nextState: 'r'
+        },
+        '*': {
+          action_: ['output', 'r='],
+          nextState: 'r'
+        }
+      },
+      '+': {
+        'o': {
+          action_: 'd= kv',
+          nextState: 'd'
+        },
+        'd|D': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qd|qD': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'dq': {
+          action_: ['output', 'd='],
+          nextState: 'd'
+        },
+        '3': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '0'
+        }
+      },
+      'amount': {
+        '0|2': {
+          action_: 'a=',
+          nextState: 'a'
+        }
+      },
+      'pm-operator': {
+        '0|1|2|a|as': {
+          action_: ['sb=false', 'output', {
+            type_: 'operator',
+            option: '\\pm'
+          }],
+          nextState: '0'
+        }
+      },
+      'operator': {
+        '0|1|2|a|as': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '0'
+        }
+      },
+      '-$': {
+        'o|q': {
+          action_: ['charge or bond', 'output'],
+          nextState: 'qd'
+        },
+        'd': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'D': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qd': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qD|dq': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        }
+      },
+      '-9': {
+        '3|o': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '3'
+        }
+      },
+      '- orbital overlap': {
+        'o': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        },
+        'd': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        }
+      },
+      '-': {
+        '0|1|2': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'beginsWithBond=true', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        '3': {
+          action_: {
+            type_: 'bond',
+            option: "-"
+          }
+        },
+        'a': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        },
+        'as': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        'b': {
+          action_: 'b='
+        },
+        'o': {
+          action_: {
+            type_: '- after o/d',
+            option: false
+          },
+          nextState: '2'
+        },
+        'q': {
+          action_: {
+            type_: '- after o/d',
+            option: false
+          },
+          nextState: '2'
+        },
+        'd|qd|dq': {
+          action_: {
+            type_: '- after o/d',
+            option: true
+          },
+          nextState: '2'
+        },
+        'D|qD|p': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        }
+      },
+      'amount2': {
+        '1|3': {
+          action_: 'a=',
+          nextState: 'a'
+        }
+      },
+      'letters': {
+        '0|1|2|3|a|as|b|p|bp|o': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        'q|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        },
+        'd|D|qd|qD': {
+          action_: 'o after d',
+          nextState: 'o'
+        }
+      },
+      'digits': {
+        'o': {
+          action_: 'q=',
+          nextState: 'q'
+        },
+        'd|D': {
+          action_: 'q=',
+          nextState: 'dq'
+        },
+        'q': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        },
+        'a': {
+          action_: 'o=',
+          nextState: 'o'
+        }
+      },
+      'space A': {
+        'b|p|bp': {}
+      },
+      'space': {
+        'a': {
+          nextState: 'as'
+        },
+        '0': {
+          action_: 'sb=false'
+        },
+        '1|2': {
+          action_: 'sb=true'
+        },
+        'r|rt|rd|rdt|rdq': {
+          action_: 'output',
+          nextState: '0'
+        },
+        '*': {
+          action_: ['output', 'sb=true'],
+          nextState: '1'
+        }
+      },
+      '1st-level escape': {
+        '1|2': {
+          action_: ['output', {
+            type_: 'insert+p1',
+            option: '1st-level escape'
+          }]
+        },
+        '*': {
+          action_: ['output', {
+            type_: 'insert+p1',
+            option: '1st-level escape'
+          }],
+          nextState: '0'
+        }
+      },
+      '[(...)]': {
+        'r|rt': {
+          action_: 'rd=',
+          nextState: 'rd'
+        },
+        'rd|rdt': {
+          action_: 'rq=',
+          nextState: 'rdq'
+        }
+      },
+      '...': {
+        'o|d|D|dq|qd|qD': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "..."
+          }],
+          nextState: '3'
+        },
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, {
+            type_: 'insert',
+            option: 'ellipsis'
+          }],
+          nextState: '1'
+        }
+      },
+      '. |* ': {
+        '*': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'addition compound'
+          }],
+          nextState: '1'
+        }
+      },
+      'state of aggregation $': {
+        '*': {
+          action_: ['output', 'state of aggregation'],
+          nextState: '1'
+        }
+      },
+      '{[(': {
+        'a|as|o': {
+          action_: ['o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        },
+        '0|1|2|3': {
+          action_: ['o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        },
+        '*': {
+          action_: ['output', 'o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        }
+      },
+      ')]}': {
+        '0|1|2|3|b|p|bp|o': {
+          action_: ['o=', 'parenthesisLevel--'],
+          nextState: 'o'
+        },
+        'a|as|d|D|q|qd|qD|dq': {
+          action_: ['output', 'o=', 'parenthesisLevel--'],
+          nextState: 'o'
+        }
+      },
+      ', ': {
+        '*': {
+          action_: ['output', 'comma'],
+          nextState: '0'
+        }
+      },
+      '^_': {
+        // ^ and _ without a sensible argument
+        '*': {}
+      },
+      '^{(...)}|^($...$)': {
+        '0|1|2|as': {
+          action_: 'b=',
+          nextState: 'b'
+        },
+        'p': {
+          action_: 'b=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'd= kv',
+          nextState: 'D'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qD'
+        },
+        'd|D|qd|qD|dq': {
+          action_: ['output', 'd='],
+          nextState: 'D'
+        }
+      },
+      '^a|^\\x{}{}|^\\x{}|^\\x|\'': {
+        '0|1|2|as': {
+          action_: 'b=',
+          nextState: 'b'
+        },
+        'p': {
+          action_: 'b=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'd= kv',
+          nextState: 'd'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'd|qd|D|qD': {
+          action_: 'd='
+        },
+        'dq': {
+          action_: ['output', 'd='],
+          nextState: 'd'
+        }
+      },
+      '_{(state of aggregation)}$': {
+        'd|D|q|qd|qD|dq': {
+          action_: ['output', 'q='],
+          nextState: 'q'
+        }
+      },
+      '_{(...)}|_($...$)|_9|_\\x{}{}|_\\x{}|_\\x': {
+        '0|1|2|as': {
+          action_: 'p=',
+          nextState: 'p'
+        },
+        'b': {
+          action_: 'p=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'q=',
+          nextState: 'q'
+        },
+        'd|D': {
+          action_: 'q=',
+          nextState: 'dq'
+        },
+        'q|qd|qD|dq': {
+          action_: ['output', 'q='],
+          nextState: 'q'
+        }
+      },
+      '=<>': {
+        '0|1|2|3|a|as|o|q|d|D|qd|qD|dq': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'bond'],
+          nextState: '3'
+        }
+      },
+      '#': {
+        '0|1|2|3|a|as|o': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, {
+            type_: 'bond',
+            option: "#"
+          }],
+          nextState: '3'
+        }
+      },
+      '{}': {
+        '*': {
+          action_: {
+            type_: 'output',
+            option: 1
+          },
+          nextState: '1'
+        }
+      },
+      '{...}': {
+        '0|1|2|3|a|as|b|p|bp': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        'o|d|D|q|qd|qD|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        }
+      },
+      '$...$': {
+        'a': {
+          action_: 'a='
+        },
+        // 2$n$
+        '0|1|2|3|as|b|p|bp|o': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        // not 'amount'
+        'as|o': {
+          action_: 'o='
+        },
+        'q|d|D|qd|qD|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        }
+      },
+      '\\bond{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'bond'],
+          nextState: "3"
+        }
+      },
+      '\\frac{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'frac-output'],
+          nextState: '3'
+        }
+      },
+      '\\overset{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'overset-output'],
+          nextState: '3'
+        }
+      },
+      "\\underset{(...)}": {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'underset-output'],
+          nextState: '3'
+        }
+      },
+      "\\underbrace{(...)}": {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'underbrace-output'],
+          nextState: '3'
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'color-output'],
+          nextState: '3'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'color0-output']
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'ce'],
+          nextState: '3'
+        }
+      },
+      '\\,': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'copy'],
+          nextState: '1'
+        }
+      },
+      '\\x{}{}|\\x{}|\\x': {
+        '0|1|2|3|a|as|b|p|bp|o|c0': {
+          action_: ['o=', 'output'],
+          nextState: '3'
+        },
+        '*': {
+          action_: ['output', 'o=', 'output'],
+          nextState: '3'
+        }
+      },
+      'others': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'copy'],
+          nextState: '3'
+        }
+      },
+      'else2': {
+        'a': {
+          action_: 'a to o',
+          nextState: 'o',
+          revisit: true
+        },
+        'as': {
+          action_: ['output', 'sb=true'],
+          nextState: '1',
+          revisit: true
+        },
+        'r|rt|rd|rdt|rdq': {
+          action_: ['output'],
+          nextState: '0',
+          revisit: true
+        },
+        '*': {
+          action_: ['output', 'copy'],
+          nextState: '3'
+        }
+      }
+    }),
+    actions: {
+      'o after d': function oAfterD(buffer, m) {
+        var ret;
+
+        if ((buffer.d || "").match(/^[0-9]+$/)) {
+          var tmp = buffer.d;
+          buffer.d = undefined;
+          ret = this['output'](buffer);
+          buffer.b = tmp;
+        } else {
+          ret = this['output'](buffer);
+        }
+
+        mhchemParser.actions['o='](buffer, m);
+        return ret;
+      },
+      'd= kv': function dKv(buffer, m) {
+        buffer.d = m;
+        buffer.dType = 'kv';
+      },
+      'charge or bond': function chargeOrBond(buffer, m) {
+        if (buffer['beginsWithBond']) {
+          /** @type {ParserOutput[]} */
+          var ret = [];
+          mhchemParser.concatArray(ret, this['output'](buffer));
+          mhchemParser.concatArray(ret, mhchemParser.actions['bond'](buffer, m, "-"));
+          return ret;
+        } else {
+          buffer.d = m;
+        }
+      },
+      '- after o/d': function afterOD(buffer, m, isAfterD) {
+        var c1 = mhchemParser.patterns.match_('orbital', buffer.o || "");
+        var c2 = mhchemParser.patterns.match_('one lowercase greek letter $', buffer.o || "");
+        var c3 = mhchemParser.patterns.match_('one lowercase latin letter $', buffer.o || "");
+        var c4 = mhchemParser.patterns.match_('$one lowercase latin letter$ $', buffer.o || "");
+        var hyphenFollows = m === "-" && (c1 && c1.remainder === "" || c2 || c3 || c4);
+
+        if (hyphenFollows && !buffer.a && !buffer.b && !buffer.p && !buffer.d && !buffer.q && !c1 && c3) {
+          buffer.o = '$' + buffer.o + '$';
+        }
+        /** @type {ParserOutput[]} */
+
+
+        var ret = [];
+
+        if (hyphenFollows) {
+          mhchemParser.concatArray(ret, this['output'](buffer));
+          ret.push({
+            type_: 'hyphen'
+          });
+        } else {
+          c1 = mhchemParser.patterns.match_('digits', buffer.d || "");
+
+          if (isAfterD && c1 && c1.remainder === '') {
+            mhchemParser.concatArray(ret, mhchemParser.actions['d='](buffer, m));
+            mhchemParser.concatArray(ret, this['output'](buffer));
+          } else {
+            mhchemParser.concatArray(ret, this['output'](buffer));
+            mhchemParser.concatArray(ret, mhchemParser.actions['bond'](buffer, m, "-"));
+          }
+        }
+
+        return ret;
+      },
+      'a to o': function aToO(buffer) {
+        buffer.o = buffer.a;
+        buffer.a = undefined;
+      },
+      'sb=true': function sbTrue(buffer) {
+        buffer.sb = true;
+      },
+      'sb=false': function sbFalse(buffer) {
+        buffer.sb = false;
+      },
+      'beginsWithBond=true': function beginsWithBondTrue(buffer) {
+        buffer['beginsWithBond'] = true;
+      },
+      'beginsWithBond=false': function beginsWithBondFalse(buffer) {
+        buffer['beginsWithBond'] = false;
+      },
+      'parenthesisLevel++': function parenthesisLevel(buffer) {
+        buffer['parenthesisLevel']++;
+      },
+      'parenthesisLevel--': function parenthesisLevel(buffer) {
+        buffer['parenthesisLevel']--;
+      },
+      'state of aggregation': function stateOfAggregation(buffer, m) {
+        return {
+          type_: 'state of aggregation',
+          p1: mhchemParser.go(m, 'o')
+        };
+      },
+      'comma': function comma(buffer, m) {
+        var a = m.replace(/\s*$/, '');
+        var withSpace = a !== m;
+
+        if (withSpace && buffer['parenthesisLevel'] === 0) {
+          return {
+            type_: 'comma enumeration L',
+            p1: a
+          };
+        } else {
+          return {
+            type_: 'comma enumeration M',
+            p1: a
+          };
+        }
+      },
+      'output': function output(buffer, m, entityFollows) {
+        // entityFollows:
+        //   undefined = if we have nothing else to output, also ignore the just read space (buffer.sb)
+        //   1 = an entity follows, never omit the space if there was one just read before (can only apply to state 1)
+        //   2 = 1 + the entity can have an amount, so output a\, instead of converting it to o (can only apply to states a|as)
+
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret;
+
+        if (!buffer.r) {
+          ret = [];
+
+          if (!buffer.a && !buffer.b && !buffer.p && !buffer.o && !buffer.q && !buffer.d && !entityFollows) {//ret = [];
+          } else {
+            if (buffer.sb) {
+              ret.push({
+                type_: 'entitySkip'
+              });
+            }
+
+            if (!buffer.o && !buffer.q && !buffer.d && !buffer.b && !buffer.p && entityFollows !== 2) {
+              buffer.o = buffer.a;
+              buffer.a = undefined;
+            } else if (!buffer.o && !buffer.q && !buffer.d && (buffer.b || buffer.p)) {
+              buffer.o = buffer.a;
+              buffer.d = buffer.b;
+              buffer.q = buffer.p;
+              buffer.a = buffer.b = buffer.p = undefined;
+            } else {
+              if (buffer.o && buffer.dType === 'kv' && mhchemParser.patterns.match_('d-oxidation$', buffer.d || "")) {
+                buffer.dType = 'oxidation';
+              } else if (buffer.o && buffer.dType === 'kv' && !buffer.q) {
+                buffer.dType = undefined;
+              }
+            }
+
+            ret.push({
+              type_: 'chemfive',
+              a: mhchemParser.go(buffer.a, 'a'),
+              b: mhchemParser.go(buffer.b, 'bd'),
+              p: mhchemParser.go(buffer.p, 'pq'),
+              o: mhchemParser.go(buffer.o, 'o'),
+              q: mhchemParser.go(buffer.q, 'pq'),
+              d: mhchemParser.go(buffer.d, buffer.dType === 'oxidation' ? 'oxidation' : 'bd'),
+              dType: buffer.dType
+            });
+          }
+        } else {
+          // r
+
+          /** @type {ParserOutput[]} */
+          var rd;
+
+          if (buffer.rdt === 'M') {
+            rd = mhchemParser.go(buffer.rd, 'tex-math');
+          } else if (buffer.rdt === 'T') {
+            rd = [{
+              type_: 'text',
+              p1: buffer.rd || ""
+            }];
+          } else {
+            rd = mhchemParser.go(buffer.rd);
+          }
+          /** @type {ParserOutput[]} */
+
+
+          var rq;
+
+          if (buffer.rqt === 'M') {
+            rq = mhchemParser.go(buffer.rq, 'tex-math');
+          } else if (buffer.rqt === 'T') {
+            rq = [{
+              type_: 'text',
+              p1: buffer.rq || ""
+            }];
+          } else {
+            rq = mhchemParser.go(buffer.rq);
+          }
+
+          ret = {
+            type_: 'arrow',
+            r: buffer.r,
+            rd: rd,
+            rq: rq
+          };
+        }
+
+        for (var p in buffer) {
+          if (p !== 'parenthesisLevel' && p !== 'beginsWithBond') {
+            delete buffer[p];
+          }
+        }
+
+        return ret;
+      },
+      'oxidation-output': function oxidationOutput(buffer, m) {
+        var ret = ["{"];
+        mhchemParser.concatArray(ret, mhchemParser.go(m, 'oxidation'));
+        ret.push("}");
+        return ret;
+      },
+      'frac-output': function fracOutput(buffer, m) {
+        return {
+          type_: 'frac-ce',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'overset-output': function oversetOutput(buffer, m) {
+        return {
+          type_: 'overset',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'underset-output': function undersetOutput(buffer, m) {
+        return {
+          type_: 'underset',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'underbrace-output': function underbraceOutput(buffer, m) {
+        return {
+          type_: 'underbrace',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1])
+        };
+      },
+      'r=': function r(buffer, m) {
+        buffer.r = m;
+      },
+      'rdt=': function rdt(buffer, m) {
+        buffer.rdt = m;
+      },
+      'rd=': function rd(buffer, m) {
+        buffer.rd = m;
+      },
+      'rqt=': function rqt(buffer, m) {
+        buffer.rqt = m;
+      },
+      'rq=': function rq(buffer, m) {
+        buffer.rq = m;
+      },
+      'operator': function operator(buffer, m, p1) {
+        return {
+          type_: 'operator',
+          kind_: p1 || m
+        };
+      }
+    }
+  },
+  'a': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '1',
+          revisit: true
+        }
+      },
+      '$(...)$': {
+        '*': {
+          action_: 'tex-math tight',
+          nextState: '1'
+        }
+      },
+      ',': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'commaDecimal'
+          }
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {}
+  },
+  'o': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '1',
+          revisit: true
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '\\ca': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'circa'
+          }
+        }
+      },
+      '\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: '{text}'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {}
+  },
+  'text': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '{...}': {
+        '*': {
+          action_: 'text='
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '\\greek': {
+        '*': {
+          action_: ['output', 'rm']
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: ['output', 'copy']
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'text='
+        }
+      }
+    }),
+    actions: {
+      'output': function output(buffer) {
+        if (buffer.text_) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'text',
+            p1: buffer.text_
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  'pq': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'state of aggregation $': {
+        '*': {
+          action_: 'state of aggregation'
+        }
+      },
+      'i$': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '(KV letters),': {
+        '0': {
+          action_: 'rm',
+          nextState: '0'
+        }
+      },
+      'formula$': {
+        '0': {
+          nextState: 'f',
+          revisit: true
+        }
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: 'text'
+        }
+      },
+      'a-z': {
+        'f': {
+          action_: 'tex-math'
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '-9.,9': {
+        '*': {
+          action_: '9,9'
+        }
+      },
+      ',': {
+        '*': {
+          action_: {
+            type_: 'insert+p1',
+            option: 'comma enumeration S'
+          }
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: 'color-output'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: 'color0-output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: 'ce'
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'state of aggregation': function stateOfAggregation(buffer, m) {
+        return {
+          type_: 'state of aggregation subscript',
+          p1: mhchemParser.go(m, 'o')
+        };
+      },
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1], 'pq')
+        };
+      }
+    }
+  },
+  'bd': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'x$': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      'formula$': {
+        '0': {
+          nextState: 'f',
+          revisit: true
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '-9.,9 no missing 0': {
+        '*': {
+          action_: '9,9'
+        }
+      },
+      '.': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'electron dot'
+          }
+        }
+      },
+      'a-z': {
+        'f': {
+          action_: 'tex-math'
+        }
+      },
+      'x': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'KV x'
+          }
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '\'': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'prime'
+          }
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: 'text'
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: 'color-output'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: 'color0-output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: 'ce'
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1], 'bd')
+        };
+      }
+    }
+  },
+  'oxidation': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'roman numeral': {
+        '*': {
+          action_: 'roman-numeral'
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'roman-numeral': function romanNumeral(buffer, m) {
+        return {
+          type_: 'roman numeral',
+          p1: m || ""
+        };
+      }
+    }
+  },
+  'tex-math': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: ['output', 'ce']
+        }
+      },
+      '{...}|\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'o='
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'o='
+        }
+      }
+    }),
+    actions: {
+      'output': function output(buffer) {
+        if (buffer.o) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'tex-math',
+            p1: buffer.o
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  'tex-math tight': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: ['output', 'ce']
+        }
+      },
+      '{...}|\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'o='
+        }
+      },
+      '-|+': {
+        '*': {
+          action_: 'tight operator'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'o='
+        }
+      }
+    }),
+    actions: {
+      'tight operator': function tightOperator(buffer, m) {
+        buffer.o = (buffer.o || "") + "{" + m + "}";
+      },
+      'output': function output(buffer) {
+        if (buffer.o) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'tex-math',
+            p1: buffer.o
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  '9,9': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      ',': {
+        '*': {
+          action_: 'comma'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'comma': function comma() {
+        return {
+          type_: 'commaDecimal'
+        };
+      }
+    }
+  },
+  //#endregion
+  //
+  // \pu state machines
+  //
+  //#region pu
+  'pu': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      'space$': {
+        '*': {
+          action_: ['output', 'space']
+        }
+      },
+      '{[(|)]}': {
+        '0|a': {
+          action_: 'copy'
+        }
+      },
+      '(-)(9)^(-9)': {
+        '0': {
+          action_: 'number^',
+          nextState: 'a'
+        }
+      },
+      '(-)(9.,9)(e)(99)': {
+        '0': {
+          action_: 'enumber',
+          nextState: 'a'
+        }
+      },
+      'space': {
+        '0|a': {}
+      },
+      'pm-operator': {
+        '0|a': {
+          action_: {
+            type_: 'operator',
+            option: '\\pm'
+          },
+          nextState: '0'
+        }
+      },
+      'operator': {
+        '0|a': {
+          action_: 'copy',
+          nextState: '0'
+        }
+      },
+      '//': {
+        'd': {
+          action_: 'o=',
+          nextState: '/'
+        }
+      },
+      '/': {
+        'd': {
+          action_: 'o=',
+          nextState: '/'
+        }
+      },
+      '{...}|else': {
+        '0|d': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'a': {
+          action_: ['space', 'd='],
+          nextState: 'd'
+        },
+        '/|q': {
+          action_: 'q=',
+          nextState: 'q'
+        }
+      }
+    }),
+    actions: {
+      'enumber': function enumber(buffer, m) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+
+        if (m[0] === "+-" || m[0] === "+/-") {
+          ret.push("\\pm ");
+        } else if (m[0]) {
+          ret.push(m[0]);
+        }
+
+        if (m[1]) {
+          mhchemParser.concatArray(ret, mhchemParser.go(m[1], 'pu-9,9'));
+
+          if (m[2]) {
+            if (m[2].match(/[,.]/)) {
+              mhchemParser.concatArray(ret, mhchemParser.go(m[2], 'pu-9,9'));
+            } else {
+              ret.push(m[2]);
+            }
+          }
+
+          m[3] = m[4] || m[3];
+
+          if (m[3]) {
+            m[3] = m[3].trim();
+
+            if (m[3] === "e" || m[3].substr(0, 1) === "*") {
+              ret.push({
+                type_: 'cdot'
+              });
+            } else {
+              ret.push({
+                type_: 'times'
+              });
+            }
+          }
+        }
+
+        if (m[3]) {
+          ret.push("10^{" + m[5] + "}");
+        }
+
+        return ret;
+      },
+      'number^': function number(buffer, m) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+
+        if (m[0] === "+-" || m[0] === "+/-") {
+          ret.push("\\pm ");
+        } else if (m[0]) {
+          ret.push(m[0]);
+        }
+
+        mhchemParser.concatArray(ret, mhchemParser.go(m[1], 'pu-9,9'));
+        ret.push("^{" + m[2] + "}");
+        return ret;
+      },
+      'operator': function operator(buffer, m, p1) {
+        return {
+          type_: 'operator',
+          kind_: p1 || m
+        };
+      },
+      'space': function space() {
+        return {
+          type_: 'pu-space-1'
+        };
+      },
+      'output': function output(buffer) {
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret;
+        var md = mhchemParser.patterns.match_('{(...)}', buffer.d || "");
+
+        if (md && md.remainder === '') {
+          buffer.d = md.match_;
+        }
+
+        var mq = mhchemParser.patterns.match_('{(...)}', buffer.q || "");
+
+        if (mq && mq.remainder === '') {
+          buffer.q = mq.match_;
+        }
+
+        if (buffer.d) {
+          buffer.d = buffer.d.replace(/\u00B0C|\^oC|\^{o}C/g, "{}^{\\circ}C");
+          buffer.d = buffer.d.replace(/\u00B0F|\^oF|\^{o}F/g, "{}^{\\circ}F");
+        }
+
+        if (buffer.q) {
+          // fraction
+          buffer.q = buffer.q.replace(/\u00B0C|\^oC|\^{o}C/g, "{}^{\\circ}C");
+          buffer.q = buffer.q.replace(/\u00B0F|\^oF|\^{o}F/g, "{}^{\\circ}F");
+          var b5 = {
+            d: mhchemParser.go(buffer.d, 'pu'),
+            q: mhchemParser.go(buffer.q, 'pu')
+          };
+
+          if (buffer.o === '//') {
+            ret = {
+              type_: 'pu-frac',
+              p1: b5.d,
+              p2: b5.q
+            };
+          } else {
+            ret = b5.d;
+
+            if (b5.d.length > 1 || b5.q.length > 1) {
+              ret.push({
+                type_: ' / '
+              });
+            } else {
+              ret.push({
+                type_: '/'
+              });
+            }
+
+            mhchemParser.concatArray(ret, b5.q);
+          }
+        } else {
+          // no fraction
+          ret = mhchemParser.go(buffer.d, 'pu-2');
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    }
+  },
+  'pu-2': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '*': {
+        '*': {
+          action_: ['output', 'cdot'],
+          nextState: '0'
+        }
+      },
+      '\\x': {
+        '*': {
+          action_: 'rm='
+        }
+      },
+      'space': {
+        '*': {
+          action_: ['output', 'space'],
+          nextState: '0'
+        }
+      },
+      '^{(...)}|^(-1)': {
+        '1': {
+          action_: '^(-1)'
+        }
+      },
+      '-9.,9': {
+        '0': {
+          action_: 'rm=',
+          nextState: '0'
+        },
+        '1': {
+          action_: '^(-1)',
+          nextState: '0'
+        }
+      },
+      '{...}|else': {
+        '*': {
+          action_: 'rm=',
+          nextState: '1'
+        }
+      }
+    }),
+    actions: {
+      'cdot': function cdot() {
+        return {
+          type_: 'tight cdot'
+        };
+      },
+      '^(-1)': function _(buffer, m) {
+        buffer.rm += "^{" + m + "}";
+      },
+      'space': function space() {
+        return {
+          type_: 'pu-space-2'
+        };
+      },
+      'output': function output(buffer) {
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret = [];
+
+        if (buffer.rm) {
+          var mrm = mhchemParser.patterns.match_('{(...)}', buffer.rm || "");
+
+          if (mrm && mrm.remainder === '') {
+            ret = mhchemParser.go(mrm.match_, 'pu');
+          } else {
+            ret = {
+              type_: 'rm',
+              p1: buffer.rm
+            };
+          }
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    }
+  },
+  'pu-9,9': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '0': {
+          action_: 'output-0'
+        },
+        'o': {
+          action_: 'output-o'
+        }
+      },
+      ',': {
+        '0': {
+          action_: ['output-0', 'comma'],
+          nextState: 'o'
+        }
+      },
+      '.': {
+        '0': {
+          action_: ['output-0', 'copy'],
+          nextState: 'o'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'text='
+        }
+      }
+    }),
+    actions: {
+      'comma': function comma() {
+        return {
+          type_: 'commaDecimal'
+        };
+      },
+      'output-0': function output0(buffer) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+        buffer.text_ = buffer.text_ || "";
+
+        if (buffer.text_.length > 4) {
+          var a = buffer.text_.length % 3;
+
+          if (a === 0) {
+            a = 3;
+          }
+
+          for (var i = buffer.text_.length - 3; i > 0; i -= 3) {
+            ret.push(buffer.text_.substr(i, 3));
+            ret.push({
+              type_: '1000 separator'
+            });
+          }
+
+          ret.push(buffer.text_.substr(0, a));
+          ret.reverse();
+        } else {
+          ret.push(buffer.text_);
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      },
+      'output-o': function outputO(buffer) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+        buffer.text_ = buffer.text_ || "";
+
+        if (buffer.text_.length > 4) {
+          var a = buffer.text_.length - 3;
+
+          for (var i = 0; i < a; i += 3) {
+            ret.push(buffer.text_.substr(i, 3));
+            ret.push({
+              type_: '1000 separator'
+            });
+          }
+
+          ret.push(buffer.text_.substr(i));
+        } else {
+          ret.push(buffer.text_);
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    } //#endregion
+
+  }
+}; //
+// texify: Take MhchemParser output and convert it to TeX
+//
+
+/** @type {Texify} */
+
+var texify = {
+  go: function go(input, isInner) {
+    // (recursive, max 4 levels)
+    if (!input) {
+      return "";
+    }
+
+    var res = "";
+    var cee = false;
+
+    for (var i = 0; i < input.length; i++) {
+      var inputi = input[i];
+
+      if (typeof inputi === "string") {
+        res += inputi;
+      } else {
+        res += texify._go2(inputi);
+
+        if (inputi.type_ === '1st-level escape') {
+          cee = true;
+        }
+      }
+    }
+
+    if (!isInner && !cee && res) {
+      res = "{" + res + "}";
+    }
+
+    return res;
+  },
+  _goInner: function _goInner(input) {
+    if (!input) {
+      return input;
+    }
+
+    return texify.go(input, true);
+  },
+  _go2: function _go2(buf) {
+    /** @type {undefined | string} */
+    var res;
+
+    switch (buf.type_) {
+      case 'chemfive':
+        res = "";
+        var b5 = {
+          a: texify._goInner(buf.a),
+          b: texify._goInner(buf.b),
+          p: texify._goInner(buf.p),
+          o: texify._goInner(buf.o),
+          q: texify._goInner(buf.q),
+          d: texify._goInner(buf.d)
+        }; //
+        // a
+        //
+
+        if (b5.a) {
+          if (b5.a.match(/^[+\-]/)) {
+            b5.a = "{" + b5.a + "}";
+          }
+
+          res += b5.a + "\\,";
+        } //
+        // b and p
+        //
+
+
+        if (b5.b || b5.p) {
+          res += "{\\vphantom{X}}";
+          res += "^{\\hphantom{" + (b5.b || "") + "}}_{\\hphantom{" + (b5.p || "") + "}}";
+          res += "{\\vphantom{X}}";
+          res += "^{\\smash[t]{\\vphantom{2}}\\mathllap{" + (b5.b || "") + "}}";
+          res += "_{\\vphantom{2}\\mathllap{\\smash[t]{" + (b5.p || "") + "}}}";
+        } //
+        // o
+        //
+
+
+        if (b5.o) {
+          if (b5.o.match(/^[+\-]/)) {
+            b5.o = "{" + b5.o + "}";
+          }
+
+          res += b5.o;
+        } //
+        // q and d
+        //
+
+
+        if (buf.dType === 'kv') {
+          if (b5.d || b5.q) {
+            res += "{\\vphantom{X}}";
+          }
+
+          if (b5.d) {
+            res += "^{" + b5.d + "}";
+          }
+
+          if (b5.q) {
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+        } else if (buf.dType === 'oxidation') {
+          if (b5.d) {
+            res += "{\\vphantom{X}}";
+            res += "^{" + b5.d + "}";
+          }
+
+          if (b5.q) {
+            res += "{\\vphantom{X}}";
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+        } else {
+          if (b5.q) {
+            res += "{\\vphantom{X}}";
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+
+          if (b5.d) {
+            res += "{\\vphantom{X}}";
+            res += "^{" + b5.d + "}";
+          }
+        }
+
+        break;
+
+      case 'rm':
+        res = "\\mathrm{" + buf.p1 + "}";
+        break;
+
+      case 'text':
+        if (buf.p1.match(/[\^_]/)) {
+          buf.p1 = buf.p1.replace(" ", "~").replace("-", "\\text{-}");
+          res = "\\mathrm{" + buf.p1 + "}";
+        } else {
+          res = "\\text{" + buf.p1 + "}";
+        }
+
+        break;
+
+      case 'roman numeral':
+        res = "\\mathrm{" + buf.p1 + "}";
+        break;
+
+      case 'state of aggregation':
+        res = "\\mskip2mu " + texify._goInner(buf.p1);
+        break;
+
+      case 'state of aggregation subscript':
+        res = "\\mskip1mu " + texify._goInner(buf.p1);
+        break;
+
+      case 'bond':
+        res = texify._getBond(buf.kind_);
+
+        if (!res) {
+          throw ["MhchemErrorBond", "mhchem Error. Unknown bond type (" + buf.kind_ + ")"];
+        }
+
+        break;
+
+      case 'frac':
+        var c = "\\frac{" + buf.p1 + "}{" + buf.p2 + "}";
+        res = "\\mathchoice{\\textstyle" + c + "}{" + c + "}{" + c + "}{" + c + "}";
+        break;
+
+      case 'pu-frac':
+        var d = "\\frac{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        res = "\\mathchoice{\\textstyle" + d + "}{" + d + "}{" + d + "}{" + d + "}";
+        break;
+
+      case 'tex-math':
+        res = buf.p1 + " ";
+        break;
+
+      case 'frac-ce':
+        res = "\\frac{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'overset':
+        res = "\\overset{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'underset':
+        res = "\\underset{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'underbrace':
+        res = "\\underbrace{" + texify._goInner(buf.p1) + "}_{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'color':
+        res = "{\\color{" + buf.color1 + "}{" + texify._goInner(buf.color2) + "}}";
+        break;
+
+      case 'color0':
+        res = "\\color{" + buf.color + "}";
+        break;
+
+      case 'arrow':
+        var b6 = {
+          rd: texify._goInner(buf.rd),
+          rq: texify._goInner(buf.rq)
+        };
+
+        var arrow = "\\x" + texify._getArrow(buf.r);
+
+        if (b6.rq) {
+          arrow += "[{" + b6.rq + "}]";
+        }
+
+        if (b6.rd) {
+          arrow += "{" + b6.rd + "}";
+        } else {
+          arrow += "{}";
+        }
+
+        res = arrow;
+        break;
+
+      case 'operator':
+        res = texify._getOperator(buf.kind_);
+        break;
+
+      case '1st-level escape':
+        res = buf.p1 + " "; // &, \\\\, \\hlin
+
+        break;
+
+      case 'space':
+        res = " ";
+        break;
+
+      case 'entitySkip':
+        res = "~";
+        break;
+
+      case 'pu-space-1':
+        res = "~";
+        break;
+
+      case 'pu-space-2':
+        res = "\\mkern3mu ";
+        break;
+
+      case '1000 separator':
+        res = "\\mkern2mu ";
+        break;
+
+      case 'commaDecimal':
+        res = "{,}";
+        break;
+
+      case 'comma enumeration L':
+        res = "{" + buf.p1 + "}\\mkern6mu ";
+        break;
+
+      case 'comma enumeration M':
+        res = "{" + buf.p1 + "}\\mkern3mu ";
+        break;
+
+      case 'comma enumeration S':
+        res = "{" + buf.p1 + "}\\mkern1mu ";
+        break;
+
+      case 'hyphen':
+        res = "\\text{-}";
+        break;
+
+      case 'addition compound':
+        res = "\\,{\\cdot}\\,";
+        break;
+
+      case 'electron dot':
+        res = "\\mkern1mu \\bullet\\mkern1mu ";
+        break;
+
+      case 'KV x':
+        res = "{\\times}";
+        break;
+
+      case 'prime':
+        res = "\\prime ";
+        break;
+
+      case 'cdot':
+        res = "\\cdot ";
+        break;
+
+      case 'tight cdot':
+        res = "\\mkern1mu{\\cdot}\\mkern1mu ";
+        break;
+
+      case 'times':
+        res = "\\times ";
+        break;
+
+      case 'circa':
+        res = "{\\sim}";
+        break;
+
+      case '^':
+        res = "uparrow";
+        break;
+
+      case 'v':
+        res = "downarrow";
+        break;
+
+      case 'ellipsis':
+        res = "\\ldots ";
+        break;
+
+      case '/':
+        res = "/";
+        break;
+
+      case ' / ':
+        res = "\\,/\\,";
+        break;
+
+      default:
+        assertNever(buf);
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+      // Missing texify rule or unknown MhchemParser output
+    }
+
+    assertString(res);
+    return res;
+  },
+  _getArrow: function _getArrow(a) {
+    switch (a) {
+      case "->":
+        return "rightarrow";
+
+      case "\u2192":
+        return "rightarrow";
+
+      case "\u27F6":
+        return "rightarrow";
+
+      case "<-":
+        return "leftarrow";
+
+      case "<->":
+        return "leftrightarrow";
+
+      case "<-->":
+        return "rightleftarrows";
+
+      case "<=>":
+        return "rightleftharpoons";
+
+      case "\u21CC":
+        return "rightleftharpoons";
+
+      case "<=>>":
+        return "rightequilibrium";
+
+      case "<<=>":
+        return "leftequilibrium";
+
+      default:
+        assertNever(a);
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  },
+  _getBond: function _getBond(a) {
+    switch (a) {
+      case "-":
+        return "{-}";
+
+      case "1":
+        return "{-}";
+
+      case "=":
+        return "{=}";
+
+      case "2":
+        return "{=}";
+
+      case "#":
+        return "{\\equiv}";
+
+      case "3":
+        return "{\\equiv}";
+
+      case "~":
+        return "{\\tripledash}";
+
+      case "~-":
+        return "{\\mathrlap{\\raisebox{-.1em}{$-$}}\\raisebox{.1em}{$\\tripledash$}}";
+
+      case "~=":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";
+
+      case "~--":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";
+
+      case "-~-":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$-$}}\\tripledash}";
+
+      case "...":
+        return "{{\\cdot}{\\cdot}{\\cdot}}";
+
+      case "....":
+        return "{{\\cdot}{\\cdot}{\\cdot}{\\cdot}}";
+
+      case "->":
+        return "{\\rightarrow}";
+
+      case "<-":
+        return "{\\leftarrow}";
+
+      case "<":
+        return "{<}";
+
+      case ">":
+        return "{>}";
+
+      default:
+        assertNever(a);
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  },
+  _getOperator: function _getOperator(a) {
+    switch (a) {
+      case "+":
+        return " {}+{} ";
+
+      case "-":
+        return " {}-{} ";
+
+      case "=":
+        return " {}={} ";
+
+      case "<":
+        return " {}<{} ";
+
+      case ">":
+        return " {}>{} ";
+
+      case "<<":
+        return " {}\\ll{} ";
+
+      case ">>":
+        return " {}\\gg{} ";
+
+      case "\\pm":
+        return " {}\\pm{} ";
+
+      case "\\approx":
+        return " {}\\approx{} ";
+
+      case "$\\approx$":
+        return " {}\\approx{} ";
+
+      case "v":
+        return " \\downarrow{} ";
+
+      case "(v)":
+        return " \\downarrow{} ";
+
+      case "^":
+        return " \\uparrow{} ";
+
+      case "(^)":
+        return " \\uparrow{} ";
+
+      default:
+        assertNever(a);
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  }
+}; //
+// Helpers for code anaylsis
+// Will show type error at calling position
+//
+
+/** @param {number} a */
+
+function assertNever(a) {}
+/** @param {string} a */
+
+
+function assertString(a) {}
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/docs/pt/katex/contrib/mhchem.min.js b/docs/pt/katex/contrib/mhchem.min.js
new file mode 100644
index 0000000..432a76a
--- /dev/null
+++ b/docs/pt/katex/contrib/mhchem.min.js
@@ -0,0 +1 @@
+!function(t,e){if("object"==typeof exports&&"object"==typeof module)module.exports=e(require("katex"));else if("function"==typeof define&&define.amd)define(["katex"],e);else{var n="object"==typeof exports?e(require("katex")):e(t.katex);for(var o in n)("object"==typeof exports?exports:t)[o]=n[o]}}("undefined"!=typeof self?self:this,function(t){return function(t){var e={};function n(o){if(e[o])return e[o].exports;var a=e[o]={i:o,l:!1,exports:{}};return t[o].call(a.exports,a,a.exports,n),a.l=!0,a.exports}return n.m=t,n.c=e,n.d=function(t,e,o){n.o(t,e)||Object.defineProperty(t,e,{enumerable:!0,get:o})},n.r=function(t){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(t,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(t,"__esModule",{value:!0})},n.t=function(t,e){if(1&e&&(t=n(t)),8&e)return t;if(4&e&&"object"==typeof t&&t&&t.__esModule)return t;var o=Object.create(null);if(n.r(o),Object.defineProperty(o,"default",{enumerable:!0,value:t}),2&e&&"string"!=typeof t)for(var a in t)n.d(o,a,function(e){return t[e]}.bind(null,a));return o},n.n=function(t){var e=t&&t.__esModule?function(){return t.default}:function(){return t};return n.d(e,"a",e),e},n.o=function(t,e){return Object.prototype.hasOwnProperty.call(t,e)},n.p="",n(n.s=1)}([function(e,n){e.exports=t},function(t,e,n){"use strict";n.r(e);var o=n(0),a=n.n(o);a.a.__defineMacro("\\ce",function(t){return r(t.consumeArgs(1)[0],"ce")}),a.a.__defineMacro("\\pu",function(t){return r(t.consumeArgs(1)[0],"pu")}),a.a.__defineMacro("\\tripledash","{\\vphantom{-}\\raisebox{2.56mu}{$\\mkern2mu\\tiny\\text{-}\\mkern1mu\\text{-}\\mkern1mu\\text{-}\\mkern2mu$}}");var r=function(t,e){for(var n="",o=t[t.length-1].loc.start,a=t.length-1;a>=0;a--)t[a].loc.start>o&&(n+=" ",o=t[a].loc.start),n+=t[a].text,o+=t[a].text.length;return c.go(i.go(n,e))},i={go:function(t,e){if(!t)return[];void 0===e&&(e="ce");var n,o="0",a={};a.parenthesisLevel=0,t=(t=(t=t.replace(/\n/g," ")).replace(/[\u2212\u2013\u2014\u2010]/g,"-")).replace(/[\u2026]/g,"...");for(var r=10,c=[];;){n!==t?(r=10,n=t):r--;var u=i.stateMachines[e],p=u.transitions[o]||u.transitions["*"];t:for(var s=0;s<p.length;s++){var _=i.patterns.match_(p[s].pattern,t);if(_){for(var d=p[s].task,m=0;m<d.action_.length;m++){var l;if(u.actions[d.action_[m].type_])l=u.actions[d.action_[m].type_](a,_.match_,d.action_[m].option);else{if(!i.actions[d.action_[m].type_])throw["MhchemBugA","mhchem bug A. 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numeral",p1:e||""}}}},"tex-math":{transitions:i.createTransitions({empty:{"*":{action_:"output"}},"\\ce{(...)}":{"*":{action_:["output","ce"]}},"{...}|\\,|\\x{}{}|\\x{}|\\x":{"*":{action_:"o="}},else:{"*":{action_:"o="}}}),actions:{output:function(t){if(t.o){var e={type_:"tex-math",p1:t.o};for(var n in t)delete t[n];return e}}}},"tex-math tight":{transitions:i.createTransitions({empty:{"*":{action_:"output"}},"\\ce{(...)}":{"*":{action_:["output","ce"]}},"{...}|\\,|\\x{}{}|\\x{}|\\x":{"*":{action_:"o="}},"-|+":{"*":{action_:"tight operator"}},else:{"*":{action_:"o="}}}),actions:{"tight operator":function(t,e){t.o=(t.o||"")+"{"+e+"}"},output:function(t){if(t.o){var e={type_:"tex-math",p1:t.o};for(var n in t)delete t[n];return e}}}},"9,9":{transitions:i.createTransitions({empty:{"*":{}},",":{"*":{action_:"comma"}},else:{"*":{action_:"copy"}}}),actions:{comma:function(){return{type_:"commaDecimal"}}}},pu:{transitions:i.createTransitions({empty:{"*":{action_:"output"}},space$:{"*":{action_:["output","space"]}},"{[(|)]}":{"0|a":{action_:"copy"}},"(-)(9)^(-9)":{0:{action_:"number^",nextState:"a"}},"(-)(9.,9)(e)(99)":{0:{action_:"enumber",nextState:"a"}},space:{"0|a":{}},"pm-operator":{"0|a":{action_:{type_:"operator",option:"\\pm"},nextState:"0"}},operator:{"0|a":{action_:"copy",nextState:"0"}},"//":{d:{action_:"o=",nextState:"/"}},"/":{d:{action_:"o=",nextState:"/"}},"{...}|else":{"0|d":{action_:"d=",nextState:"d"},a:{action_:["space","d="],nextState:"d"},"/|q":{action_:"q=",nextState:"q"}}}),actions:{enumber:function(t,e){var n=[];return"+-"===e[0]||"+/-"===e[0]?n.push("\\pm "):e[0]&&n.push(e[0]),e[1]&&(i.concatArray(n,i.go(e[1],"pu-9,9")),e[2]&&(e[2].match(/[,.]/)?i.concatArray(n,i.go(e[2],"pu-9,9")):n.push(e[2])),e[3]=e[4]||e[3],e[3]&&(e[3]=e[3].trim(),"e"===e[3]||"*"===e[3].substr(0,1)?n.push({type_:"cdot"}):n.push({type_:"times"}))),e[3]&&n.push("10^{"+e[5]+"}"),n},"number^":function(t,e){var n=[];return"+-"===e[0]||"+/-"===e[0]?n.push("\\pm "):e[0]&&n.push(e[0]),i.concatArray(n,i.go(e[1],"pu-9,9")),n.push("^{"+e[2]+"}"),n},operator:function(t,e,n){return{type_:"operator",kind_:n||e}},space:function(){return{type_:"pu-space-1"}},output:function(t){var e,n=i.patterns.match_("{(...)}",t.d||"");n&&""===n.remainder&&(t.d=n.match_);var o=i.patterns.match_("{(...)}",t.q||"");if(o&&""===o.remainder&&(t.q=o.match_),t.d&&(t.d=t.d.replace(/\u00B0C|\^oC|\^{o}C/g,"{}^{\\circ}C"),t.d=t.d.replace(/\u00B0F|\^oF|\^{o}F/g,"{}^{\\circ}F")),t.q){t.q=t.q.replace(/\u00B0C|\^oC|\^{o}C/g,"{}^{\\circ}C"),t.q=t.q.replace(/\u00B0F|\^oF|\^{o}F/g,"{}^{\\circ}F");var a={d:i.go(t.d,"pu"),q:i.go(t.q,"pu")};"//"===t.o?e={type_:"pu-frac",p1:a.d,p2:a.q}:(e=a.d,a.d.length>1||a.q.length>1?e.push({type_:" / "}):e.push({type_:"/"}),i.concatArray(e,a.q))}else e=i.go(t.d,"pu-2");for(var r in t)delete t[r];return e}}},"pu-2":{transitions:i.createTransitions({empty:{"*":{action_:"output"}},"*":{"*":{action_:["output","cdot"],nextState:"0"}},"\\x":{"*":{action_:"rm="}},space:{"*":{action_:["output","space"],nextState:"0"}},"^{(...)}|^(-1)":{1:{action_:"^(-1)"}},"-9.,9":{0:{action_:"rm=",nextState:"0"},1:{action_:"^(-1)",nextState:"0"}},"{...}|else":{"*":{action_:"rm=",nextState:"1"}}}),actions:{cdot:function(){return{type_:"tight cdot"}},"^(-1)":function(t,e){t.rm+="^{"+e+"}"},space:function(){return{type_:"pu-space-2"}},output:function(t){var e=[];if(t.rm){var n=i.patterns.match_("{(...)}",t.rm||"");e=n&&""===n.remainder?i.go(n.match_,"pu"):{type_:"rm",p1:t.rm}}for(var o in t)delete t[o];return e}}},"pu-9,9":{transitions:i.createTransitions({empty:{0:{action_:"output-0"},o:{action_:"output-o"}},",":{0:{action_:["output-0","comma"],nextState:"o"}},".":{0:{action_:["output-0","copy"],nextState:"o"}},else:{"*":{action_:"text="}}}),actions:{comma:function(){return{type_:"commaDecimal"}},"output-0":function(t){var e=[];if(t.text_=t.text_||"",t.text_.length>4){var n=t.text_.length%3;0===n&&(n=3);for(var o=t.text_.length-3;o>0;o-=3)e.push(t.text_.substr(o,3)),e.push({type_:"1000 separator"});e.push(t.text_.substr(0,n)),e.reverse()}else e.push(t.text_);for(var a in t)delete t[a];return e},"output-o":function(t){var e=[];if(t.text_=t.text_||"",t.text_.length>4){for(var n=t.text_.length-3,o=0;o<n;o+=3)e.push(t.text_.substr(o,3)),e.push({type_:"1000 separator"});e.push(t.text_.substr(o))}else e.push(t.text_);for(var a in t)delete t[a];return e}}}};var c={go:function(t,e){if(!t)return"";for(var n="",o=!1,a=0;a<t.length;a++){var r=t[a];"string"==typeof r?n+=r:(n+=c._go2(r),"1st-level escape"===r.type_&&(o=!0))}return e||o||!n||(n="{"+n+"}"),n},_goInner:function(t){return t?c.go(t,!0):t},_go2:function(t){var e;switch(t.type_){case"chemfive":e="";var n={a:c._goInner(t.a),b:c._goInner(t.b),p:c._goInner(t.p),o:c._goInner(t.o),q:c._goInner(t.q),d:c._goInner(t.d)};n.a&&(n.a.match(/^[+\-]/)&&(n.a="{"+n.a+"}"),e+=n.a+"\\,"),(n.b||n.p)&&(e+="{\\vphantom{X}}",e+="^{\\hphantom{"+(n.b||"")+"}}_{\\hphantom{"+(n.p||"")+"}}",e+="{\\vphantom{X}}",e+="^{\\smash[t]{\\vphantom{2}}\\mathllap{"+(n.b||"")+"}}",e+="_{\\vphantom{2}\\mathllap{\\smash[t]{"+(n.p||"")+"}}}"),n.o&&(n.o.match(/^[+\-]/)&&(n.o="{"+n.o+"}"),e+=n.o),"kv"===t.dType?((n.d||n.q)&&(e+="{\\vphantom{X}}"),n.d&&(e+="^{"+n.d+"}"),n.q&&(e+="_{\\smash[t]{"+n.q+"}}")):"oxidation"===t.dType?(n.d&&(e+="{\\vphantom{X}}",e+="^{"+n.d+"}"),n.q&&(e+="{\\vphantom{X}}",e+="_{\\smash[t]{"+n.q+"}}")):(n.q&&(e+="{\\vphantom{X}}",e+="_{\\smash[t]{"+n.q+"}}"),n.d&&(e+="{\\vphantom{X}}",e+="^{"+n.d+"}"));break;case"rm":e="\\mathrm{"+t.p1+"}";break;case"text":t.p1.match(/[\^_]/)?(t.p1=t.p1.replace(" ","~").replace("-","\\text{-}"),e="\\mathrm{"+t.p1+"}"):e="\\text{"+t.p1+"}";break;case"roman numeral":e="\\mathrm{"+t.p1+"}";break;case"state of aggregation":e="\\mskip2mu "+c._goInner(t.p1);break;case"state of aggregation subscript":e="\\mskip1mu "+c._goInner(t.p1);break;case"bond":if(!(e=c._getBond(t.kind_)))throw["MhchemErrorBond","mhchem Error. Unknown bond type ("+t.kind_+")"];break;case"frac":var o="\\frac{"+t.p1+"}{"+t.p2+"}";e="\\mathchoice{\\textstyle"+o+"}{"+o+"}{"+o+"}{"+o+"}";break;case"pu-frac":var a="\\frac{"+c._goInner(t.p1)+"}{"+c._goInner(t.p2)+"}";e="\\mathchoice{\\textstyle"+a+"}{"+a+"}{"+a+"}{"+a+"}";break;case"tex-math":e=t.p1+" ";break;case"frac-ce":e="\\frac{"+c._goInner(t.p1)+"}{"+c._goInner(t.p2)+"}";break;case"overset":e="\\overset{"+c._goInner(t.p1)+"}{"+c._goInner(t.p2)+"}";break;case"underset":e="\\underset{"+c._goInner(t.p1)+"}{"+c._goInner(t.p2)+"}";break;case"underbrace":e="\\underbrace{"+c._goInner(t.p1)+"}_{"+c._goInner(t.p2)+"}";break;case"color":e="{\\color{"+t.color1+"}{"+c._goInner(t.color2)+"}}";break;case"color0":e="\\color{"+t.color+"}";break;case"arrow":var r={rd:c._goInner(t.rd),rq:c._goInner(t.rq)},i="\\x"+c._getArrow(t.r);r.rq&&(i+="[{"+r.rq+"}]"),e=i+=r.rd?"{"+r.rd+"}":"{}";break;case"operator":e=c._getOperator(t.kind_);break;case"1st-level escape":e=t.p1+" ";break;case"space":e=" ";break;case"entitySkip":case"pu-space-1":e="~";break;case"pu-space-2":e="\\mkern3mu ";break;case"1000 separator":e="\\mkern2mu ";break;case"commaDecimal":e="{,}";break;case"comma enumeration L":e="{"+t.p1+"}\\mkern6mu ";break;case"comma enumeration M":e="{"+t.p1+"}\\mkern3mu ";break;case"comma enumeration S":e="{"+t.p1+"}\\mkern1mu ";break;case"hyphen":e="\\text{-}";break;case"addition compound":e="\\,{\\cdot}\\,";break;case"electron dot":e="\\mkern1mu \\bullet\\mkern1mu ";break;case"KV x":e="{\\times}";break;case"prime":e="\\prime ";break;case"cdot":e="\\cdot ";break;case"tight cdot":e="\\mkern1mu{\\cdot}\\mkern1mu ";break;case"times":e="\\times ";break;case"circa":e="{\\sim}";break;case"^":e="uparrow";break;case"v":e="downarrow";break;case"ellipsis":e="\\ldots ";break;case"/":e="/";break;case" / ":e="\\,/\\,";break;default:throw["MhchemBugT","mhchem bug T. Please report."]}return e},_getArrow:function(t){switch(t){case"->":case"\u2192":case"\u27f6":return"rightarrow";case"<-":return"leftarrow";case"<->":return"leftrightarrow";case"<--\x3e":return"rightleftarrows";case"<=>":case"\u21cc":return"rightleftharpoons";case"<=>>":return"rightequilibrium";case"<<=>":return"leftequilibrium";default:throw["MhchemBugT","mhchem bug T. Please report."]}},_getBond:function(t){switch(t){case"-":case"1":return"{-}";case"=":case"2":return"{=}";case"#":case"3":return"{\\equiv}";case"~":return"{\\tripledash}";case"~-":return"{\\mathrlap{\\raisebox{-.1em}{$-$}}\\raisebox{.1em}{$\\tripledash$}}";case"~=":case"~--":return"{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";case"-~-":return"{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$-$}}\\tripledash}";case"...":return"{{\\cdot}{\\cdot}{\\cdot}}";case"....":return"{{\\cdot}{\\cdot}{\\cdot}{\\cdot}}";case"->":return"{\\rightarrow}";case"<-":return"{\\leftarrow}";case"<":return"{<}";case">":return"{>}";default:throw["MhchemBugT","mhchem bug T. Please report."]}},_getOperator:function(t){switch(t){case"+":return" {}+{} ";case"-":return" {}-{} ";case"=":return" {}={} ";case"<":return" {}<{} ";case">":return" {}>{} ";case"<<":return" {}\\ll{} ";case">>":return" {}\\gg{} ";case"\\pm":return" {}\\pm{} ";case"\\approx":case"$\\approx$":return" {}\\approx{} ";case"v":case"(v)":return" \\downarrow{} ";case"^":case"(^)":return" \\uparrow{} ";default:throw["MhchemBugT","mhchem bug T. Please report."]}}}}]).default});
\ No newline at end of file
diff --git a/docs/pt/katex/contrib/mhchem.mjs b/docs/pt/katex/contrib/mhchem.mjs
new file mode 100644
index 0000000..46241b5
--- /dev/null
+++ b/docs/pt/katex/contrib/mhchem.mjs
@@ -0,0 +1,3109 @@
+import katex from '../katex.mjs';
+
+/* eslint-disable */
+
+/* -*- Mode: Javascript; indent-tabs-mode:nil; js-indent-level: 2 -*- */
+
+/* vim: set ts=2 et sw=2 tw=80: */
+
+/*************************************************************
+ *
+ *  KaTeX mhchem.js
+ *
+ *  This file implements a KaTeX version of mhchem version 3.3.0.
+ *  It is adapted from MathJax/extensions/TeX/mhchem.js
+ *  It differs from the MathJax version as follows:
+ *    1. The interface is changed so that it can be called from KaTeX, not MathJax.
+ *    2. \rlap and \llap are replaced with \mathrlap and \mathllap.
+ *    3. Four lines of code are edited in order to use \raisebox instead of \raise.
+ *    4. The reaction arrow code is simplified. All reaction arrows are rendered
+ *       using KaTeX extensible arrows instead of building non-extensible arrows.
+ *    5. \tripledash vertical alignment is slightly adjusted.
+ *
+ *    This code, as other KaTeX code, is released under the MIT license.
+ * 
+ * /*************************************************************
+ *
+ *  MathJax/extensions/TeX/mhchem.js
+ *
+ *  Implements the \ce command for handling chemical formulas
+ *  from the mhchem LaTeX package.
+ *
+ *  ---------------------------------------------------------------------
+ *
+ *  Copyright (c) 2011-2015 The MathJax Consortium
+ *  Copyright (c) 2015-2018 Martin Hensel
+ *
+ *  Licensed under the Apache License, Version 2.0 (the "License");
+ *  you may not use this file except in compliance with the License.
+ *  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ *  Unless required by applicable law or agreed to in writing, software
+ *  distributed under the License is distributed on an "AS IS" BASIS,
+ *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ *  See the License for the specific language governing permissions and
+ *  limitations under the License.
+ */
+//
+// Coding Style
+//   - use '' for identifiers that can by minified/uglified
+//   - use "" for strings that need to stay untouched
+// version: "3.3.0" for MathJax and KaTeX
+// Add \ce, \pu, and \tripledash to the KaTeX macros.
+katex.__defineMacro("\\ce", function (context) {
+  return chemParse(context.consumeArgs(1)[0], "ce");
+});
+
+katex.__defineMacro("\\pu", function (context) {
+  return chemParse(context.consumeArgs(1)[0], "pu");
+}); //  Needed for \bond for the ~ forms
+//  Raise by 2.56mu, not 2mu. We're raising a hyphen-minus, U+002D, not 
+//  a mathematical minus, U+2212. So we need that extra 0.56.
+
+
+katex.__defineMacro("\\tripledash", "{\\vphantom{-}\\raisebox{2.56mu}{$\\mkern2mu" + "\\tiny\\text{-}\\mkern1mu\\text{-}\\mkern1mu\\text{-}\\mkern2mu$}}");
+//  This is the main function for handing the \ce and \pu commands.
+//  It takes the argument to \ce or \pu and returns the corresponding TeX string.
+//
+
+var chemParse = function chemParse(tokens, stateMachine) {
+  // Recreate the argument string from KaTeX's array of tokens.
+  var str = "";
+  var expectedLoc = tokens[tokens.length - 1].loc.start;
+
+  for (var i = tokens.length - 1; i >= 0; i--) {
+    if (tokens[i].loc.start > expectedLoc) {
+      // context.consumeArgs has eaten a space.
+      str += " ";
+      expectedLoc = tokens[i].loc.start;
+    }
+
+    str += tokens[i].text;
+    expectedLoc += tokens[i].text.length;
+  }
+
+  var tex = texify.go(mhchemParser.go(str, stateMachine));
+  return tex;
+}; //
+// Core parser for mhchem syntax  (recursive)
+//
+
+/** @type {MhchemParser} */
+
+
+var mhchemParser = {
+  //
+  // Parses mchem \ce syntax
+  //
+  // Call like
+  //   go("H2O");
+  //
+  go: function go(input, stateMachine) {
+    if (!input) {
+      return [];
+    }
+
+    if (stateMachine === undefined) {
+      stateMachine = 'ce';
+    }
+
+    var state = '0'; //
+    // String buffers for parsing:
+    //
+    // buffer.a == amount
+    // buffer.o == element
+    // buffer.b == left-side superscript
+    // buffer.p == left-side subscript
+    // buffer.q == right-side subscript
+    // buffer.d == right-side superscript
+    //
+    // buffer.r == arrow
+    // buffer.rdt == arrow, script above, type
+    // buffer.rd == arrow, script above, content
+    // buffer.rqt == arrow, script below, type
+    // buffer.rq == arrow, script below, content
+    //
+    // buffer.text_
+    // buffer.rm
+    // etc.
+    //
+    // buffer.parenthesisLevel == int, starting at 0
+    // buffer.sb == bool, space before
+    // buffer.beginsWithBond == bool
+    //
+    // These letters are also used as state names.
+    //
+    // Other states:
+    // 0 == begin of main part (arrow/operator unlikely)
+    // 1 == next entity
+    // 2 == next entity (arrow/operator unlikely)
+    // 3 == next atom
+    // c == macro
+    //
+
+    /** @type {Buffer} */
+
+    var buffer = {};
+    buffer['parenthesisLevel'] = 0;
+    input = input.replace(/\n/g, " ");
+    input = input.replace(/[\u2212\u2013\u2014\u2010]/g, "-");
+    input = input.replace(/[\u2026]/g, "..."); //
+    // Looks through mhchemParser.transitions, to execute a matching action
+    // (recursive)
+    //
+
+    var lastInput;
+    var watchdog = 10;
+    /** @type {ParserOutput[]} */
+
+    var output = [];
+
+    while (true) {
+      if (lastInput !== input) {
+        watchdog = 10;
+        lastInput = input;
+      } else {
+        watchdog--;
+      } //
+      // Find actions in transition table
+      //
+
+
+      var machine = mhchemParser.stateMachines[stateMachine];
+      var t = machine.transitions[state] || machine.transitions['*'];
+
+      iterateTransitions: for (var i = 0; i < t.length; i++) {
+        var matches = mhchemParser.patterns.match_(t[i].pattern, input);
+
+        if (matches) {
+          //
+          // Execute actions
+          //
+          var task = t[i].task;
+
+          for (var iA = 0; iA < task.action_.length; iA++) {
+            var o; //
+            // Find and execute action
+            //
+
+            if (machine.actions[task.action_[iA].type_]) {
+              o = machine.actions[task.action_[iA].type_](buffer, matches.match_, task.action_[iA].option);
+            } else if (mhchemParser.actions[task.action_[iA].type_]) {
+              o = mhchemParser.actions[task.action_[iA].type_](buffer, matches.match_, task.action_[iA].option);
+            } else {
+              throw ["MhchemBugA", "mhchem bug A. Please report. (" + task.action_[iA].type_ + ")"]; // Trying to use non-existing action
+            } //
+            // Add output
+            //
+
+
+            mhchemParser.concatArray(output, o);
+          } //
+          // Set next state,
+          // Shorten input,
+          // Continue with next character
+          //   (= apply only one transition per position)
+          //
+
+
+          state = task.nextState || state;
+
+          if (input.length > 0) {
+            if (!task.revisit) {
+              input = matches.remainder;
+            }
+
+            if (!task.toContinue) {
+              break iterateTransitions;
+            }
+          } else {
+            return output;
+          }
+        }
+      } //
+      // Prevent infinite loop
+      //
+
+
+      if (watchdog <= 0) {
+        throw ["MhchemBugU", "mhchem bug U. Please report."]; // Unexpected character
+      }
+    }
+  },
+  concatArray: function concatArray(a, b) {
+    if (b) {
+      if (Array.isArray(b)) {
+        for (var iB = 0; iB < b.length; iB++) {
+          a.push(b[iB]);
+        }
+      } else {
+        a.push(b);
+      }
+    }
+  },
+  patterns: {
+    //
+    // Matching patterns
+    // either regexps or function that return null or {match_:"a", remainder:"bc"}
+    //
+    patterns: {
+      // property names must not look like integers ("2") for correct property traversal order, later on
+      'empty': /^$/,
+      'else': /^./,
+      'else2': /^./,
+      'space': /^\s/,
+      'space A': /^\s(?=[A-Z\\$])/,
+      'space$': /^\s$/,
+      'a-z': /^[a-z]/,
+      'x': /^x/,
+      'x$': /^x$/,
+      'i$': /^i$/,
+      'letters': /^(?:[a-zA-Z\u03B1-\u03C9\u0391-\u03A9?@]|(?:\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega|Gamma|Delta|Theta|Lambda|Xi|Pi|Sigma|Upsilon|Phi|Psi|Omega)(?:\s+|\{\}|(?![a-zA-Z]))))+/,
+      '\\greek': /^\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega|Gamma|Delta|Theta|Lambda|Xi|Pi|Sigma|Upsilon|Phi|Psi|Omega)(?:\s+|\{\}|(?![a-zA-Z]))/,
+      'one lowercase latin letter $': /^(?:([a-z])(?:$|[^a-zA-Z]))$/,
+      '$one lowercase latin letter$ $': /^\$(?:([a-z])(?:$|[^a-zA-Z]))\$$/,
+      'one lowercase greek letter $': /^(?:\$?[\u03B1-\u03C9]\$?|\$?\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega)\s*\$?)(?:\s+|\{\}|(?![a-zA-Z]))$/,
+      'digits': /^[0-9]+/,
+      '-9.,9': /^[+\-]?(?:[0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))/,
+      '-9.,9 no missing 0': /^[+\-]?[0-9]+(?:[.,][0-9]+)?/,
+      '(-)(9.,9)(e)(99)': function e99(input) {
+        var m = input.match(/^(\+\-|\+\/\-|\+|\-|\\pm\s?)?([0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))?(\((?:[0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))\))?(?:([eE]|\s*(\*|x|\\times|\u00D7)\s*10\^)([+\-]?[0-9]+|\{[+\-]?[0-9]+\}))?/);
+
+        if (m && m[0]) {
+          return {
+            match_: m.splice(1),
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      '(-)(9)^(-9)': function _(input) {
+        var m = input.match(/^(\+\-|\+\/\-|\+|\-|\\pm\s?)?([0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+)?)\^([+\-]?[0-9]+|\{[+\-]?[0-9]+\})/);
+
+        if (m && m[0]) {
+          return {
+            match_: m.splice(1),
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      'state of aggregation $': function stateOfAggregation$(input) {
+        // ... or crystal system
+        var a = mhchemParser.patterns.findObserveGroups(input, "", /^\([a-z]{1,3}(?=[\),])/, ")", ""); // (aq), (aq,$\infty$), (aq, sat)
+
+        if (a && a.remainder.match(/^($|[\s,;\)\]\}])/)) {
+          return a;
+        } //  AND end of 'phrase'
+
+
+        var m = input.match(/^(?:\((?:\\ca\s?)?\$[amothc]\$\))/); // OR crystal system ($o$) (\ca$c$)
+
+        if (m) {
+          return {
+            match_: m[0],
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      '_{(state of aggregation)}$': /^_\{(\([a-z]{1,3}\))\}/,
+      '{[(': /^(?:\\\{|\[|\()/,
+      ')]}': /^(?:\)|\]|\\\})/,
+      ', ': /^[,;]\s*/,
+      ',': /^[,;]/,
+      '.': /^[.]/,
+      '. ': /^([.\u22C5\u00B7\u2022])\s*/,
+      '...': /^\.\.\.(?=$|[^.])/,
+      '* ': /^([*])\s*/,
+      '^{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^{", "", "", "}");
+      },
+      '^($...$)': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", "$", "$", "");
+      },
+      '^a': /^\^([0-9]+|[^\\_])/,
+      '^\\x{}{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '^\\x{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '^\\x': /^\^(\\[a-zA-Z]+)\s*/,
+      '^(-1)': /^\^(-?\d+)/,
+      '\'': /^'/,
+      '_{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_{", "", "", "}");
+      },
+      '_($...$)': function _$$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", "$", "$", "");
+      },
+      '_9': /^_([+\-]?[0-9]+|[^\\])/,
+      '_\\x{}{}': function _X(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '_\\x{}': function _X(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '_\\x': /^_(\\[a-zA-Z]+)\s*/,
+      '^_': /^(?:\^(?=_)|\_(?=\^)|[\^_]$)/,
+      '{}': /^\{\}/,
+      '{...}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", "{", "}", "");
+      },
+      '{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "{", "", "", "}");
+      },
+      '$...$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", "$", "$", "");
+      },
+      '${(...)}$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "${", "", "", "}$");
+      },
+      '$(...)$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "$", "", "", "$");
+      },
+      '=<>': /^[=<>]/,
+      '#': /^[#\u2261]/,
+      '+': /^\+/,
+      '-$': /^-(?=[\s_},;\]/]|$|\([a-z]+\))/,
+      // -space -, -; -] -/ -$ -state-of-aggregation
+      '-9': /^-(?=[0-9])/,
+      '- orbital overlap': /^-(?=(?:[spd]|sp)(?:$|[\s,;\)\]\}]))/,
+      '-': /^-/,
+      'pm-operator': /^(?:\\pm|\$\\pm\$|\+-|\+\/-)/,
+      'operator': /^(?:\+|(?:[\-=<>]|<<|>>|\\approx|\$\\approx\$)(?=\s|$|-?[0-9]))/,
+      'arrowUpDown': /^(?:v|\(v\)|\^|\(\^\))(?=$|[\s,;\)\]\}])/,
+      '\\bond{(...)}': function bond(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\bond{", "", "", "}");
+      },
+      '->': /^(?:<->|<-->|->|<-|<=>>|<<=>|<=>|[\u2192\u27F6\u21CC])/,
+      'CMT': /^[CMT](?=\[)/,
+      '[(...)]': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "[", "", "", "]");
+      },
+      '1st-level escape': /^(&|\\\\|\\hline)\s*/,
+      '\\,': /^(?:\\[,\ ;:])/,
+      // \\x - but output no space before
+      '\\x{}{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '\\x{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '\\ca': /^\\ca(?:\s+|(?![a-zA-Z]))/,
+      '\\x': /^(?:\\[a-zA-Z]+\s*|\\[_&{}%])/,
+      'orbital': /^(?:[0-9]{1,2}[spdfgh]|[0-9]{0,2}sp)(?=$|[^a-zA-Z])/,
+      // only those with numbers in front, because the others will be formatted correctly anyway
+      'others': /^[\/~|]/,
+      '\\frac{(...)}': function frac(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\frac{", "", "", "}", "{", "", "", "}");
+      },
+      '\\overset{(...)}': function overset(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\overset{", "", "", "}", "{", "", "", "}");
+      },
+      '\\underset{(...)}': function underset(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\underset{", "", "", "}", "{", "", "", "}");
+      },
+      '\\underbrace{(...)}': function underbrace(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\underbrace{", "", "", "}_", "{", "", "", "}");
+      },
+      '\\color{(...)}0': function color0(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color{", "", "", "}");
+      },
+      '\\color{(...)}{(...)}1': function color1(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color{", "", "", "}", "{", "", "", "}");
+      },
+      '\\color(...){(...)}2': function color2(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color", "\\", "", /^(?=\{)/, "{", "", "", "}");
+      },
+      '\\ce{(...)}': function ce(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\ce{", "", "", "}");
+      },
+      'oxidation$': /^(?:[+-][IVX]+|\\pm\s*0|\$\\pm\$\s*0)$/,
+      'd-oxidation$': /^(?:[+-]?\s?[IVX]+|\\pm\s*0|\$\\pm\$\s*0)$/,
+      // 0 could be oxidation or charge
+      'roman numeral': /^[IVX]+/,
+      '1/2$': /^[+\-]?(?:[0-9]+|\$[a-z]\$|[a-z])\/[0-9]+(?:\$[a-z]\$|[a-z])?$/,
+      'amount': function amount(input) {
+        var match; // e.g. 2, 0.5, 1/2, -2, n/2, +;  $a$ could be added later in parsing
+
+        match = input.match(/^(?:(?:(?:\([+\-]?[0-9]+\/[0-9]+\)|[+\-]?(?:[0-9]+|\$[a-z]\$|[a-z])\/[0-9]+|[+\-]?[0-9]+[.,][0-9]+|[+\-]?\.[0-9]+|[+\-]?[0-9]+)(?:[a-z](?=\s*[A-Z]))?)|[+\-]?[a-z](?=\s*[A-Z])|\+(?!\s))/);
+
+        if (match) {
+          return {
+            match_: match[0],
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        var a = mhchemParser.patterns.findObserveGroups(input, "", "$", "$", "");
+
+        if (a) {
+          // e.g. $2n-1$, $-$
+          match = a.match_.match(/^\$(?:\(?[+\-]?(?:[0-9]*[a-z]?[+\-])?[0-9]*[a-z](?:[+\-][0-9]*[a-z]?)?\)?|\+|-)\$$/);
+
+          if (match) {
+            return {
+              match_: match[0],
+              remainder: input.substr(match[0].length)
+            };
+          }
+        }
+
+        return null;
+      },
+      'amount2': function amount2(input) {
+        return this['amount'](input);
+      },
+      '(KV letters),': /^(?:[A-Z][a-z]{0,2}|i)(?=,)/,
+      'formula$': function formula$(input) {
+        if (input.match(/^\([a-z]+\)$/)) {
+          return null;
+        } // state of aggregation = no formula
+
+
+        var match = input.match(/^(?:[a-z]|(?:[0-9\ \+\-\,\.\(\)]+[a-z])+[0-9\ \+\-\,\.\(\)]*|(?:[a-z][0-9\ \+\-\,\.\(\)]+)+[a-z]?)$/);
+
+        if (match) {
+          return {
+            match_: match[0],
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        return null;
+      },
+      'uprightEntities': /^(?:pH|pOH|pC|pK|iPr|iBu)(?=$|[^a-zA-Z])/,
+      '/': /^\s*(\/)\s*/,
+      '//': /^\s*(\/\/)\s*/,
+      '*': /^\s*[*.]\s*/
+    },
+    findObserveGroups: function findObserveGroups(input, begExcl, begIncl, endIncl, endExcl, beg2Excl, beg2Incl, end2Incl, end2Excl, combine) {
+      /** @type {{(input: string, pattern: string | RegExp): string | string[] | null;}} */
+      var _match = function _match(input, pattern) {
+        if (typeof pattern === "string") {
+          if (input.indexOf(pattern) !== 0) {
+            return null;
+          }
+
+          return pattern;
+        } else {
+          var match = input.match(pattern);
+
+          if (!match) {
+            return null;
+          }
+
+          return match[0];
+        }
+      };
+      /** @type {{(input: string, i: number, endChars: string | RegExp): {endMatchBegin: number, endMatchEnd: number} | null;}} */
+
+
+      var _findObserveGroups = function _findObserveGroups(input, i, endChars) {
+        var braces = 0;
+
+        while (i < input.length) {
+          var a = input.charAt(i);
+
+          var match = _match(input.substr(i), endChars);
+
+          if (match !== null && braces === 0) {
+            return {
+              endMatchBegin: i,
+              endMatchEnd: i + match.length
+            };
+          } else if (a === "{") {
+            braces++;
+          } else if (a === "}") {
+            if (braces === 0) {
+              throw ["ExtraCloseMissingOpen", "Extra close brace or missing open brace"];
+            } else {
+              braces--;
+            }
+          }
+
+          i++;
+        }
+
+        if (braces > 0) {
+          return null;
+        }
+
+        return null;
+      };
+
+      var match = _match(input, begExcl);
+
+      if (match === null) {
+        return null;
+      }
+
+      input = input.substr(match.length);
+      match = _match(input, begIncl);
+
+      if (match === null) {
+        return null;
+      }
+
+      var e = _findObserveGroups(input, match.length, endIncl || endExcl);
+
+      if (e === null) {
+        return null;
+      }
+
+      var match1 = input.substring(0, endIncl ? e.endMatchEnd : e.endMatchBegin);
+
+      if (!(beg2Excl || beg2Incl)) {
+        return {
+          match_: match1,
+          remainder: input.substr(e.endMatchEnd)
+        };
+      } else {
+        var group2 = this.findObserveGroups(input.substr(e.endMatchEnd), beg2Excl, beg2Incl, end2Incl, end2Excl);
+
+        if (group2 === null) {
+          return null;
+        }
+        /** @type {string[]} */
+
+
+        var matchRet = [match1, group2.match_];
+        return {
+          match_: combine ? matchRet.join("") : matchRet,
+          remainder: group2.remainder
+        };
+      }
+    },
+    //
+    // Matching function
+    // e.g. match("a", input) will look for the regexp called "a" and see if it matches
+    // returns null or {match_:"a", remainder:"bc"}
+    //
+    match_: function match_(m, input) {
+      var pattern = mhchemParser.patterns.patterns[m];
+
+      if (pattern === undefined) {
+        throw ["MhchemBugP", "mhchem bug P. Please report. (" + m + ")"]; // Trying to use non-existing pattern
+      } else if (typeof pattern === "function") {
+        return mhchemParser.patterns.patterns[m](input); // cannot use cached var pattern here, because some pattern functions need this===mhchemParser
+      } else {
+        // RegExp
+        var match = input.match(pattern);
+
+        if (match) {
+          var mm;
+
+          if (match[2]) {
+            mm = [match[1], match[2]];
+          } else if (match[1]) {
+            mm = match[1];
+          } else {
+            mm = match[0];
+          }
+
+          return {
+            match_: mm,
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        return null;
+      }
+    }
+  },
+  //
+  // Generic state machine actions
+  //
+  actions: {
+    'a=': function a(buffer, m) {
+      buffer.a = (buffer.a || "") + m;
+    },
+    'b=': function b(buffer, m) {
+      buffer.b = (buffer.b || "") + m;
+    },
+    'p=': function p(buffer, m) {
+      buffer.p = (buffer.p || "") + m;
+    },
+    'o=': function o(buffer, m) {
+      buffer.o = (buffer.o || "") + m;
+    },
+    'q=': function q(buffer, m) {
+      buffer.q = (buffer.q || "") + m;
+    },
+    'd=': function d(buffer, m) {
+      buffer.d = (buffer.d || "") + m;
+    },
+    'rm=': function rm(buffer, m) {
+      buffer.rm = (buffer.rm || "") + m;
+    },
+    'text=': function text(buffer, m) {
+      buffer.text_ = (buffer.text_ || "") + m;
+    },
+    'insert': function insert(buffer, m, a) {
+      return {
+        type_: a
+      };
+    },
+    'insert+p1': function insertP1(buffer, m, a) {
+      return {
+        type_: a,
+        p1: m
+      };
+    },
+    'insert+p1+p2': function insertP1P2(buffer, m, a) {
+      return {
+        type_: a,
+        p1: m[0],
+        p2: m[1]
+      };
+    },
+    'copy': function copy(buffer, m) {
+      return m;
+    },
+    'rm': function rm(buffer, m) {
+      return {
+        type_: 'rm',
+        p1: m || ""
+      };
+    },
+    'text': function text(buffer, m) {
+      return mhchemParser.go(m, 'text');
+    },
+    '{text}': function text(buffer, m) {
+      var ret = ["{"];
+      mhchemParser.concatArray(ret, mhchemParser.go(m, 'text'));
+      ret.push("}");
+      return ret;
+    },
+    'tex-math': function texMath(buffer, m) {
+      return mhchemParser.go(m, 'tex-math');
+    },
+    'tex-math tight': function texMathTight(buffer, m) {
+      return mhchemParser.go(m, 'tex-math tight');
+    },
+    'bond': function bond(buffer, m, k) {
+      return {
+        type_: 'bond',
+        kind_: k || m
+      };
+    },
+    'color0-output': function color0Output(buffer, m) {
+      return {
+        type_: 'color0',
+        color: m[0]
+      };
+    },
+    'ce': function ce(buffer, m) {
+      return mhchemParser.go(m);
+    },
+    '1/2': function _(buffer, m) {
+      /** @type {ParserOutput[]} */
+      var ret = [];
+
+      if (m.match(/^[+\-]/)) {
+        ret.push(m.substr(0, 1));
+        m = m.substr(1);
+      }
+
+      var n = m.match(/^([0-9]+|\$[a-z]\$|[a-z])\/([0-9]+)(\$[a-z]\$|[a-z])?$/);
+      n[1] = n[1].replace(/\$/g, "");
+      ret.push({
+        type_: 'frac',
+        p1: n[1],
+        p2: n[2]
+      });
+
+      if (n[3]) {
+        n[3] = n[3].replace(/\$/g, "");
+        ret.push({
+          type_: 'tex-math',
+          p1: n[3]
+        });
+      }
+
+      return ret;
+    },
+    '9,9': function _(buffer, m) {
+      return mhchemParser.go(m, '9,9');
+    }
+  },
+  //
+  // createTransitions
+  // convert  { 'letter': { 'state': { action_: 'output' } } }  to  { 'state' => [ { pattern: 'letter', task: { action_: [{type_: 'output'}] } } ] }
+  // with expansion of 'a|b' to 'a' and 'b' (at 2 places)
+  //
+  createTransitions: function createTransitions(o) {
+    var pattern, state;
+    /** @type {string[]} */
+
+    var stateArray;
+    var i; //
+    // 1. Collect all states
+    //
+
+    /** @type {Transitions} */
+
+    var transitions = {};
+
+    for (pattern in o) {
+      for (state in o[pattern]) {
+        stateArray = state.split("|");
+        o[pattern][state].stateArray = stateArray;
+
+        for (i = 0; i < stateArray.length; i++) {
+          transitions[stateArray[i]] = [];
+        }
+      }
+    } //
+    // 2. Fill states
+    //
+
+
+    for (pattern in o) {
+      for (state in o[pattern]) {
+        stateArray = o[pattern][state].stateArray || [];
+
+        for (i = 0; i < stateArray.length; i++) {
+          //
+          // 2a. Normalize actions into array:  'text=' ==> [{type_:'text='}]
+          // (Note to myself: Resolving the function here would be problematic. It would need .bind (for *this*) and currying (for *option*).)
+          //
+
+          /** @type {any} */
+          var p = o[pattern][state];
+
+          if (p.action_) {
+            p.action_ = [].concat(p.action_);
+
+            for (var k = 0; k < p.action_.length; k++) {
+              if (typeof p.action_[k] === "string") {
+                p.action_[k] = {
+                  type_: p.action_[k]
+                };
+              }
+            }
+          } else {
+            p.action_ = [];
+          } //
+          // 2.b Multi-insert
+          //
+
+
+          var patternArray = pattern.split("|");
+
+          for (var j = 0; j < patternArray.length; j++) {
+            if (stateArray[i] === '*') {
+              // insert into all
+              for (var t in transitions) {
+                transitions[t].push({
+                  pattern: patternArray[j],
+                  task: p
+                });
+              }
+            } else {
+              transitions[stateArray[i]].push({
+                pattern: patternArray[j],
+                task: p
+              });
+            }
+          }
+        }
+      }
+    }
+
+    return transitions;
+  },
+  stateMachines: {}
+}; //
+// Definition of state machines
+//
+
+mhchemParser.stateMachines = {
+  //
+  // \ce state machines
+  //
+  //#region ce
+  'ce': {
+    // main parser
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      'else': {
+        '0|1|2': {
+          action_: 'beginsWithBond=false',
+          revisit: true,
+          toContinue: true
+        }
+      },
+      'oxidation$': {
+        '0': {
+          action_: 'oxidation-output'
+        }
+      },
+      'CMT': {
+        'r': {
+          action_: 'rdt=',
+          nextState: 'rt'
+        },
+        'rd': {
+          action_: 'rqt=',
+          nextState: 'rdt'
+        }
+      },
+      'arrowUpDown': {
+        '0|1|2|as': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '1'
+        }
+      },
+      'uprightEntities': {
+        '0|1|2': {
+          action_: ['o=', 'output'],
+          nextState: '1'
+        }
+      },
+      'orbital': {
+        '0|1|2|3': {
+          action_: 'o=',
+          nextState: 'o'
+        }
+      },
+      '->': {
+        '0|1|2|3': {
+          action_: 'r=',
+          nextState: 'r'
+        },
+        'a|as': {
+          action_: ['output', 'r='],
+          nextState: 'r'
+        },
+        '*': {
+          action_: ['output', 'r='],
+          nextState: 'r'
+        }
+      },
+      '+': {
+        'o': {
+          action_: 'd= kv',
+          nextState: 'd'
+        },
+        'd|D': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qd|qD': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'dq': {
+          action_: ['output', 'd='],
+          nextState: 'd'
+        },
+        '3': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '0'
+        }
+      },
+      'amount': {
+        '0|2': {
+          action_: 'a=',
+          nextState: 'a'
+        }
+      },
+      'pm-operator': {
+        '0|1|2|a|as': {
+          action_: ['sb=false', 'output', {
+            type_: 'operator',
+            option: '\\pm'
+          }],
+          nextState: '0'
+        }
+      },
+      'operator': {
+        '0|1|2|a|as': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '0'
+        }
+      },
+      '-$': {
+        'o|q': {
+          action_: ['charge or bond', 'output'],
+          nextState: 'qd'
+        },
+        'd': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'D': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qd': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qD|dq': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        }
+      },
+      '-9': {
+        '3|o': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '3'
+        }
+      },
+      '- orbital overlap': {
+        'o': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        },
+        'd': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        }
+      },
+      '-': {
+        '0|1|2': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'beginsWithBond=true', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        '3': {
+          action_: {
+            type_: 'bond',
+            option: "-"
+          }
+        },
+        'a': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        },
+        'as': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        'b': {
+          action_: 'b='
+        },
+        'o': {
+          action_: {
+            type_: '- after o/d',
+            option: false
+          },
+          nextState: '2'
+        },
+        'q': {
+          action_: {
+            type_: '- after o/d',
+            option: false
+          },
+          nextState: '2'
+        },
+        'd|qd|dq': {
+          action_: {
+            type_: '- after o/d',
+            option: true
+          },
+          nextState: '2'
+        },
+        'D|qD|p': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        }
+      },
+      'amount2': {
+        '1|3': {
+          action_: 'a=',
+          nextState: 'a'
+        }
+      },
+      'letters': {
+        '0|1|2|3|a|as|b|p|bp|o': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        'q|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        },
+        'd|D|qd|qD': {
+          action_: 'o after d',
+          nextState: 'o'
+        }
+      },
+      'digits': {
+        'o': {
+          action_: 'q=',
+          nextState: 'q'
+        },
+        'd|D': {
+          action_: 'q=',
+          nextState: 'dq'
+        },
+        'q': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        },
+        'a': {
+          action_: 'o=',
+          nextState: 'o'
+        }
+      },
+      'space A': {
+        'b|p|bp': {}
+      },
+      'space': {
+        'a': {
+          nextState: 'as'
+        },
+        '0': {
+          action_: 'sb=false'
+        },
+        '1|2': {
+          action_: 'sb=true'
+        },
+        'r|rt|rd|rdt|rdq': {
+          action_: 'output',
+          nextState: '0'
+        },
+        '*': {
+          action_: ['output', 'sb=true'],
+          nextState: '1'
+        }
+      },
+      '1st-level escape': {
+        '1|2': {
+          action_: ['output', {
+            type_: 'insert+p1',
+            option: '1st-level escape'
+          }]
+        },
+        '*': {
+          action_: ['output', {
+            type_: 'insert+p1',
+            option: '1st-level escape'
+          }],
+          nextState: '0'
+        }
+      },
+      '[(...)]': {
+        'r|rt': {
+          action_: 'rd=',
+          nextState: 'rd'
+        },
+        'rd|rdt': {
+          action_: 'rq=',
+          nextState: 'rdq'
+        }
+      },
+      '...': {
+        'o|d|D|dq|qd|qD': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "..."
+          }],
+          nextState: '3'
+        },
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, {
+            type_: 'insert',
+            option: 'ellipsis'
+          }],
+          nextState: '1'
+        }
+      },
+      '. |* ': {
+        '*': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'addition compound'
+          }],
+          nextState: '1'
+        }
+      },
+      'state of aggregation $': {
+        '*': {
+          action_: ['output', 'state of aggregation'],
+          nextState: '1'
+        }
+      },
+      '{[(': {
+        'a|as|o': {
+          action_: ['o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        },
+        '0|1|2|3': {
+          action_: ['o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        },
+        '*': {
+          action_: ['output', 'o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        }
+      },
+      ')]}': {
+        '0|1|2|3|b|p|bp|o': {
+          action_: ['o=', 'parenthesisLevel--'],
+          nextState: 'o'
+        },
+        'a|as|d|D|q|qd|qD|dq': {
+          action_: ['output', 'o=', 'parenthesisLevel--'],
+          nextState: 'o'
+        }
+      },
+      ', ': {
+        '*': {
+          action_: ['output', 'comma'],
+          nextState: '0'
+        }
+      },
+      '^_': {
+        // ^ and _ without a sensible argument
+        '*': {}
+      },
+      '^{(...)}|^($...$)': {
+        '0|1|2|as': {
+          action_: 'b=',
+          nextState: 'b'
+        },
+        'p': {
+          action_: 'b=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'd= kv',
+          nextState: 'D'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qD'
+        },
+        'd|D|qd|qD|dq': {
+          action_: ['output', 'd='],
+          nextState: 'D'
+        }
+      },
+      '^a|^\\x{}{}|^\\x{}|^\\x|\'': {
+        '0|1|2|as': {
+          action_: 'b=',
+          nextState: 'b'
+        },
+        'p': {
+          action_: 'b=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'd= kv',
+          nextState: 'd'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'd|qd|D|qD': {
+          action_: 'd='
+        },
+        'dq': {
+          action_: ['output', 'd='],
+          nextState: 'd'
+        }
+      },
+      '_{(state of aggregation)}$': {
+        'd|D|q|qd|qD|dq': {
+          action_: ['output', 'q='],
+          nextState: 'q'
+        }
+      },
+      '_{(...)}|_($...$)|_9|_\\x{}{}|_\\x{}|_\\x': {
+        '0|1|2|as': {
+          action_: 'p=',
+          nextState: 'p'
+        },
+        'b': {
+          action_: 'p=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'q=',
+          nextState: 'q'
+        },
+        'd|D': {
+          action_: 'q=',
+          nextState: 'dq'
+        },
+        'q|qd|qD|dq': {
+          action_: ['output', 'q='],
+          nextState: 'q'
+        }
+      },
+      '=<>': {
+        '0|1|2|3|a|as|o|q|d|D|qd|qD|dq': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'bond'],
+          nextState: '3'
+        }
+      },
+      '#': {
+        '0|1|2|3|a|as|o': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, {
+            type_: 'bond',
+            option: "#"
+          }],
+          nextState: '3'
+        }
+      },
+      '{}': {
+        '*': {
+          action_: {
+            type_: 'output',
+            option: 1
+          },
+          nextState: '1'
+        }
+      },
+      '{...}': {
+        '0|1|2|3|a|as|b|p|bp': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        'o|d|D|q|qd|qD|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        }
+      },
+      '$...$': {
+        'a': {
+          action_: 'a='
+        },
+        // 2$n$
+        '0|1|2|3|as|b|p|bp|o': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        // not 'amount'
+        'as|o': {
+          action_: 'o='
+        },
+        'q|d|D|qd|qD|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        }
+      },
+      '\\bond{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'bond'],
+          nextState: "3"
+        }
+      },
+      '\\frac{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'frac-output'],
+          nextState: '3'
+        }
+      },
+      '\\overset{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'overset-output'],
+          nextState: '3'
+        }
+      },
+      '\\underset{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'underset-output'],
+          nextState: '3'
+        }
+      },
+      '\\underbrace{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'underbrace-output'],
+          nextState: '3'
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'color-output'],
+          nextState: '3'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'color0-output']
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'ce'],
+          nextState: '3'
+        }
+      },
+      '\\,': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'copy'],
+          nextState: '1'
+        }
+      },
+      '\\x{}{}|\\x{}|\\x': {
+        '0|1|2|3|a|as|b|p|bp|o|c0': {
+          action_: ['o=', 'output'],
+          nextState: '3'
+        },
+        '*': {
+          action_: ['output', 'o=', 'output'],
+          nextState: '3'
+        }
+      },
+      'others': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'copy'],
+          nextState: '3'
+        }
+      },
+      'else2': {
+        'a': {
+          action_: 'a to o',
+          nextState: 'o',
+          revisit: true
+        },
+        'as': {
+          action_: ['output', 'sb=true'],
+          nextState: '1',
+          revisit: true
+        },
+        'r|rt|rd|rdt|rdq': {
+          action_: ['output'],
+          nextState: '0',
+          revisit: true
+        },
+        '*': {
+          action_: ['output', 'copy'],
+          nextState: '3'
+        }
+      }
+    }),
+    actions: {
+      'o after d': function oAfterD(buffer, m) {
+        var ret;
+
+        if ((buffer.d || "").match(/^[0-9]+$/)) {
+          var tmp = buffer.d;
+          buffer.d = undefined;
+          ret = this['output'](buffer);
+          buffer.b = tmp;
+        } else {
+          ret = this['output'](buffer);
+        }
+
+        mhchemParser.actions['o='](buffer, m);
+        return ret;
+      },
+      'd= kv': function dKv(buffer, m) {
+        buffer.d = m;
+        buffer.dType = 'kv';
+      },
+      'charge or bond': function chargeOrBond(buffer, m) {
+        if (buffer['beginsWithBond']) {
+          /** @type {ParserOutput[]} */
+          var ret = [];
+          mhchemParser.concatArray(ret, this['output'](buffer));
+          mhchemParser.concatArray(ret, mhchemParser.actions['bond'](buffer, m, "-"));
+          return ret;
+        } else {
+          buffer.d = m;
+        }
+      },
+      '- after o/d': function afterOD(buffer, m, isAfterD) {
+        var c1 = mhchemParser.patterns.match_('orbital', buffer.o || "");
+        var c2 = mhchemParser.patterns.match_('one lowercase greek letter $', buffer.o || "");
+        var c3 = mhchemParser.patterns.match_('one lowercase latin letter $', buffer.o || "");
+        var c4 = mhchemParser.patterns.match_('$one lowercase latin letter$ $', buffer.o || "");
+        var hyphenFollows = m === "-" && (c1 && c1.remainder === "" || c2 || c3 || c4);
+
+        if (hyphenFollows && !buffer.a && !buffer.b && !buffer.p && !buffer.d && !buffer.q && !c1 && c3) {
+          buffer.o = '$' + buffer.o + '$';
+        }
+        /** @type {ParserOutput[]} */
+
+
+        var ret = [];
+
+        if (hyphenFollows) {
+          mhchemParser.concatArray(ret, this['output'](buffer));
+          ret.push({
+            type_: 'hyphen'
+          });
+        } else {
+          c1 = mhchemParser.patterns.match_('digits', buffer.d || "");
+
+          if (isAfterD && c1 && c1.remainder === '') {
+            mhchemParser.concatArray(ret, mhchemParser.actions['d='](buffer, m));
+            mhchemParser.concatArray(ret, this['output'](buffer));
+          } else {
+            mhchemParser.concatArray(ret, this['output'](buffer));
+            mhchemParser.concatArray(ret, mhchemParser.actions['bond'](buffer, m, "-"));
+          }
+        }
+
+        return ret;
+      },
+      'a to o': function aToO(buffer) {
+        buffer.o = buffer.a;
+        buffer.a = undefined;
+      },
+      'sb=true': function sbTrue(buffer) {
+        buffer.sb = true;
+      },
+      'sb=false': function sbFalse(buffer) {
+        buffer.sb = false;
+      },
+      'beginsWithBond=true': function beginsWithBondTrue(buffer) {
+        buffer['beginsWithBond'] = true;
+      },
+      'beginsWithBond=false': function beginsWithBondFalse(buffer) {
+        buffer['beginsWithBond'] = false;
+      },
+      'parenthesisLevel++': function parenthesisLevel(buffer) {
+        buffer['parenthesisLevel']++;
+      },
+      'parenthesisLevel--': function parenthesisLevel(buffer) {
+        buffer['parenthesisLevel']--;
+      },
+      'state of aggregation': function stateOfAggregation(buffer, m) {
+        return {
+          type_: 'state of aggregation',
+          p1: mhchemParser.go(m, 'o')
+        };
+      },
+      'comma': function comma(buffer, m) {
+        var a = m.replace(/\s*$/, '');
+        var withSpace = a !== m;
+
+        if (withSpace && buffer['parenthesisLevel'] === 0) {
+          return {
+            type_: 'comma enumeration L',
+            p1: a
+          };
+        } else {
+          return {
+            type_: 'comma enumeration M',
+            p1: a
+          };
+        }
+      },
+      'output': function output(buffer, m, entityFollows) {
+        // entityFollows:
+        //   undefined = if we have nothing else to output, also ignore the just read space (buffer.sb)
+        //   1 = an entity follows, never omit the space if there was one just read before (can only apply to state 1)
+        //   2 = 1 + the entity can have an amount, so output a\, instead of converting it to o (can only apply to states a|as)
+
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret;
+
+        if (!buffer.r) {
+          ret = [];
+
+          if (!buffer.a && !buffer.b && !buffer.p && !buffer.o && !buffer.q && !buffer.d && !entityFollows) ; else {
+            if (buffer.sb) {
+              ret.push({
+                type_: 'entitySkip'
+              });
+            }
+
+            if (!buffer.o && !buffer.q && !buffer.d && !buffer.b && !buffer.p && entityFollows !== 2) {
+              buffer.o = buffer.a;
+              buffer.a = undefined;
+            } else if (!buffer.o && !buffer.q && !buffer.d && (buffer.b || buffer.p)) {
+              buffer.o = buffer.a;
+              buffer.d = buffer.b;
+              buffer.q = buffer.p;
+              buffer.a = buffer.b = buffer.p = undefined;
+            } else {
+              if (buffer.o && buffer.dType === 'kv' && mhchemParser.patterns.match_('d-oxidation$', buffer.d || "")) {
+                buffer.dType = 'oxidation';
+              } else if (buffer.o && buffer.dType === 'kv' && !buffer.q) {
+                buffer.dType = undefined;
+              }
+            }
+
+            ret.push({
+              type_: 'chemfive',
+              a: mhchemParser.go(buffer.a, 'a'),
+              b: mhchemParser.go(buffer.b, 'bd'),
+              p: mhchemParser.go(buffer.p, 'pq'),
+              o: mhchemParser.go(buffer.o, 'o'),
+              q: mhchemParser.go(buffer.q, 'pq'),
+              d: mhchemParser.go(buffer.d, buffer.dType === 'oxidation' ? 'oxidation' : 'bd'),
+              dType: buffer.dType
+            });
+          }
+        } else {
+          // r
+
+          /** @type {ParserOutput[]} */
+          var rd;
+
+          if (buffer.rdt === 'M') {
+            rd = mhchemParser.go(buffer.rd, 'tex-math');
+          } else if (buffer.rdt === 'T') {
+            rd = [{
+              type_: 'text',
+              p1: buffer.rd || ""
+            }];
+          } else {
+            rd = mhchemParser.go(buffer.rd);
+          }
+          /** @type {ParserOutput[]} */
+
+
+          var rq;
+
+          if (buffer.rqt === 'M') {
+            rq = mhchemParser.go(buffer.rq, 'tex-math');
+          } else if (buffer.rqt === 'T') {
+            rq = [{
+              type_: 'text',
+              p1: buffer.rq || ""
+            }];
+          } else {
+            rq = mhchemParser.go(buffer.rq);
+          }
+
+          ret = {
+            type_: 'arrow',
+            r: buffer.r,
+            rd: rd,
+            rq: rq
+          };
+        }
+
+        for (var p in buffer) {
+          if (p !== 'parenthesisLevel' && p !== 'beginsWithBond') {
+            delete buffer[p];
+          }
+        }
+
+        return ret;
+      },
+      'oxidation-output': function oxidationOutput(buffer, m) {
+        var ret = ["{"];
+        mhchemParser.concatArray(ret, mhchemParser.go(m, 'oxidation'));
+        ret.push("}");
+        return ret;
+      },
+      'frac-output': function fracOutput(buffer, m) {
+        return {
+          type_: 'frac-ce',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'overset-output': function oversetOutput(buffer, m) {
+        return {
+          type_: 'overset',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'underset-output': function undersetOutput(buffer, m) {
+        return {
+          type_: 'underset',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'underbrace-output': function underbraceOutput(buffer, m) {
+        return {
+          type_: 'underbrace',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1])
+        };
+      },
+      'r=': function r(buffer, m) {
+        buffer.r = m;
+      },
+      'rdt=': function rdt(buffer, m) {
+        buffer.rdt = m;
+      },
+      'rd=': function rd(buffer, m) {
+        buffer.rd = m;
+      },
+      'rqt=': function rqt(buffer, m) {
+        buffer.rqt = m;
+      },
+      'rq=': function rq(buffer, m) {
+        buffer.rq = m;
+      },
+      'operator': function operator(buffer, m, p1) {
+        return {
+          type_: 'operator',
+          kind_: p1 || m
+        };
+      }
+    }
+  },
+  'a': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '1',
+          revisit: true
+        }
+      },
+      '$(...)$': {
+        '*': {
+          action_: 'tex-math tight',
+          nextState: '1'
+        }
+      },
+      ',': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'commaDecimal'
+          }
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {}
+  },
+  'o': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '1',
+          revisit: true
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '\\ca': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'circa'
+          }
+        }
+      },
+      '\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: '{text}'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {}
+  },
+  'text': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '{...}': {
+        '*': {
+          action_: 'text='
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '\\greek': {
+        '*': {
+          action_: ['output', 'rm']
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: ['output', 'copy']
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'text='
+        }
+      }
+    }),
+    actions: {
+      'output': function output(buffer) {
+        if (buffer.text_) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'text',
+            p1: buffer.text_
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  'pq': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'state of aggregation $': {
+        '*': {
+          action_: 'state of aggregation'
+        }
+      },
+      'i$': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '(KV letters),': {
+        '0': {
+          action_: 'rm',
+          nextState: '0'
+        }
+      },
+      'formula$': {
+        '0': {
+          nextState: 'f',
+          revisit: true
+        }
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: 'text'
+        }
+      },
+      'a-z': {
+        'f': {
+          action_: 'tex-math'
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '-9.,9': {
+        '*': {
+          action_: '9,9'
+        }
+      },
+      ',': {
+        '*': {
+          action_: {
+            type_: 'insert+p1',
+            option: 'comma enumeration S'
+          }
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: 'color-output'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: 'color0-output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: 'ce'
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'state of aggregation': function stateOfAggregation(buffer, m) {
+        return {
+          type_: 'state of aggregation subscript',
+          p1: mhchemParser.go(m, 'o')
+        };
+      },
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1], 'pq')
+        };
+      }
+    }
+  },
+  'bd': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'x$': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      'formula$': {
+        '0': {
+          nextState: 'f',
+          revisit: true
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '-9.,9 no missing 0': {
+        '*': {
+          action_: '9,9'
+        }
+      },
+      '.': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'electron dot'
+          }
+        }
+      },
+      'a-z': {
+        'f': {
+          action_: 'tex-math'
+        }
+      },
+      'x': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'KV x'
+          }
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '\'': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'prime'
+          }
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: 'text'
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: 'color-output'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: 'color0-output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: 'ce'
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1], 'bd')
+        };
+      }
+    }
+  },
+  'oxidation': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'roman numeral': {
+        '*': {
+          action_: 'roman-numeral'
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'roman-numeral': function romanNumeral(buffer, m) {
+        return {
+          type_: 'roman numeral',
+          p1: m || ""
+        };
+      }
+    }
+  },
+  'tex-math': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: ['output', 'ce']
+        }
+      },
+      '{...}|\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'o='
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'o='
+        }
+      }
+    }),
+    actions: {
+      'output': function output(buffer) {
+        if (buffer.o) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'tex-math',
+            p1: buffer.o
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  'tex-math tight': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: ['output', 'ce']
+        }
+      },
+      '{...}|\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'o='
+        }
+      },
+      '-|+': {
+        '*': {
+          action_: 'tight operator'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'o='
+        }
+      }
+    }),
+    actions: {
+      'tight operator': function tightOperator(buffer, m) {
+        buffer.o = (buffer.o || "") + "{" + m + "}";
+      },
+      'output': function output(buffer) {
+        if (buffer.o) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'tex-math',
+            p1: buffer.o
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  '9,9': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      ',': {
+        '*': {
+          action_: 'comma'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'comma': function comma() {
+        return {
+          type_: 'commaDecimal'
+        };
+      }
+    }
+  },
+  //#endregion
+  //
+  // \pu state machines
+  //
+  //#region pu
+  'pu': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      'space$': {
+        '*': {
+          action_: ['output', 'space']
+        }
+      },
+      '{[(|)]}': {
+        '0|a': {
+          action_: 'copy'
+        }
+      },
+      '(-)(9)^(-9)': {
+        '0': {
+          action_: 'number^',
+          nextState: 'a'
+        }
+      },
+      '(-)(9.,9)(e)(99)': {
+        '0': {
+          action_: 'enumber',
+          nextState: 'a'
+        }
+      },
+      'space': {
+        '0|a': {}
+      },
+      'pm-operator': {
+        '0|a': {
+          action_: {
+            type_: 'operator',
+            option: '\\pm'
+          },
+          nextState: '0'
+        }
+      },
+      'operator': {
+        '0|a': {
+          action_: 'copy',
+          nextState: '0'
+        }
+      },
+      '//': {
+        'd': {
+          action_: 'o=',
+          nextState: '/'
+        }
+      },
+      '/': {
+        'd': {
+          action_: 'o=',
+          nextState: '/'
+        }
+      },
+      '{...}|else': {
+        '0|d': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'a': {
+          action_: ['space', 'd='],
+          nextState: 'd'
+        },
+        '/|q': {
+          action_: 'q=',
+          nextState: 'q'
+        }
+      }
+    }),
+    actions: {
+      'enumber': function enumber(buffer, m) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+
+        if (m[0] === "+-" || m[0] === "+/-") {
+          ret.push("\\pm ");
+        } else if (m[0]) {
+          ret.push(m[0]);
+        }
+
+        if (m[1]) {
+          mhchemParser.concatArray(ret, mhchemParser.go(m[1], 'pu-9,9'));
+
+          if (m[2]) {
+            if (m[2].match(/[,.]/)) {
+              mhchemParser.concatArray(ret, mhchemParser.go(m[2], 'pu-9,9'));
+            } else {
+              ret.push(m[2]);
+            }
+          }
+
+          m[3] = m[4] || m[3];
+
+          if (m[3]) {
+            m[3] = m[3].trim();
+
+            if (m[3] === "e" || m[3].substr(0, 1) === "*") {
+              ret.push({
+                type_: 'cdot'
+              });
+            } else {
+              ret.push({
+                type_: 'times'
+              });
+            }
+          }
+        }
+
+        if (m[3]) {
+          ret.push("10^{" + m[5] + "}");
+        }
+
+        return ret;
+      },
+      'number^': function number(buffer, m) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+
+        if (m[0] === "+-" || m[0] === "+/-") {
+          ret.push("\\pm ");
+        } else if (m[0]) {
+          ret.push(m[0]);
+        }
+
+        mhchemParser.concatArray(ret, mhchemParser.go(m[1], 'pu-9,9'));
+        ret.push("^{" + m[2] + "}");
+        return ret;
+      },
+      'operator': function operator(buffer, m, p1) {
+        return {
+          type_: 'operator',
+          kind_: p1 || m
+        };
+      },
+      'space': function space() {
+        return {
+          type_: 'pu-space-1'
+        };
+      },
+      'output': function output(buffer) {
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret;
+        var md = mhchemParser.patterns.match_('{(...)}', buffer.d || "");
+
+        if (md && md.remainder === '') {
+          buffer.d = md.match_;
+        }
+
+        var mq = mhchemParser.patterns.match_('{(...)}', buffer.q || "");
+
+        if (mq && mq.remainder === '') {
+          buffer.q = mq.match_;
+        }
+
+        if (buffer.d) {
+          buffer.d = buffer.d.replace(/\u00B0C|\^oC|\^{o}C/g, "{}^{\\circ}C");
+          buffer.d = buffer.d.replace(/\u00B0F|\^oF|\^{o}F/g, "{}^{\\circ}F");
+        }
+
+        if (buffer.q) {
+          // fraction
+          buffer.q = buffer.q.replace(/\u00B0C|\^oC|\^{o}C/g, "{}^{\\circ}C");
+          buffer.q = buffer.q.replace(/\u00B0F|\^oF|\^{o}F/g, "{}^{\\circ}F");
+          var b5 = {
+            d: mhchemParser.go(buffer.d, 'pu'),
+            q: mhchemParser.go(buffer.q, 'pu')
+          };
+
+          if (buffer.o === '//') {
+            ret = {
+              type_: 'pu-frac',
+              p1: b5.d,
+              p2: b5.q
+            };
+          } else {
+            ret = b5.d;
+
+            if (b5.d.length > 1 || b5.q.length > 1) {
+              ret.push({
+                type_: ' / '
+              });
+            } else {
+              ret.push({
+                type_: '/'
+              });
+            }
+
+            mhchemParser.concatArray(ret, b5.q);
+          }
+        } else {
+          // no fraction
+          ret = mhchemParser.go(buffer.d, 'pu-2');
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    }
+  },
+  'pu-2': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '*': {
+        '*': {
+          action_: ['output', 'cdot'],
+          nextState: '0'
+        }
+      },
+      '\\x': {
+        '*': {
+          action_: 'rm='
+        }
+      },
+      'space': {
+        '*': {
+          action_: ['output', 'space'],
+          nextState: '0'
+        }
+      },
+      '^{(...)}|^(-1)': {
+        '1': {
+          action_: '^(-1)'
+        }
+      },
+      '-9.,9': {
+        '0': {
+          action_: 'rm=',
+          nextState: '0'
+        },
+        '1': {
+          action_: '^(-1)',
+          nextState: '0'
+        }
+      },
+      '{...}|else': {
+        '*': {
+          action_: 'rm=',
+          nextState: '1'
+        }
+      }
+    }),
+    actions: {
+      'cdot': function cdot() {
+        return {
+          type_: 'tight cdot'
+        };
+      },
+      '^(-1)': function _(buffer, m) {
+        buffer.rm += "^{" + m + "}";
+      },
+      'space': function space() {
+        return {
+          type_: 'pu-space-2'
+        };
+      },
+      'output': function output(buffer) {
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret = [];
+
+        if (buffer.rm) {
+          var mrm = mhchemParser.patterns.match_('{(...)}', buffer.rm || "");
+
+          if (mrm && mrm.remainder === '') {
+            ret = mhchemParser.go(mrm.match_, 'pu');
+          } else {
+            ret = {
+              type_: 'rm',
+              p1: buffer.rm
+            };
+          }
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    }
+  },
+  'pu-9,9': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '0': {
+          action_: 'output-0'
+        },
+        'o': {
+          action_: 'output-o'
+        }
+      },
+      ',': {
+        '0': {
+          action_: ['output-0', 'comma'],
+          nextState: 'o'
+        }
+      },
+      '.': {
+        '0': {
+          action_: ['output-0', 'copy'],
+          nextState: 'o'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'text='
+        }
+      }
+    }),
+    actions: {
+      'comma': function comma() {
+        return {
+          type_: 'commaDecimal'
+        };
+      },
+      'output-0': function output0(buffer) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+        buffer.text_ = buffer.text_ || "";
+
+        if (buffer.text_.length > 4) {
+          var a = buffer.text_.length % 3;
+
+          if (a === 0) {
+            a = 3;
+          }
+
+          for (var i = buffer.text_.length - 3; i > 0; i -= 3) {
+            ret.push(buffer.text_.substr(i, 3));
+            ret.push({
+              type_: '1000 separator'
+            });
+          }
+
+          ret.push(buffer.text_.substr(0, a));
+          ret.reverse();
+        } else {
+          ret.push(buffer.text_);
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      },
+      'output-o': function outputO(buffer) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+        buffer.text_ = buffer.text_ || "";
+
+        if (buffer.text_.length > 4) {
+          var a = buffer.text_.length - 3;
+
+          for (var i = 0; i < a; i += 3) {
+            ret.push(buffer.text_.substr(i, 3));
+            ret.push({
+              type_: '1000 separator'
+            });
+          }
+
+          ret.push(buffer.text_.substr(i));
+        } else {
+          ret.push(buffer.text_);
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    } //#endregion
+
+  }
+}; //
+// texify: Take MhchemParser output and convert it to TeX
+//
+
+/** @type {Texify} */
+
+var texify = {
+  go: function go(input, isInner) {
+    // (recursive, max 4 levels)
+    if (!input) {
+      return "";
+    }
+
+    var res = "";
+    var cee = false;
+
+    for (var i = 0; i < input.length; i++) {
+      var inputi = input[i];
+
+      if (typeof inputi === "string") {
+        res += inputi;
+      } else {
+        res += texify._go2(inputi);
+
+        if (inputi.type_ === '1st-level escape') {
+          cee = true;
+        }
+      }
+    }
+
+    if (!isInner && !cee && res) {
+      res = "{" + res + "}";
+    }
+
+    return res;
+  },
+  _goInner: function _goInner(input) {
+    if (!input) {
+      return input;
+    }
+
+    return texify.go(input, true);
+  },
+  _go2: function _go2(buf) {
+    /** @type {undefined | string} */
+    var res;
+
+    switch (buf.type_) {
+      case 'chemfive':
+        res = "";
+        var b5 = {
+          a: texify._goInner(buf.a),
+          b: texify._goInner(buf.b),
+          p: texify._goInner(buf.p),
+          o: texify._goInner(buf.o),
+          q: texify._goInner(buf.q),
+          d: texify._goInner(buf.d)
+        }; //
+        // a
+        //
+
+        if (b5.a) {
+          if (b5.a.match(/^[+\-]/)) {
+            b5.a = "{" + b5.a + "}";
+          }
+
+          res += b5.a + "\\,";
+        } //
+        // b and p
+        //
+
+
+        if (b5.b || b5.p) {
+          res += "{\\vphantom{X}}";
+          res += "^{\\hphantom{" + (b5.b || "") + "}}_{\\hphantom{" + (b5.p || "") + "}}";
+          res += "{\\vphantom{X}}";
+          res += "^{\\smash[t]{\\vphantom{2}}\\mathllap{" + (b5.b || "") + "}}";
+          res += "_{\\vphantom{2}\\mathllap{\\smash[t]{" + (b5.p || "") + "}}}";
+        } //
+        // o
+        //
+
+
+        if (b5.o) {
+          if (b5.o.match(/^[+\-]/)) {
+            b5.o = "{" + b5.o + "}";
+          }
+
+          res += b5.o;
+        } //
+        // q and d
+        //
+
+
+        if (buf.dType === 'kv') {
+          if (b5.d || b5.q) {
+            res += "{\\vphantom{X}}";
+          }
+
+          if (b5.d) {
+            res += "^{" + b5.d + "}";
+          }
+
+          if (b5.q) {
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+        } else if (buf.dType === 'oxidation') {
+          if (b5.d) {
+            res += "{\\vphantom{X}}";
+            res += "^{" + b5.d + "}";
+          }
+
+          if (b5.q) {
+            res += "{\\vphantom{X}}";
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+        } else {
+          if (b5.q) {
+            res += "{\\vphantom{X}}";
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+
+          if (b5.d) {
+            res += "{\\vphantom{X}}";
+            res += "^{" + b5.d + "}";
+          }
+        }
+
+        break;
+
+      case 'rm':
+        res = "\\mathrm{" + buf.p1 + "}";
+        break;
+
+      case 'text':
+        if (buf.p1.match(/[\^_]/)) {
+          buf.p1 = buf.p1.replace(" ", "~").replace("-", "\\text{-}");
+          res = "\\mathrm{" + buf.p1 + "}";
+        } else {
+          res = "\\text{" + buf.p1 + "}";
+        }
+
+        break;
+
+      case 'roman numeral':
+        res = "\\mathrm{" + buf.p1 + "}";
+        break;
+
+      case 'state of aggregation':
+        res = "\\mskip2mu " + texify._goInner(buf.p1);
+        break;
+
+      case 'state of aggregation subscript':
+        res = "\\mskip1mu " + texify._goInner(buf.p1);
+        break;
+
+      case 'bond':
+        res = texify._getBond(buf.kind_);
+
+        if (!res) {
+          throw ["MhchemErrorBond", "mhchem Error. Unknown bond type (" + buf.kind_ + ")"];
+        }
+
+        break;
+
+      case 'frac':
+        var c = "\\frac{" + buf.p1 + "}{" + buf.p2 + "}";
+        res = "\\mathchoice{\\textstyle" + c + "}{" + c + "}{" + c + "}{" + c + "}";
+        break;
+
+      case 'pu-frac':
+        var d = "\\frac{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        res = "\\mathchoice{\\textstyle" + d + "}{" + d + "}{" + d + "}{" + d + "}";
+        break;
+
+      case 'tex-math':
+        res = buf.p1 + " ";
+        break;
+
+      case 'frac-ce':
+        res = "\\frac{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'overset':
+        res = "\\overset{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'underset':
+        res = "\\underset{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'underbrace':
+        res = "\\underbrace{" + texify._goInner(buf.p1) + "}_{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'color':
+        res = "{\\color{" + buf.color1 + "}{" + texify._goInner(buf.color2) + "}}";
+        break;
+
+      case 'color0':
+        res = "\\color{" + buf.color + "}";
+        break;
+
+      case 'arrow':
+        var b6 = {
+          rd: texify._goInner(buf.rd),
+          rq: texify._goInner(buf.rq)
+        };
+
+        var arrow = "\\x" + texify._getArrow(buf.r);
+
+        if (b6.rq) {
+          arrow += "[{" + b6.rq + "}]";
+        }
+
+        if (b6.rd) {
+          arrow += "{" + b6.rd + "}";
+        } else {
+          arrow += "{}";
+        }
+
+        res = arrow;
+        break;
+
+      case 'operator':
+        res = texify._getOperator(buf.kind_);
+        break;
+
+      case '1st-level escape':
+        res = buf.p1 + " "; // &, \\\\, \\hlin
+
+        break;
+
+      case 'space':
+        res = " ";
+        break;
+
+      case 'entitySkip':
+        res = "~";
+        break;
+
+      case 'pu-space-1':
+        res = "~";
+        break;
+
+      case 'pu-space-2':
+        res = "\\mkern3mu ";
+        break;
+
+      case '1000 separator':
+        res = "\\mkern2mu ";
+        break;
+
+      case 'commaDecimal':
+        res = "{,}";
+        break;
+
+      case 'comma enumeration L':
+        res = "{" + buf.p1 + "}\\mkern6mu ";
+        break;
+
+      case 'comma enumeration M':
+        res = "{" + buf.p1 + "}\\mkern3mu ";
+        break;
+
+      case 'comma enumeration S':
+        res = "{" + buf.p1 + "}\\mkern1mu ";
+        break;
+
+      case 'hyphen':
+        res = "\\text{-}";
+        break;
+
+      case 'addition compound':
+        res = "\\,{\\cdot}\\,";
+        break;
+
+      case 'electron dot':
+        res = "\\mkern1mu \\bullet\\mkern1mu ";
+        break;
+
+      case 'KV x':
+        res = "{\\times}";
+        break;
+
+      case 'prime':
+        res = "\\prime ";
+        break;
+
+      case 'cdot':
+        res = "\\cdot ";
+        break;
+
+      case 'tight cdot':
+        res = "\\mkern1mu{\\cdot}\\mkern1mu ";
+        break;
+
+      case 'times':
+        res = "\\times ";
+        break;
+
+      case 'circa':
+        res = "{\\sim}";
+        break;
+
+      case '^':
+        res = "uparrow";
+        break;
+
+      case 'v':
+        res = "downarrow";
+        break;
+
+      case 'ellipsis':
+        res = "\\ldots ";
+        break;
+
+      case '/':
+        res = "/";
+        break;
+
+      case ' / ':
+        res = "\\,/\\,";
+        break;
+
+      default:
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+      // Missing texify rule or unknown MhchemParser output
+    }
+    return res;
+  },
+  _getArrow: function _getArrow(a) {
+    switch (a) {
+      case "->":
+        return "rightarrow";
+
+      case "\u2192":
+        return "rightarrow";
+
+      case "\u27F6":
+        return "rightarrow";
+
+      case "<-":
+        return "leftarrow";
+
+      case "<->":
+        return "leftrightarrow";
+
+      case "<-->":
+        return "rightleftarrows";
+
+      case "<=>":
+        return "rightleftharpoons";
+
+      case "\u21CC":
+        return "rightleftharpoons";
+
+      case "<=>>":
+        return "rightequilibrium";
+
+      case "<<=>":
+        return "leftequilibrium";
+
+      default:
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  },
+  _getBond: function _getBond(a) {
+    switch (a) {
+      case "-":
+        return "{-}";
+
+      case "1":
+        return "{-}";
+
+      case "=":
+        return "{=}";
+
+      case "2":
+        return "{=}";
+
+      case "#":
+        return "{\\equiv}";
+
+      case "3":
+        return "{\\equiv}";
+
+      case "~":
+        return "{\\tripledash}";
+
+      case "~-":
+        return "{\\mathrlap{\\raisebox{-.1em}{$-$}}\\raisebox{.1em}{$\\tripledash$}}";
+
+      case "~=":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";
+
+      case "~--":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";
+
+      case "-~-":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$-$}}\\tripledash}";
+
+      case "...":
+        return "{{\\cdot}{\\cdot}{\\cdot}}";
+
+      case "....":
+        return "{{\\cdot}{\\cdot}{\\cdot}{\\cdot}}";
+
+      case "->":
+        return "{\\rightarrow}";
+
+      case "<-":
+        return "{\\leftarrow}";
+
+      case "<":
+        return "{<}";
+
+      case ">":
+        return "{>}";
+
+      default:
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  },
+  _getOperator: function _getOperator(a) {
+    switch (a) {
+      case "+":
+        return " {}+{} ";
+
+      case "-":
+        return " {}-{} ";
+
+      case "=":
+        return " {}={} ";
+
+      case "<":
+        return " {}<{} ";
+
+      case ">":
+        return " {}>{} ";
+
+      case "<<":
+        return " {}\\ll{} ";
+
+      case ">>":
+        return " {}\\gg{} ";
+
+      case "\\pm":
+        return " {}\\pm{} ";
+
+      case "\\approx":
+        return " {}\\approx{} ";
+
+      case "$\\approx$":
+        return " {}\\approx{} ";
+
+      case "v":
+        return " \\downarrow{} ";
+
+      case "(v)":
+        return " \\downarrow{} ";
+
+      case "^":
+        return " \\uparrow{} ";
+
+      case "(^)":
+        return " \\uparrow{} ";
+
+      default:
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  }
+}; //
diff --git a/docs/pt/katex/contrib/render-a11y-string.js b/docs/pt/katex/contrib/render-a11y-string.js
new file mode 100644
index 0000000..d8b27c7
--- /dev/null
+++ b/docs/pt/katex/contrib/render-a11y-string.js
@@ -0,0 +1,870 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory(require("katex"));
+	else if(typeof define === 'function' && define.amd)
+		define(["katex"], factory);
+	else {
+		var a = typeof exports === 'object' ? factory(require("katex")) : factory(root["katex"]);
+		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
+	}
+})((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__0__) {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
+/******/ 	function __webpack_require__(moduleId) {
+/******/
+/******/ 		// Check if module is in cache
+/******/ 		if(installedModules[moduleId]) {
+/******/ 			return installedModules[moduleId].exports;
+/******/ 		}
+/******/ 		// Create a new module (and put it into the cache)
+/******/ 		var module = installedModules[moduleId] = {
+/******/ 			i: moduleId,
+/******/ 			l: false,
+/******/ 			exports: {}
+/******/ 		};
+/******/
+/******/ 		// Execute the module function
+/******/ 		modules[moduleId].call(module.exports, module, module.exports, __webpack_require__);
+/******/
+/******/ 		// Flag the module as loaded
+/******/ 		module.l = true;
+/******/
+/******/ 		// Return the exports of the module
+/******/ 		return module.exports;
+/******/ 	}
+/******/
+/******/
+/******/ 	// expose the modules object (__webpack_modules__)
+/******/ 	__webpack_require__.m = modules;
+/******/
+/******/ 	// expose the module cache
+/******/ 	__webpack_require__.c = installedModules;
+/******/
+/******/ 	// define getter function for harmony exports
+/******/ 	__webpack_require__.d = function(exports, name, getter) {
+/******/ 		if(!__webpack_require__.o(exports, name)) {
+/******/ 			Object.defineProperty(exports, name, { enumerable: true, get: getter });
+/******/ 		}
+/******/ 	};
+/******/
+/******/ 	// define __esModule on exports
+/******/ 	__webpack_require__.r = function(exports) {
+/******/ 		if(typeof Symbol !== 'undefined' && Symbol.toStringTag) {
+/******/ 			Object.defineProperty(exports, Symbol.toStringTag, { value: 'Module' });
+/******/ 		}
+/******/ 		Object.defineProperty(exports, '__esModule', { value: true });
+/******/ 	};
+/******/
+/******/ 	// create a fake namespace object
+/******/ 	// mode & 1: value is a module id, require it
+/******/ 	// mode & 2: merge all properties of value into the ns
+/******/ 	// mode & 4: return value when already ns object
+/******/ 	// mode & 8|1: behave like require
+/******/ 	__webpack_require__.t = function(value, mode) {
+/******/ 		if(mode & 1) value = __webpack_require__(value);
+/******/ 		if(mode & 8) return value;
+/******/ 		if((mode & 4) && typeof value === 'object' && value && value.__esModule) return value;
+/******/ 		var ns = Object.create(null);
+/******/ 		__webpack_require__.r(ns);
+/******/ 		Object.defineProperty(ns, 'default', { enumerable: true, value: value });
+/******/ 		if(mode & 2 && typeof value != 'string') for(var key in value) __webpack_require__.d(ns, key, function(key) { return value[key]; }.bind(null, key));
+/******/ 		return ns;
+/******/ 	};
+/******/
+/******/ 	// getDefaultExport function for compatibility with non-harmony modules
+/******/ 	__webpack_require__.n = function(module) {
+/******/ 		var getter = module && module.__esModule ?
+/******/ 			function getDefault() { return module['default']; } :
+/******/ 			function getModuleExports() { return module; };
+/******/ 		__webpack_require__.d(getter, 'a', getter);
+/******/ 		return getter;
+/******/ 	};
+/******/
+/******/ 	// Object.prototype.hasOwnProperty.call
+/******/ 	__webpack_require__.o = function(object, property) { return Object.prototype.hasOwnProperty.call(object, property); };
+/******/
+/******/ 	// __webpack_public_path__
+/******/ 	__webpack_require__.p = "";
+/******/
+/******/
+/******/ 	// Load entry module and return exports
+/******/ 	return __webpack_require__(__webpack_require__.s = 1);
+/******/ })
+/************************************************************************/
+/******/ ([
+/* 0 */
+/***/ (function(module, exports) {
+
+module.exports = __WEBPACK_EXTERNAL_MODULE__0__;
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0__ = __webpack_require__(0);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0___default = /*#__PURE__*/__webpack_require__.n(katex__WEBPACK_IMPORTED_MODULE_0__);
+/**
+ * renderA11yString returns a readable string.
+ *
+ * In some cases the string will have the proper semantic math
+ * meaning,:
+ *   renderA11yString("\\frac{1}{2}"")
+ *   -> "start fraction, 1, divided by, 2, end fraction"
+ *
+ * However, other cases do not:
+ *   renderA11yString("f(x) = x^2")
+ *   -> "f, left parenthesis, x, right parenthesis, equals, x, squared"
+ *
+ * The commas in the string aim to increase ease of understanding
+ * when read by a screenreader.
+ */
+// NOTE: since we're importing types here these files won't actually be
+// included in the build.
+// $FlowIgnore: we import the types directly anyways
+
+var stringMap = {
+  "(": "left parenthesis",
+  ")": "right parenthesis",
+  "[": "open bracket",
+  "]": "close bracket",
+  "\\{": "left brace",
+  "\\}": "right brace",
+  "\\lvert": "open vertical bar",
+  "\\rvert": "close vertical bar",
+  "|": "vertical bar",
+  "\\uparrow": "up arrow",
+  "\\Uparrow": "up arrow",
+  "\\downarrow": "down arrow",
+  "\\Downarrow": "down arrow",
+  "\\updownarrow": "up down arrow",
+  "\\leftarrow": "left arrow",
+  "\\Leftarrow": "left arrow",
+  "\\rightarrow": "right arrow",
+  "\\Rightarrow": "right arrow",
+  "\\langle": "open angle",
+  "\\rangle": "close angle",
+  "\\lfloor": "open floor",
+  "\\rfloor": "close floor",
+  "\\int": "integral",
+  "\\intop": "integral",
+  "\\lim": "limit",
+  "\\ln": "natural log",
+  "\\log": "log",
+  "\\sin": "sine",
+  "\\cos": "cosine",
+  "\\tan": "tangent",
+  "\\cot": "cotangent",
+  "\\sum": "sum",
+  "/": "slash",
+  ",": "comma",
+  ".": "point",
+  "-": "negative",
+  "+": "plus",
+  "~": "tilde",
+  ":": "colon",
+  "?": "question mark",
+  "'": "apostrophe",
+  "\\%": "percent",
+  " ": "space",
+  "\\ ": "space",
+  "\\$": "dollar sign",
+  "\\angle": "angle",
+  "\\degree": "degree",
+  "\\circ": "circle",
+  "\\vec": "vector",
+  "\\triangle": "triangle",
+  "\\pi": "pi",
+  "\\prime": "prime",
+  "\\infty": "infinity",
+  "\\alpha": "alpha",
+  "\\beta": "beta",
+  "\\gamma": "gamma",
+  "\\omega": "omega",
+  "\\theta": "theta",
+  "\\sigma": "sigma",
+  "\\lambda": "lambda",
+  "\\tau": "tau",
+  "\\Delta": "delta",
+  "\\delta": "delta",
+  "\\mu": "mu",
+  "\\rho": "rho",
+  "\\nabla": "del",
+  "\\ell": "ell",
+  "\\ldots": "dots",
+  // TODO: add entries for all accents
+  "\\hat": "hat",
+  "\\acute": "acute"
+};
+var powerMap = {
+  "prime": "prime",
+  "degree": "degrees",
+  "circle": "degrees",
+  "2": "squared",
+  "3": "cubed"
+};
+var openMap = {
+  "|": "open vertical bar",
+  ".": ""
+};
+var closeMap = {
+  "|": "close vertical bar",
+  ".": ""
+};
+var binMap = {
+  "+": "plus",
+  "-": "minus",
+  "\\pm": "plus minus",
+  "\\cdot": "dot",
+  "*": "times",
+  "/": "divided by",
+  "\\times": "times",
+  "\\div": "divided by",
+  "\\circ": "circle",
+  "\\bullet": "bullet"
+};
+var relMap = {
+  "=": "equals",
+  "\\approx": "approximately equals",
+  "≠": "does not equal",
+  "\\geq": "is greater than or equal to",
+  "\\ge": "is greater than or equal to",
+  "\\leq": "is less than or equal to",
+  "\\le": "is less than or equal to",
+  ">": "is greater than",
+  "<": "is less than",
+  "\\leftarrow": "left arrow",
+  "\\Leftarrow": "left arrow",
+  "\\rightarrow": "right arrow",
+  "\\Rightarrow": "right arrow",
+  ":": "colon"
+};
+var accentUnderMap = {
+  "\\underleftarrow": "left arrow",
+  "\\underrightarrow": "right arrow",
+  "\\underleftrightarrow": "left-right arrow",
+  "\\undergroup": "group",
+  "\\underlinesegment": "line segment",
+  "\\utilde": "tilde"
+};
+
+var buildString = function buildString(str, type, a11yStrings) {
+  if (!str) {
+    return;
+  }
+
+  var ret;
+
+  if (type === "open") {
+    ret = str in openMap ? openMap[str] : stringMap[str] || str;
+  } else if (type === "close") {
+    ret = str in closeMap ? closeMap[str] : stringMap[str] || str;
+  } else if (type === "bin") {
+    ret = binMap[str] || str;
+  } else if (type === "rel") {
+    ret = relMap[str] || str;
+  } else {
+    ret = stringMap[str] || str;
+  } // If the text to add is a number and there is already a string
+  // in the list and the last string is a number then we should
+  // combine them into a single number
+
+
+  if (/^\d+$/.test(ret) && a11yStrings.length > 0 && // TODO(kevinb): check that the last item in a11yStrings is a string
+  // I think we might be able to drop the nested arrays, which would make
+  // this easier to type - $FlowFixMe
+  /^\d+$/.test(a11yStrings[a11yStrings.length - 1])) {
+    a11yStrings[a11yStrings.length - 1] += ret;
+  } else if (ret) {
+    a11yStrings.push(ret);
+  }
+};
+
+var buildRegion = function buildRegion(a11yStrings, callback) {
+  var regionStrings = [];
+  a11yStrings.push(regionStrings);
+  callback(regionStrings);
+};
+
+var handleObject = function handleObject(tree, a11yStrings, atomType) {
+  // Everything else is assumed to be an object...
+  switch (tree.type) {
+    case "accent":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          buildA11yStrings(tree.base, a11yStrings, atomType);
+          a11yStrings.push("with");
+          buildString(tree.label, "normal", a11yStrings);
+          a11yStrings.push("on top");
+        });
+        break;
+      }
+
+    case "accentUnder":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          buildA11yStrings(tree.base, a11yStrings, atomType);
+          a11yStrings.push("with");
+          buildString(accentUnderMap[tree.label], "normal", a11yStrings);
+          a11yStrings.push("underneath");
+        });
+        break;
+      }
+
+    case "accent-token":
+      {
+        // Used internally by accent symbols.
+        break;
+      }
+
+    case "atom":
+      {
+        var text = tree.text;
+
+        switch (tree.family) {
+          case "bin":
+            {
+              buildString(text, "bin", a11yStrings);
+              break;
+            }
+
+          case "close":
+            {
+              buildString(text, "close", a11yStrings);
+              break;
+            }
+          // TODO(kevinb): figure out what should be done for inner
+
+          case "inner":
+            {
+              buildString(tree.text, "inner", a11yStrings);
+              break;
+            }
+
+          case "open":
+            {
+              buildString(text, "open", a11yStrings);
+              break;
+            }
+
+          case "punct":
+            {
+              buildString(text, "punct", a11yStrings);
+              break;
+            }
+
+          case "rel":
+            {
+              buildString(text, "rel", a11yStrings);
+              break;
+            }
+
+          default:
+            {
+              tree.family;
+              throw new Error("\"" + tree.family + "\" is not a valid atom type");
+            }
+        }
+
+        break;
+      }
+
+    case "color":
+      {
+        var color = tree.color.replace(/katex-/, "");
+        buildRegion(a11yStrings, function (regionStrings) {
+          regionStrings.push("start color " + color);
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          regionStrings.push("end color " + color);
+        });
+        break;
+      }
+
+    case "color-token":
+      {
+        // Used by \color, \colorbox, and \fcolorbox but not directly rendered.
+        // It's a leaf node and has no children so just break.
+        break;
+      }
+
+    case "delimsizing":
+      {
+        if (tree.delim && tree.delim !== ".") {
+          buildString(tree.delim, "normal", a11yStrings);
+        }
+
+        break;
+      }
+
+    case "genfrac":
+      {
+        buildRegion(a11yStrings, function (regionStrings) {
+          // genfrac can have unbalanced delimiters
+          var leftDelim = tree.leftDelim,
+              rightDelim = tree.rightDelim; // NOTE: Not sure if this is a safe assumption
+          // hasBarLine true -> fraction, false -> binomial
+
+          if (tree.hasBarLine) {
+            regionStrings.push("start fraction");
+            leftDelim && buildString(leftDelim, "open", regionStrings);
+            buildA11yStrings(tree.numer, regionStrings, atomType);
+            regionStrings.push("divided by");
+            buildA11yStrings(tree.denom, regionStrings, atomType);
+            rightDelim && buildString(rightDelim, "close", regionStrings);
+            regionStrings.push("end fraction");
+          } else {
+            regionStrings.push("start binomial");
+            leftDelim && buildString(leftDelim, "open", regionStrings);
+            buildA11yStrings(tree.numer, regionStrings, atomType);
+            regionStrings.push("over");
+            buildA11yStrings(tree.denom, regionStrings, atomType);
+            rightDelim && buildString(rightDelim, "close", regionStrings);
+            regionStrings.push("end binomial");
+          }
+        });
+        break;
+      }
+
+    case "kern":
+      {
+        // No op: we don't attempt to present kerning information
+        // to the screen reader.
+        break;
+      }
+
+    case "leftright":
+      {
+        buildRegion(a11yStrings, function (regionStrings) {
+          buildString(tree.left, "open", regionStrings);
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          buildString(tree.right, "close", regionStrings);
+        });
+        break;
+      }
+
+    case "leftright-right":
+      {
+        // TODO: double check that this is a no-op
+        break;
+      }
+
+    case "lap":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "mathord":
+      {
+        buildString(tree.text, "normal", a11yStrings);
+        break;
+      }
+
+    case "op":
+      {
+        var body = tree.body,
+            name = tree.name;
+
+        if (body) {
+          buildA11yStrings(body, a11yStrings, atomType);
+        } else if (name) {
+          buildString(name, "normal", a11yStrings);
+        }
+
+        break;
+      }
+
+    case "op-token":
+      {
+        // Used internally by operator symbols.
+        buildString(tree.text, atomType, a11yStrings);
+        break;
+      }
+
+    case "ordgroup":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "overline":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          a11yStrings.push("start overline");
+          buildA11yStrings(tree.body, a11yStrings, atomType);
+          a11yStrings.push("end overline");
+        });
+        break;
+      }
+
+    case "phantom":
+      {
+        a11yStrings.push("empty space");
+        break;
+      }
+
+    case "raisebox":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "rule":
+      {
+        a11yStrings.push("rectangle");
+        break;
+      }
+
+    case "sizing":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "spacing":
+      {
+        a11yStrings.push("space");
+        break;
+      }
+
+    case "styling":
+      {
+        // We ignore the styling and just pass through the contents
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "sqrt":
+      {
+        buildRegion(a11yStrings, function (regionStrings) {
+          var body = tree.body,
+              index = tree.index;
+
+          if (index) {
+            var indexString = flatten(buildA11yStrings(index, [], atomType)).join(",");
+
+            if (indexString === "3") {
+              regionStrings.push("cube root of");
+              buildA11yStrings(body, regionStrings, atomType);
+              regionStrings.push("end cube root");
+              return;
+            }
+
+            regionStrings.push("root");
+            regionStrings.push("start index");
+            buildA11yStrings(index, regionStrings, atomType);
+            regionStrings.push("end index");
+            return;
+          }
+
+          regionStrings.push("square root of");
+          buildA11yStrings(body, regionStrings, atomType);
+          regionStrings.push("end square root");
+        });
+        break;
+      }
+
+    case "supsub":
+      {
+        var base = tree.base,
+            sub = tree.sub,
+            sup = tree.sup;
+        var isLog = false;
+
+        if (base) {
+          buildA11yStrings(base, a11yStrings, atomType);
+          isLog = base.type === "op" && base.name === "\\log";
+        }
+
+        if (sub) {
+          var regionName = isLog ? "base" : "subscript";
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start " + regionName);
+            buildA11yStrings(sub, regionStrings, atomType);
+            regionStrings.push("end " + regionName);
+          });
+        }
+
+        if (sup) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            var supString = flatten(buildA11yStrings(sup, [], atomType)).join(",");
+
+            if (supString in powerMap) {
+              regionStrings.push(powerMap[supString]);
+              return;
+            }
+
+            regionStrings.push("start superscript");
+            buildA11yStrings(sup, regionStrings, atomType);
+            regionStrings.push("end superscript");
+          });
+        }
+
+        break;
+      }
+
+    case "text":
+      {
+        // TODO: handle other fonts
+        if (tree.font === "\\textbf") {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start bold text");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end bold text");
+          });
+          break;
+        }
+
+        buildRegion(a11yStrings, function (regionStrings) {
+          regionStrings.push("start text");
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          regionStrings.push("end text");
+        });
+        break;
+      }
+
+    case "textord":
+      {
+        buildString(tree.text, atomType, a11yStrings);
+        break;
+      }
+
+    case "smash":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "enclose":
+      {
+        // TODO: create a map for these.
+        // TODO: differentiate between a body with a single atom, e.g.
+        // "cancel a" instead of "start cancel, a, end cancel"
+        if (/cancel/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start cancel");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end cancel");
+          });
+          break;
+        } else if (/box/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start box");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end box");
+          });
+          break;
+        } else if (/sout/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start strikeout");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end strikeout");
+          });
+          break;
+        }
+
+        throw new Error("KaTeX-a11y: enclose node with " + tree.label + " not supported yet");
+      }
+
+    case "vphantom":
+      {
+        throw new Error("KaTeX-a11y: vphantom not implemented yet");
+      }
+
+    case "hphantom":
+      {
+        throw new Error("KaTeX-a11y: hphantom not implemented yet");
+      }
+
+    case "operatorname":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "array":
+      {
+        throw new Error("KaTeX-a11y: array not implemented yet");
+      }
+
+    case "raw":
+      {
+        throw new Error("KaTeX-a11y: raw not implemented yet");
+      }
+
+    case "size":
+      {
+        // Although there are nodes of type "size" in the parse tree, they have
+        // no semantic meaning and should be ignored.
+        break;
+      }
+
+    case "url":
+      {
+        throw new Error("KaTeX-a11y: url not implemented yet");
+      }
+
+    case "tag":
+      {
+        throw new Error("KaTeX-a11y: tag not implemented yet");
+      }
+
+    case "verb":
+      {
+        buildString("start verbatim", "normal", a11yStrings);
+        buildString(tree.body, "normal", a11yStrings);
+        buildString("end verbatim", "normal", a11yStrings);
+        break;
+      }
+
+    case "environment":
+      {
+        throw new Error("KaTeX-a11y: environment not implemented yet");
+      }
+
+    case "horizBrace":
+      {
+        buildString("start " + tree.label.slice(1), "normal", a11yStrings);
+        buildA11yStrings(tree.base, a11yStrings, atomType);
+        buildString("end " + tree.label.slice(1), "normal", a11yStrings);
+        break;
+      }
+
+    case "infix":
+      {
+        // All infix nodes are replace with other nodes.
+        break;
+      }
+
+    case "includegraphics":
+      {
+        throw new Error("KaTeX-a11y: includegraphics not implemented yet");
+      }
+
+    case "font":
+      {
+        // TODO: callout the start/end of specific fonts
+        // TODO: map \BBb{N} to "the naturals" or something like that
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "href":
+      {
+        throw new Error("KaTeX-a11y: href not implemented yet");
+      }
+
+    case "cr":
+      {
+        // This is used by environments.
+        throw new Error("KaTeX-a11y: cr not implemented yet");
+      }
+
+    case "underline":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          a11yStrings.push("start underline");
+          buildA11yStrings(tree.body, a11yStrings, atomType);
+          a11yStrings.push("end underline");
+        });
+        break;
+      }
+
+    case "xArrow":
+      {
+        throw new Error("KaTeX-a11y: xArrow not implemented yet");
+      }
+
+    case "mclass":
+      {
+        // \neq and \ne are macros so we let "htmlmathml" render the mathmal
+        // side of things and extract the text from that.
+        var _atomType = tree.mclass.slice(1); // $FlowFixMe: drop the leading "m" from the values in mclass
+
+
+        buildA11yStrings(tree.body, a11yStrings, _atomType);
+        break;
+      }
+
+    case "mathchoice":
+      {
+        // TODO: track which which style we're using, e.g. dispaly, text, etc.
+        // default to text style if even that may not be the correct style
+        buildA11yStrings(tree.text, a11yStrings, atomType);
+        break;
+      }
+
+    case "htmlmathml":
+      {
+        buildA11yStrings(tree.mathml, a11yStrings, atomType);
+        break;
+      }
+
+    case "middle":
+      {
+        buildString(tree.delim, atomType, a11yStrings);
+        break;
+      }
+
+    case "internal":
+      {
+        // internal nodes are never included in the parse tree
+        break;
+      }
+
+    case "html":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    default:
+      tree.type;
+      throw new Error("KaTeX a11y un-recognized type: " + tree.type);
+  }
+};
+
+var buildA11yStrings = function buildA11yStrings(tree, a11yStrings, atomType) {
+  if (a11yStrings === void 0) {
+    a11yStrings = [];
+  }
+
+  if (tree instanceof Array) {
+    for (var i = 0; i < tree.length; i++) {
+      buildA11yStrings(tree[i], a11yStrings, atomType);
+    }
+  } else {
+    handleObject(tree, a11yStrings, atomType);
+  }
+
+  return a11yStrings;
+};
+
+var flatten = function flatten(array) {
+  var result = [];
+  array.forEach(function (item) {
+    if (item instanceof Array) {
+      result = result.concat(flatten(item));
+    } else {
+      result.push(item);
+    }
+  });
+  return result;
+};
+
+var renderA11yString = function renderA11yString(text, settings) {
+  var tree = katex__WEBPACK_IMPORTED_MODULE_0___default.a.__parse(text, settings);
+
+  var a11yStrings = buildA11yStrings(tree, [], "normal");
+  return flatten(a11yStrings).join(", ");
+};
+
+/* harmony default export */ __webpack_exports__["default"] = (renderA11yString);
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/docs/pt/katex/contrib/render-a11y-string.min.js b/docs/pt/katex/contrib/render-a11y-string.min.js
new file mode 100644
index 0000000..5d406a6
--- /dev/null
+++ b/docs/pt/katex/contrib/render-a11y-string.min.js
@@ -0,0 +1 @@
+!function(e,r){if("object"==typeof exports&&"object"==typeof module)module.exports=r(require("katex"));else if("function"==typeof define&&define.amd)define(["katex"],r);else{var t="object"==typeof exports?r(require("katex")):r(e.katex);for(var a in t)("object"==typeof exports?exports:e)[a]=t[a]}}("undefined"!=typeof self?self:this,function(e){return function(e){var r={};function t(a){if(r[a])return r[a].exports;var o=r[a]={i:a,l:!1,exports:{}};return e[a].call(o.exports,o,o.exports,t),o.l=!0,o.exports}return t.m=e,t.c=r,t.d=function(e,r,a){t.o(e,r)||Object.defineProperty(e,r,{enumerable:!0,get:a})},t.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},t.t=function(e,r){if(1&r&&(e=t(e)),8&r)return e;if(4&r&&"object"==typeof e&&e&&e.__esModule)return e;var a=Object.create(null);if(t.r(a),Object.defineProperty(a,"default",{enumerable:!0,value:e}),2&r&&"string"!=typeof e)for(var o in e)t.d(a,o,function(r){return e[r]}.bind(null,o));return a},t.n=function(e){var r=e&&e.__esModule?function(){return e.default}:function(){return e};return t.d(r,"a",r),r},t.o=function(e,r){return Object.prototype.hasOwnProperty.call(e,r)},t.p="",t(t.s=1)}([function(r,t){r.exports=e},function(e,r,t){"use strict";t.r(r);var a=t(0),o=t.n(a),n={"(":"left parenthesis",")":"right parenthesis","[":"open bracket","]":"close bracket","\\{":"left brace","\\}":"right brace","\\lvert":"open vertical bar","\\rvert":"close vertical bar","|":"vertical bar","\\uparrow":"up arrow","\\Uparrow":"up arrow","\\downarrow":"down arrow","\\Downarrow":"down arrow","\\updownarrow":"up down arrow","\\leftarrow":"left arrow","\\Leftarrow":"left arrow","\\rightarrow":"right arrow","\\Rightarrow":"right arrow","\\langle":"open angle","\\rangle":"close angle","\\lfloor":"open floor","\\rfloor":"close floor","\\int":"integral","\\intop":"integral","\\lim":"limit","\\ln":"natural log","\\log":"log","\\sin":"sine","\\cos":"cosine","\\tan":"tangent","\\cot":"cotangent","\\sum":"sum","/":"slash",",":"comma",".":"point","-":"negative","+":"plus","~":"tilde",":":"colon","?":"question mark","'":"apostrophe","\\%":"percent"," ":"space","\\ ":"space","\\$":"dollar sign","\\angle":"angle","\\degree":"degree","\\circ":"circle","\\vec":"vector","\\triangle":"triangle","\\pi":"pi","\\prime":"prime","\\infty":"infinity","\\alpha":"alpha","\\beta":"beta","\\gamma":"gamma","\\omega":"omega","\\theta":"theta","\\sigma":"sigma","\\lambda":"lambda","\\tau":"tau","\\Delta":"delta","\\delta":"delta","\\mu":"mu","\\rho":"rho","\\nabla":"del","\\ell":"ell","\\ldots":"dots","\\hat":"hat","\\acute":"acute"},s={prime:"prime",degree:"degrees",circle:"degrees",2:"squared",3:"cubed"},i={"|":"open vertical bar",".":""},l={"|":"close vertical bar",".":""},c={"+":"plus","-":"minus","\\pm":"plus minus","\\cdot":"dot","*":"times","/":"divided by","\\times":"times","\\div":"divided by","\\circ":"circle","\\bullet":"bullet"},u={"=":"equals","\\approx":"approximately equals","\u2260":"does not equal","\\geq":"is greater than or equal to","\\ge":"is greater than or equal to","\\leq":"is less than or equal to","\\le":"is less than or equal to",">":"is greater than","<":"is less than","\\leftarrow":"left arrow","\\Leftarrow":"left arrow","\\rightarrow":"right arrow","\\Rightarrow":"right arrow",":":"colon"},p={"\\underleftarrow":"left arrow","\\underrightarrow":"right arrow","\\underleftrightarrow":"left-right arrow","\\undergroup":"group","\\underlinesegment":"line segment","\\utilde":"tilde"},d=function(e,r,t){var a;e&&(/^\d+$/.test(a="open"===r?e in i?i[e]:n[e]||e:"close"===r?e in l?l[e]:n[e]||e:"bin"===r?c[e]||e:"rel"===r?u[e]||e:n[e]||e)&&t.length>0&&/^\d+$/.test(t[t.length-1])?t[t.length-1]+=a:a&&t.push(a))},b=function(e,r){var t=[];e.push(t),r(t)},h=function(e,r,t){switch(e.type){case"accent":b(r,function(r){f(e.base,r,t),r.push("with"),d(e.label,"normal",r),r.push("on top")});break;case"accentUnder":b(r,function(r){f(e.base,r,t),r.push("with"),d(p[e.label],"normal",r),r.push("underneath")});break;case"accent-token":break;case"atom":var a=e.text;switch(e.family){case"bin":d(a,"bin",r);break;case"close":d(a,"close",r);break;case"inner":d(e.text,"inner",r);break;case"open":d(a,"open",r);break;case"punct":d(a,"punct",r);break;case"rel":d(a,"rel",r);break;default:throw e.family,new Error('"'+e.family+'" is not a valid atom type')}break;case"color":var o=e.color.replace(/katex-/,"");b(r,function(r){r.push("start color "+o),f(e.body,r,t),r.push("end color "+o)});break;case"color-token":break;case"delimsizing":e.delim&&"."!==e.delim&&d(e.delim,"normal",r);break;case"genfrac":b(r,function(r){var a=e.leftDelim,o=e.rightDelim;e.hasBarLine?(r.push("start fraction"),a&&d(a,"open",r),f(e.numer,r,t),r.push("divided by"),f(e.denom,r,t),o&&d(o,"close",r),r.push("end fraction")):(r.push("start binomial"),a&&d(a,"open",r),f(e.numer,r,t),r.push("over"),f(e.denom,r,t),o&&d(o,"close",r),r.push("end binomial"))});break;case"kern":break;case"leftright":b(r,function(r){d(e.left,"open",r),f(e.body,r,t),d(e.right,"close",r)});break;case"leftright-right":break;case"lap":f(e.body,r,t);break;case"mathord":d(e.text,"normal",r);break;case"op":var n=e.body,i=e.name;n?f(n,r,t):i&&d(i,"normal",r);break;case"op-token":d(e.text,t,r);break;case"ordgroup":f(e.body,r,t);break;case"overline":b(r,function(r){r.push("start overline"),f(e.body,r,t),r.push("end overline")});break;case"phantom":r.push("empty space");break;case"raisebox":f(e.body,r,t);break;case"rule":r.push("rectangle");break;case"sizing":f(e.body,r,t);break;case"spacing":r.push("space");break;case"styling":f(e.body,r,t);break;case"sqrt":b(r,function(r){var a=e.body,o=e.index;if(o)return"3"===m(f(o,[],t)).join(",")?(r.push("cube root of"),f(a,r,t),void r.push("end cube root")):(r.push("root"),r.push("start index"),f(o,r,t),void r.push("end index"));r.push("square root of"),f(a,r,t),r.push("end square root")});break;case"supsub":var l=e.base,c=e.sub,u=e.sup,h=!1;if(l&&(f(l,r,t),h="op"===l.type&&"\\log"===l.name),c){var y=h?"base":"subscript";b(r,function(e){e.push("start "+y),f(c,e,t),e.push("end "+y)})}u&&b(r,function(e){var r=m(f(u,[],t)).join(",");r in s?e.push(s[r]):(e.push("start superscript"),f(u,e,t),e.push("end superscript"))});break;case"text":if("\\textbf"===e.font){b(r,function(r){r.push("start bold text"),f(e.body,r,t),r.push("end bold text")});break}b(r,function(r){r.push("start text"),f(e.body,r,t),r.push("end text")});break;case"textord":d(e.text,t,r);break;case"smash":f(e.body,r,t);break;case"enclose":if(/cancel/.test(e.label)){b(r,function(r){r.push("start cancel"),f(e.body,r,t),r.push("end cancel")});break}if(/box/.test(e.label)){b(r,function(r){r.push("start box"),f(e.body,r,t),r.push("end box")});break}if(/sout/.test(e.label)){b(r,function(r){r.push("start strikeout"),f(e.body,r,t),r.push("end strikeout")});break}throw new Error("KaTeX-a11y: enclose node with "+e.label+" not supported yet");case"vphantom":throw new Error("KaTeX-a11y: vphantom not implemented yet");case"hphantom":throw new Error("KaTeX-a11y: hphantom not implemented yet");case"operatorname":f(e.body,r,t);break;case"array":throw new Error("KaTeX-a11y: array not implemented yet");case"raw":throw new Error("KaTeX-a11y: raw not implemented yet");case"size":break;case"url":throw new Error("KaTeX-a11y: url not implemented yet");case"tag":throw new Error("KaTeX-a11y: tag not implemented yet");case"verb":d("start verbatim","normal",r),d(e.body,"normal",r),d("end verbatim","normal",r);break;case"environment":throw new Error("KaTeX-a11y: environment not implemented yet");case"horizBrace":d("start "+e.label.slice(1),"normal",r),f(e.base,r,t),d("end "+e.label.slice(1),"normal",r);break;case"infix":break;case"includegraphics":throw new Error("KaTeX-a11y: includegraphics not implemented yet");case"font":f(e.body,r,t);break;case"href":throw new Error("KaTeX-a11y: href not implemented yet");case"cr":throw new Error("KaTeX-a11y: cr not implemented yet");case"underline":b(r,function(r){r.push("start underline"),f(e.body,r,t),r.push("end underline")});break;case"xArrow":throw new Error("KaTeX-a11y: xArrow not implemented yet");case"mclass":var g=e.mclass.slice(1);f(e.body,r,g);break;case"mathchoice":f(e.text,r,t);break;case"htmlmathml":f(e.mathml,r,t);break;case"middle":d(e.delim,t,r);break;case"internal":break;case"html":f(e.body,r,t);break;default:throw e.type,new Error("KaTeX a11y un-recognized type: "+e.type)}},f=function e(r,t,a){if(void 0===t&&(t=[]),r instanceof Array)for(var o=0;o<r.length;o++)e(r[o],t,a);else h(r,t,a);return t},m=function e(r){var t=[];return r.forEach(function(r){r instanceof Array?t=t.concat(e(r)):t.push(r)}),t};r.default=function(e,r){var t=o.a.__parse(e,r),a=f(t,[],"normal");return m(a).join(", ")}}]).default});
\ No newline at end of file
diff --git a/docs/pt/katex/contrib/render-a11y-string.mjs b/docs/pt/katex/contrib/render-a11y-string.mjs
new file mode 100644
index 0000000..4eabb7b
--- /dev/null
+++ b/docs/pt/katex/contrib/render-a11y-string.mjs
@@ -0,0 +1,753 @@
+import katex from '../katex.mjs';
+
+/**
+ * renderA11yString returns a readable string.
+ *
+ * In some cases the string will have the proper semantic math
+ * meaning,:
+ *   renderA11yString("\\frac{1}{2}"")
+ *   -> "start fraction, 1, divided by, 2, end fraction"
+ *
+ * However, other cases do not:
+ *   renderA11yString("f(x) = x^2")
+ *   -> "f, left parenthesis, x, right parenthesis, equals, x, squared"
+ *
+ * The commas in the string aim to increase ease of understanding
+ * when read by a screenreader.
+ */
+const stringMap = {
+  "(": "left parenthesis",
+  ")": "right parenthesis",
+  "[": "open bracket",
+  "]": "close bracket",
+  "\\{": "left brace",
+  "\\}": "right brace",
+  "\\lvert": "open vertical bar",
+  "\\rvert": "close vertical bar",
+  "|": "vertical bar",
+  "\\uparrow": "up arrow",
+  "\\Uparrow": "up arrow",
+  "\\downarrow": "down arrow",
+  "\\Downarrow": "down arrow",
+  "\\updownarrow": "up down arrow",
+  "\\leftarrow": "left arrow",
+  "\\Leftarrow": "left arrow",
+  "\\rightarrow": "right arrow",
+  "\\Rightarrow": "right arrow",
+  "\\langle": "open angle",
+  "\\rangle": "close angle",
+  "\\lfloor": "open floor",
+  "\\rfloor": "close floor",
+  "\\int": "integral",
+  "\\intop": "integral",
+  "\\lim": "limit",
+  "\\ln": "natural log",
+  "\\log": "log",
+  "\\sin": "sine",
+  "\\cos": "cosine",
+  "\\tan": "tangent",
+  "\\cot": "cotangent",
+  "\\sum": "sum",
+  "/": "slash",
+  ",": "comma",
+  ".": "point",
+  "-": "negative",
+  "+": "plus",
+  "~": "tilde",
+  ":": "colon",
+  "?": "question mark",
+  "'": "apostrophe",
+  "\\%": "percent",
+  " ": "space",
+  "\\ ": "space",
+  "\\$": "dollar sign",
+  "\\angle": "angle",
+  "\\degree": "degree",
+  "\\circ": "circle",
+  "\\vec": "vector",
+  "\\triangle": "triangle",
+  "\\pi": "pi",
+  "\\prime": "prime",
+  "\\infty": "infinity",
+  "\\alpha": "alpha",
+  "\\beta": "beta",
+  "\\gamma": "gamma",
+  "\\omega": "omega",
+  "\\theta": "theta",
+  "\\sigma": "sigma",
+  "\\lambda": "lambda",
+  "\\tau": "tau",
+  "\\Delta": "delta",
+  "\\delta": "delta",
+  "\\mu": "mu",
+  "\\rho": "rho",
+  "\\nabla": "del",
+  "\\ell": "ell",
+  "\\ldots": "dots",
+  // TODO: add entries for all accents
+  "\\hat": "hat",
+  "\\acute": "acute"
+};
+const powerMap = {
+  "prime": "prime",
+  "degree": "degrees",
+  "circle": "degrees",
+  "2": "squared",
+  "3": "cubed"
+};
+const openMap = {
+  "|": "open vertical bar",
+  ".": ""
+};
+const closeMap = {
+  "|": "close vertical bar",
+  ".": ""
+};
+const binMap = {
+  "+": "plus",
+  "-": "minus",
+  "\\pm": "plus minus",
+  "\\cdot": "dot",
+  "*": "times",
+  "/": "divided by",
+  "\\times": "times",
+  "\\div": "divided by",
+  "\\circ": "circle",
+  "\\bullet": "bullet"
+};
+const relMap = {
+  "=": "equals",
+  "\\approx": "approximately equals",
+  "≠": "does not equal",
+  "\\geq": "is greater than or equal to",
+  "\\ge": "is greater than or equal to",
+  "\\leq": "is less than or equal to",
+  "\\le": "is less than or equal to",
+  ">": "is greater than",
+  "<": "is less than",
+  "\\leftarrow": "left arrow",
+  "\\Leftarrow": "left arrow",
+  "\\rightarrow": "right arrow",
+  "\\Rightarrow": "right arrow",
+  ":": "colon"
+};
+const accentUnderMap = {
+  "\\underleftarrow": "left arrow",
+  "\\underrightarrow": "right arrow",
+  "\\underleftrightarrow": "left-right arrow",
+  "\\undergroup": "group",
+  "\\underlinesegment": "line segment",
+  "\\utilde": "tilde"
+};
+
+const buildString = (str, type, a11yStrings) => {
+  if (!str) {
+    return;
+  }
+
+  let ret;
+
+  if (type === "open") {
+    ret = str in openMap ? openMap[str] : stringMap[str] || str;
+  } else if (type === "close") {
+    ret = str in closeMap ? closeMap[str] : stringMap[str] || str;
+  } else if (type === "bin") {
+    ret = binMap[str] || str;
+  } else if (type === "rel") {
+    ret = relMap[str] || str;
+  } else {
+    ret = stringMap[str] || str;
+  } // If the text to add is a number and there is already a string
+  // in the list and the last string is a number then we should
+  // combine them into a single number
+
+
+  if (/^\d+$/.test(ret) && a11yStrings.length > 0 && // TODO(kevinb): check that the last item in a11yStrings is a string
+  // I think we might be able to drop the nested arrays, which would make
+  // this easier to type - $FlowFixMe
+  /^\d+$/.test(a11yStrings[a11yStrings.length - 1])) {
+    a11yStrings[a11yStrings.length - 1] += ret;
+  } else if (ret) {
+    a11yStrings.push(ret);
+  }
+};
+
+const buildRegion = (a11yStrings, callback) => {
+  const regionStrings = [];
+  a11yStrings.push(regionStrings);
+  callback(regionStrings);
+};
+
+const handleObject = (tree, a11yStrings, atomType) => {
+  // Everything else is assumed to be an object...
+  switch (tree.type) {
+    case "accent":
+      {
+        buildRegion(a11yStrings, a11yStrings => {
+          buildA11yStrings(tree.base, a11yStrings, atomType);
+          a11yStrings.push("with");
+          buildString(tree.label, "normal", a11yStrings);
+          a11yStrings.push("on top");
+        });
+        break;
+      }
+
+    case "accentUnder":
+      {
+        buildRegion(a11yStrings, a11yStrings => {
+          buildA11yStrings(tree.base, a11yStrings, atomType);
+          a11yStrings.push("with");
+          buildString(accentUnderMap[tree.label], "normal", a11yStrings);
+          a11yStrings.push("underneath");
+        });
+        break;
+      }
+
+    case "accent-token":
+      {
+        // Used internally by accent symbols.
+        break;
+      }
+
+    case "atom":
+      {
+        const text = tree.text;
+
+        switch (tree.family) {
+          case "bin":
+            {
+              buildString(text, "bin", a11yStrings);
+              break;
+            }
+
+          case "close":
+            {
+              buildString(text, "close", a11yStrings);
+              break;
+            }
+          // TODO(kevinb): figure out what should be done for inner
+
+          case "inner":
+            {
+              buildString(tree.text, "inner", a11yStrings);
+              break;
+            }
+
+          case "open":
+            {
+              buildString(text, "open", a11yStrings);
+              break;
+            }
+
+          case "punct":
+            {
+              buildString(text, "punct", a11yStrings);
+              break;
+            }
+
+          case "rel":
+            {
+              buildString(text, "rel", a11yStrings);
+              break;
+            }
+
+          default:
+            {
+              tree.family;
+              throw new Error(`"${tree.family}" is not a valid atom type`);
+            }
+        }
+
+        break;
+      }
+
+    case "color":
+      {
+        const color = tree.color.replace(/katex-/, "");
+        buildRegion(a11yStrings, regionStrings => {
+          regionStrings.push("start color " + color);
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          regionStrings.push("end color " + color);
+        });
+        break;
+      }
+
+    case "color-token":
+      {
+        // Used by \color, \colorbox, and \fcolorbox but not directly rendered.
+        // It's a leaf node and has no children so just break.
+        break;
+      }
+
+    case "delimsizing":
+      {
+        if (tree.delim && tree.delim !== ".") {
+          buildString(tree.delim, "normal", a11yStrings);
+        }
+
+        break;
+      }
+
+    case "genfrac":
+      {
+        buildRegion(a11yStrings, regionStrings => {
+          // genfrac can have unbalanced delimiters
+          const leftDelim = tree.leftDelim,
+                rightDelim = tree.rightDelim; // NOTE: Not sure if this is a safe assumption
+          // hasBarLine true -> fraction, false -> binomial
+
+          if (tree.hasBarLine) {
+            regionStrings.push("start fraction");
+            leftDelim && buildString(leftDelim, "open", regionStrings);
+            buildA11yStrings(tree.numer, regionStrings, atomType);
+            regionStrings.push("divided by");
+            buildA11yStrings(tree.denom, regionStrings, atomType);
+            rightDelim && buildString(rightDelim, "close", regionStrings);
+            regionStrings.push("end fraction");
+          } else {
+            regionStrings.push("start binomial");
+            leftDelim && buildString(leftDelim, "open", regionStrings);
+            buildA11yStrings(tree.numer, regionStrings, atomType);
+            regionStrings.push("over");
+            buildA11yStrings(tree.denom, regionStrings, atomType);
+            rightDelim && buildString(rightDelim, "close", regionStrings);
+            regionStrings.push("end binomial");
+          }
+        });
+        break;
+      }
+
+    case "kern":
+      {
+        // No op: we don't attempt to present kerning information
+        // to the screen reader.
+        break;
+      }
+
+    case "leftright":
+      {
+        buildRegion(a11yStrings, regionStrings => {
+          buildString(tree.left, "open", regionStrings);
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          buildString(tree.right, "close", regionStrings);
+        });
+        break;
+      }
+
+    case "leftright-right":
+      {
+        // TODO: double check that this is a no-op
+        break;
+      }
+
+    case "lap":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "mathord":
+      {
+        buildString(tree.text, "normal", a11yStrings);
+        break;
+      }
+
+    case "op":
+      {
+        const body = tree.body,
+              name = tree.name;
+
+        if (body) {
+          buildA11yStrings(body, a11yStrings, atomType);
+        } else if (name) {
+          buildString(name, "normal", a11yStrings);
+        }
+
+        break;
+      }
+
+    case "op-token":
+      {
+        // Used internally by operator symbols.
+        buildString(tree.text, atomType, a11yStrings);
+        break;
+      }
+
+    case "ordgroup":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "overline":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          a11yStrings.push("start overline");
+          buildA11yStrings(tree.body, a11yStrings, atomType);
+          a11yStrings.push("end overline");
+        });
+        break;
+      }
+
+    case "phantom":
+      {
+        a11yStrings.push("empty space");
+        break;
+      }
+
+    case "raisebox":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "rule":
+      {
+        a11yStrings.push("rectangle");
+        break;
+      }
+
+    case "sizing":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "spacing":
+      {
+        a11yStrings.push("space");
+        break;
+      }
+
+    case "styling":
+      {
+        // We ignore the styling and just pass through the contents
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "sqrt":
+      {
+        buildRegion(a11yStrings, regionStrings => {
+          const body = tree.body,
+                index = tree.index;
+
+          if (index) {
+            const indexString = flatten(buildA11yStrings(index, [], atomType)).join(",");
+
+            if (indexString === "3") {
+              regionStrings.push("cube root of");
+              buildA11yStrings(body, regionStrings, atomType);
+              regionStrings.push("end cube root");
+              return;
+            }
+
+            regionStrings.push("root");
+            regionStrings.push("start index");
+            buildA11yStrings(index, regionStrings, atomType);
+            regionStrings.push("end index");
+            return;
+          }
+
+          regionStrings.push("square root of");
+          buildA11yStrings(body, regionStrings, atomType);
+          regionStrings.push("end square root");
+        });
+        break;
+      }
+
+    case "supsub":
+      {
+        const base = tree.base,
+              sub = tree.sub,
+              sup = tree.sup;
+        let isLog = false;
+
+        if (base) {
+          buildA11yStrings(base, a11yStrings, atomType);
+          isLog = base.type === "op" && base.name === "\\log";
+        }
+
+        if (sub) {
+          const regionName = isLog ? "base" : "subscript";
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push(`start ${regionName}`);
+            buildA11yStrings(sub, regionStrings, atomType);
+            regionStrings.push(`end ${regionName}`);
+          });
+        }
+
+        if (sup) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            const supString = flatten(buildA11yStrings(sup, [], atomType)).join(",");
+
+            if (supString in powerMap) {
+              regionStrings.push(powerMap[supString]);
+              return;
+            }
+
+            regionStrings.push("start superscript");
+            buildA11yStrings(sup, regionStrings, atomType);
+            regionStrings.push("end superscript");
+          });
+        }
+
+        break;
+      }
+
+    case "text":
+      {
+        // TODO: handle other fonts
+        if (tree.font === "\\textbf") {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start bold text");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end bold text");
+          });
+          break;
+        }
+
+        buildRegion(a11yStrings, function (regionStrings) {
+          regionStrings.push("start text");
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          regionStrings.push("end text");
+        });
+        break;
+      }
+
+    case "textord":
+      {
+        buildString(tree.text, atomType, a11yStrings);
+        break;
+      }
+
+    case "smash":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "enclose":
+      {
+        // TODO: create a map for these.
+        // TODO: differentiate between a body with a single atom, e.g.
+        // "cancel a" instead of "start cancel, a, end cancel"
+        if (/cancel/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start cancel");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end cancel");
+          });
+          break;
+        } else if (/box/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start box");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end box");
+          });
+          break;
+        } else if (/sout/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start strikeout");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end strikeout");
+          });
+          break;
+        }
+
+        throw new Error(`KaTeX-a11y: enclose node with ${tree.label} not supported yet`);
+      }
+
+    case "vphantom":
+      {
+        throw new Error("KaTeX-a11y: vphantom not implemented yet");
+      }
+
+    case "hphantom":
+      {
+        throw new Error("KaTeX-a11y: hphantom not implemented yet");
+      }
+
+    case "operatorname":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "array":
+      {
+        throw new Error("KaTeX-a11y: array not implemented yet");
+      }
+
+    case "raw":
+      {
+        throw new Error("KaTeX-a11y: raw not implemented yet");
+      }
+
+    case "size":
+      {
+        // Although there are nodes of type "size" in the parse tree, they have
+        // no semantic meaning and should be ignored.
+        break;
+      }
+
+    case "url":
+      {
+        throw new Error("KaTeX-a11y: url not implemented yet");
+      }
+
+    case "tag":
+      {
+        throw new Error("KaTeX-a11y: tag not implemented yet");
+      }
+
+    case "verb":
+      {
+        buildString(`start verbatim`, "normal", a11yStrings);
+        buildString(tree.body, "normal", a11yStrings);
+        buildString(`end verbatim`, "normal", a11yStrings);
+        break;
+      }
+
+    case "environment":
+      {
+        throw new Error("KaTeX-a11y: environment not implemented yet");
+      }
+
+    case "horizBrace":
+      {
+        buildString(`start ${tree.label.slice(1)}`, "normal", a11yStrings);
+        buildA11yStrings(tree.base, a11yStrings, atomType);
+        buildString(`end ${tree.label.slice(1)}`, "normal", a11yStrings);
+        break;
+      }
+
+    case "infix":
+      {
+        // All infix nodes are replace with other nodes.
+        break;
+      }
+
+    case "includegraphics":
+      {
+        throw new Error("KaTeX-a11y: includegraphics not implemented yet");
+      }
+
+    case "font":
+      {
+        // TODO: callout the start/end of specific fonts
+        // TODO: map \BBb{N} to "the naturals" or something like that
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "href":
+      {
+        throw new Error("KaTeX-a11y: href not implemented yet");
+      }
+
+    case "cr":
+      {
+        // This is used by environments.
+        throw new Error("KaTeX-a11y: cr not implemented yet");
+      }
+
+    case "underline":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          a11yStrings.push("start underline");
+          buildA11yStrings(tree.body, a11yStrings, atomType);
+          a11yStrings.push("end underline");
+        });
+        break;
+      }
+
+    case "xArrow":
+      {
+        throw new Error("KaTeX-a11y: xArrow not implemented yet");
+      }
+
+    case "mclass":
+      {
+        // \neq and \ne are macros so we let "htmlmathml" render the mathmal
+        // side of things and extract the text from that.
+        const atomType = tree.mclass.slice(1); // $FlowFixMe: drop the leading "m" from the values in mclass
+
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "mathchoice":
+      {
+        // TODO: track which which style we're using, e.g. dispaly, text, etc.
+        // default to text style if even that may not be the correct style
+        buildA11yStrings(tree.text, a11yStrings, atomType);
+        break;
+      }
+
+    case "htmlmathml":
+      {
+        buildA11yStrings(tree.mathml, a11yStrings, atomType);
+        break;
+      }
+
+    case "middle":
+      {
+        buildString(tree.delim, atomType, a11yStrings);
+        break;
+      }
+
+    case "internal":
+      {
+        // internal nodes are never included in the parse tree
+        break;
+      }
+
+    case "html":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    default:
+      tree.type;
+      throw new Error("KaTeX a11y un-recognized type: " + tree.type);
+  }
+};
+
+const buildA11yStrings = function buildA11yStrings(tree, a11yStrings, atomType) {
+  if (a11yStrings === void 0) {
+    a11yStrings = [];
+  }
+
+  if (tree instanceof Array) {
+    for (let i = 0; i < tree.length; i++) {
+      buildA11yStrings(tree[i], a11yStrings, atomType);
+    }
+  } else {
+    handleObject(tree, a11yStrings, atomType);
+  }
+
+  return a11yStrings;
+};
+
+const flatten = function flatten(array) {
+  let result = [];
+  array.forEach(function (item) {
+    if (item instanceof Array) {
+      result = result.concat(flatten(item));
+    } else {
+      result.push(item);
+    }
+  });
+  return result;
+};
+
+const renderA11yString = function renderA11yString(text, settings) {
+  const tree = katex.__parse(text, settings);
+
+  const a11yStrings = buildA11yStrings(tree, [], "normal");
+  return flatten(a11yStrings).join(", ");
+};
+
+export default renderA11yString;
diff --git a/docs/pt/katex/fonts/KaTeX_AMS-Regular.ttf b/docs/pt/katex/fonts/KaTeX_AMS-Regular.ttf
new file mode 100644
index 0000000..737cf8e
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_AMS-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_AMS-Regular.woff b/docs/pt/katex/fonts/KaTeX_AMS-Regular.woff
new file mode 100644
index 0000000..38378bf
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_AMS-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_AMS-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_AMS-Regular.woff2
new file mode 100644
index 0000000..a4d1ba6
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_AMS-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Caligraphic-Bold.ttf b/docs/pt/katex/fonts/KaTeX_Caligraphic-Bold.ttf
new file mode 100644
index 0000000..04d28ab
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Caligraphic-Bold.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Caligraphic-Bold.woff b/docs/pt/katex/fonts/KaTeX_Caligraphic-Bold.woff
new file mode 100644
index 0000000..a01ce90
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Caligraphic-Bold.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Caligraphic-Bold.woff2 b/docs/pt/katex/fonts/KaTeX_Caligraphic-Bold.woff2
new file mode 100644
index 0000000..3792727
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Caligraphic-Bold.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Caligraphic-Regular.ttf b/docs/pt/katex/fonts/KaTeX_Caligraphic-Regular.ttf
new file mode 100644
index 0000000..b2ce555
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Caligraphic-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Caligraphic-Regular.woff b/docs/pt/katex/fonts/KaTeX_Caligraphic-Regular.woff
new file mode 100644
index 0000000..bc169b7
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Caligraphic-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Caligraphic-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_Caligraphic-Regular.woff2
new file mode 100644
index 0000000..f1e38bb
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Caligraphic-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Fraktur-Bold.ttf b/docs/pt/katex/fonts/KaTeX_Fraktur-Bold.ttf
new file mode 100644
index 0000000..c42d169
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Fraktur-Bold.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Fraktur-Bold.woff b/docs/pt/katex/fonts/KaTeX_Fraktur-Bold.woff
new file mode 100644
index 0000000..f30b54b
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Fraktur-Bold.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Fraktur-Bold.woff2 b/docs/pt/katex/fonts/KaTeX_Fraktur-Bold.woff2
new file mode 100644
index 0000000..b7a8359
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Fraktur-Bold.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Fraktur-Regular.ttf b/docs/pt/katex/fonts/KaTeX_Fraktur-Regular.ttf
new file mode 100644
index 0000000..4133228
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Fraktur-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Fraktur-Regular.woff b/docs/pt/katex/fonts/KaTeX_Fraktur-Regular.woff
new file mode 100644
index 0000000..5af51de
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Fraktur-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Fraktur-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_Fraktur-Regular.woff2
new file mode 100644
index 0000000..3874f93
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Fraktur-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Main-Bold.ttf b/docs/pt/katex/fonts/KaTeX_Main-Bold.ttf
new file mode 100644
index 0000000..14390e0
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-Bold.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Main-Bold.woff b/docs/pt/katex/fonts/KaTeX_Main-Bold.woff
new file mode 100644
index 0000000..33b4199
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-Bold.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Main-Bold.woff2 b/docs/pt/katex/fonts/KaTeX_Main-Bold.woff2
new file mode 100644
index 0000000..f9b71cb
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-Bold.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Main-BoldItalic.ttf b/docs/pt/katex/fonts/KaTeX_Main-BoldItalic.ttf
new file mode 100644
index 0000000..ad0761f
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-BoldItalic.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Main-BoldItalic.woff b/docs/pt/katex/fonts/KaTeX_Main-BoldItalic.woff
new file mode 100644
index 0000000..115af4f
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-BoldItalic.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Main-BoldItalic.woff2 b/docs/pt/katex/fonts/KaTeX_Main-BoldItalic.woff2
new file mode 100644
index 0000000..5c500c2
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-BoldItalic.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Main-Italic.ttf b/docs/pt/katex/fonts/KaTeX_Main-Italic.ttf
new file mode 100644
index 0000000..fc8625c
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-Italic.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Main-Italic.woff b/docs/pt/katex/fonts/KaTeX_Main-Italic.woff
new file mode 100644
index 0000000..2d3087a
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-Italic.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Main-Italic.woff2 b/docs/pt/katex/fonts/KaTeX_Main-Italic.woff2
new file mode 100644
index 0000000..08510d8
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-Italic.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Main-Regular.ttf b/docs/pt/katex/fonts/KaTeX_Main-Regular.ttf
new file mode 100644
index 0000000..5115a04
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Main-Regular.woff b/docs/pt/katex/fonts/KaTeX_Main-Regular.woff
new file mode 100644
index 0000000..42b74ab
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Main-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_Main-Regular.woff2
new file mode 100644
index 0000000..18647fa
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Main-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Math-BoldItalic.ttf b/docs/pt/katex/fonts/KaTeX_Math-BoldItalic.ttf
new file mode 100644
index 0000000..326b523
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Math-BoldItalic.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Math-BoldItalic.woff b/docs/pt/katex/fonts/KaTeX_Math-BoldItalic.woff
new file mode 100644
index 0000000..5b4041a
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Math-BoldItalic.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Math-BoldItalic.woff2 b/docs/pt/katex/fonts/KaTeX_Math-BoldItalic.woff2
new file mode 100644
index 0000000..ba55276
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Math-BoldItalic.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Math-Italic.ttf b/docs/pt/katex/fonts/KaTeX_Math-Italic.ttf
new file mode 100644
index 0000000..f148fce
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Math-Italic.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Math-Italic.woff b/docs/pt/katex/fonts/KaTeX_Math-Italic.woff
new file mode 100644
index 0000000..31d0038
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Math-Italic.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Math-Italic.woff2 b/docs/pt/katex/fonts/KaTeX_Math-Italic.woff2
new file mode 100644
index 0000000..9871ab6
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Math-Italic.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_SansSerif-Bold.ttf b/docs/pt/katex/fonts/KaTeX_SansSerif-Bold.ttf
new file mode 100644
index 0000000..dce35c8
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_SansSerif-Bold.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_SansSerif-Bold.woff b/docs/pt/katex/fonts/KaTeX_SansSerif-Bold.woff
new file mode 100644
index 0000000..992cb3d
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_SansSerif-Bold.woff
diff --git a/docs/pt/katex/fonts/KaTeX_SansSerif-Bold.woff2 b/docs/pt/katex/fonts/KaTeX_SansSerif-Bold.woff2
new file mode 100644
index 0000000..6dd1038
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_SansSerif-Bold.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_SansSerif-Italic.ttf b/docs/pt/katex/fonts/KaTeX_SansSerif-Italic.ttf
new file mode 100644
index 0000000..a3eb86c
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_SansSerif-Italic.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_SansSerif-Italic.woff b/docs/pt/katex/fonts/KaTeX_SansSerif-Italic.woff
new file mode 100644
index 0000000..f4fa252
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_SansSerif-Italic.woff
diff --git a/docs/pt/katex/fonts/KaTeX_SansSerif-Italic.woff2 b/docs/pt/katex/fonts/KaTeX_SansSerif-Italic.woff2
new file mode 100644
index 0000000..9f2501a
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_SansSerif-Italic.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_SansSerif-Regular.ttf b/docs/pt/katex/fonts/KaTeX_SansSerif-Regular.ttf
new file mode 100644
index 0000000..3be73ce
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_SansSerif-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_SansSerif-Regular.woff b/docs/pt/katex/fonts/KaTeX_SansSerif-Regular.woff
new file mode 100644
index 0000000..ec283f4
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_SansSerif-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_SansSerif-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_SansSerif-Regular.woff2
new file mode 100644
index 0000000..e46094f
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_SansSerif-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Script-Regular.ttf b/docs/pt/katex/fonts/KaTeX_Script-Regular.ttf
new file mode 100644
index 0000000..40c8a99
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Script-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Script-Regular.woff b/docs/pt/katex/fonts/KaTeX_Script-Regular.woff
new file mode 100644
index 0000000..4eafae7
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Script-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Script-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_Script-Regular.woff2
new file mode 100644
index 0000000..69b1754
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Script-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Size1-Regular.ttf b/docs/pt/katex/fonts/KaTeX_Size1-Regular.ttf
new file mode 100644
index 0000000..f0aff83
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size1-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Size1-Regular.woff b/docs/pt/katex/fonts/KaTeX_Size1-Regular.woff
new file mode 100644
index 0000000..0358ee4
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size1-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Size1-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_Size1-Regular.woff2
new file mode 100644
index 0000000..f951ed0
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size1-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Size2-Regular.ttf b/docs/pt/katex/fonts/KaTeX_Size2-Regular.ttf
new file mode 100644
index 0000000..4f72f16
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size2-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Size2-Regular.woff b/docs/pt/katex/fonts/KaTeX_Size2-Regular.woff
new file mode 100644
index 0000000..8a053d2
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size2-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Size2-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_Size2-Regular.woff2
new file mode 100644
index 0000000..181d962
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size2-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Size3-Regular.ttf b/docs/pt/katex/fonts/KaTeX_Size3-Regular.ttf
new file mode 100644
index 0000000..56d2dc6
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size3-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Size3-Regular.woff b/docs/pt/katex/fonts/KaTeX_Size3-Regular.woff
new file mode 100644
index 0000000..0ec99ad
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size3-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Size3-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_Size3-Regular.woff2
new file mode 100644
index 0000000..c2985cd
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size3-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Size4-Regular.ttf b/docs/pt/katex/fonts/KaTeX_Size4-Regular.ttf
new file mode 100644
index 0000000..baf0209
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size4-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Size4-Regular.woff b/docs/pt/katex/fonts/KaTeX_Size4-Regular.woff
new file mode 100644
index 0000000..ff67319
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size4-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Size4-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_Size4-Regular.woff2
new file mode 100644
index 0000000..a4e810d
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Size4-Regular.woff2
diff --git a/docs/pt/katex/fonts/KaTeX_Typewriter-Regular.ttf b/docs/pt/katex/fonts/KaTeX_Typewriter-Regular.ttf
new file mode 100644
index 0000000..e66c218
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Typewriter-Regular.ttf
diff --git a/docs/pt/katex/fonts/KaTeX_Typewriter-Regular.woff b/docs/pt/katex/fonts/KaTeX_Typewriter-Regular.woff
new file mode 100644
index 0000000..c66d149
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Typewriter-Regular.woff
diff --git a/docs/pt/katex/fonts/KaTeX_Typewriter-Regular.woff2 b/docs/pt/katex/fonts/KaTeX_Typewriter-Regular.woff2
new file mode 100644
index 0000000..e5bf2ce
--- /dev/null
+++ b/docs/pt/katex/fonts/KaTeX_Typewriter-Regular.woff2
diff --git a/docs/pt/katex/katex.css b/docs/pt/katex/katex.css
new file mode 100644
index 0000000..ba88758
--- /dev/null
+++ b/docs/pt/katex/katex.css
@@ -0,0 +1,1035 @@
+/* stylelint-disable font-family-no-missing-generic-family-keyword */
+@font-face {
+  font-family: 'KaTeX_AMS';
+  src: url(fonts/KaTeX_AMS-Regular.woff2) format('woff2'), url(fonts/KaTeX_AMS-Regular.woff) format('woff'), url(fonts/KaTeX_AMS-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Caligraphic';
+  src: url(fonts/KaTeX_Caligraphic-Bold.woff2) format('woff2'), url(fonts/KaTeX_Caligraphic-Bold.woff) format('woff'), url(fonts/KaTeX_Caligraphic-Bold.ttf) format('truetype');
+  font-weight: bold;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Caligraphic';
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+}
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+  font-size: 0.66666667em;
+}
+.katex .sizing.reset-size7.size5,
+.katex .fontsize-ensurer.reset-size7.size5 {
+  font-size: 0.75em;
+}
+.katex .sizing.reset-size7.size6,
+.katex .fontsize-ensurer.reset-size7.size6 {
+  font-size: 0.83333333em;
+}
+.katex .sizing.reset-size7.size7,
+.katex .fontsize-ensurer.reset-size7.size7 {
+  font-size: 1em;
+}
+.katex .sizing.reset-size7.size8,
+.katex .fontsize-ensurer.reset-size7.size8 {
+  font-size: 1.2em;
+}
+.katex .sizing.reset-size7.size9,
+.katex .fontsize-ensurer.reset-size7.size9 {
+  font-size: 1.44em;
+}
+.katex .sizing.reset-size7.size10,
+.katex .fontsize-ensurer.reset-size7.size10 {
+  font-size: 1.72833333em;
+}
+.katex .sizing.reset-size7.size11,
+.katex .fontsize-ensurer.reset-size7.size11 {
+  font-size: 2.07333333em;
+}
+.katex .sizing.reset-size8.size1,
+.katex .fontsize-ensurer.reset-size8.size1 {
+  font-size: 0.34722222em;
+}
+.katex .sizing.reset-size8.size2,
+.katex .fontsize-ensurer.reset-size8.size2 {
+  font-size: 0.41666667em;
+}
+.katex .sizing.reset-size8.size3,
+.katex .fontsize-ensurer.reset-size8.size3 {
+  font-size: 0.48611111em;
+}
+.katex .sizing.reset-size8.size4,
+.katex .fontsize-ensurer.reset-size8.size4 {
+  font-size: 0.55555556em;
+}
+.katex .sizing.reset-size8.size5,
+.katex .fontsize-ensurer.reset-size8.size5 {
+  font-size: 0.625em;
+}
+.katex .sizing.reset-size8.size6,
+.katex .fontsize-ensurer.reset-size8.size6 {
+  font-size: 0.69444444em;
+}
+.katex .sizing.reset-size8.size7,
+.katex .fontsize-ensurer.reset-size8.size7 {
+  font-size: 0.83333333em;
+}
+.katex .sizing.reset-size8.size8,
+.katex .fontsize-ensurer.reset-size8.size8 {
+  font-size: 1em;
+}
+.katex .sizing.reset-size8.size9,
+.katex .fontsize-ensurer.reset-size8.size9 {
+  font-size: 1.2em;
+}
+.katex .sizing.reset-size8.size10,
+.katex .fontsize-ensurer.reset-size8.size10 {
+  font-size: 1.44027778em;
+}
+.katex .sizing.reset-size8.size11,
+.katex .fontsize-ensurer.reset-size8.size11 {
+  font-size: 1.72777778em;
+}
+.katex .sizing.reset-size9.size1,
+.katex .fontsize-ensurer.reset-size9.size1 {
+  font-size: 0.28935185em;
+}
+.katex .sizing.reset-size9.size2,
+.katex .fontsize-ensurer.reset-size9.size2 {
+  font-size: 0.34722222em;
+}
+.katex .sizing.reset-size9.size3,
+.katex .fontsize-ensurer.reset-size9.size3 {
+  font-size: 0.40509259em;
+}
+.katex .sizing.reset-size9.size4,
+.katex .fontsize-ensurer.reset-size9.size4 {
+  font-size: 0.46296296em;
+}
+.katex .sizing.reset-size9.size5,
+.katex .fontsize-ensurer.reset-size9.size5 {
+  font-size: 0.52083333em;
+}
+.katex .sizing.reset-size9.size6,
+.katex .fontsize-ensurer.reset-size9.size6 {
+  font-size: 0.5787037em;
+}
+.katex .sizing.reset-size9.size7,
+.katex .fontsize-ensurer.reset-size9.size7 {
+  font-size: 0.69444444em;
+}
+.katex .sizing.reset-size9.size8,
+.katex .fontsize-ensurer.reset-size9.size8 {
+  font-size: 0.83333333em;
+}
+.katex .sizing.reset-size9.size9,
+.katex .fontsize-ensurer.reset-size9.size9 {
+  font-size: 1em;
+}
+.katex .sizing.reset-size9.size10,
+.katex .fontsize-ensurer.reset-size9.size10 {
+  font-size: 1.20023148em;
+}
+.katex .sizing.reset-size9.size11,
+.katex .fontsize-ensurer.reset-size9.size11 {
+  font-size: 1.43981481em;
+}
+.katex .sizing.reset-size10.size1,
+.katex .fontsize-ensurer.reset-size10.size1 {
+  font-size: 0.24108004em;
+}
+.katex .sizing.reset-size10.size2,
+.katex .fontsize-ensurer.reset-size10.size2 {
+  font-size: 0.28929605em;
+}
+.katex .sizing.reset-size10.size3,
+.katex .fontsize-ensurer.reset-size10.size3 {
+  font-size: 0.33751205em;
+}
+.katex .sizing.reset-size10.size4,
+.katex .fontsize-ensurer.reset-size10.size4 {
+  font-size: 0.38572806em;
+}
+.katex .sizing.reset-size10.size5,
+.katex .fontsize-ensurer.reset-size10.size5 {
+  font-size: 0.43394407em;
+}
+.katex .sizing.reset-size10.size6,
+.katex .fontsize-ensurer.reset-size10.size6 {
+  font-size: 0.48216008em;
+}
+.katex .sizing.reset-size10.size7,
+.katex .fontsize-ensurer.reset-size10.size7 {
+  font-size: 0.57859209em;
+}
+.katex .sizing.reset-size10.size8,
+.katex .fontsize-ensurer.reset-size10.size8 {
+  font-size: 0.69431051em;
+}
+.katex .sizing.reset-size10.size9,
+.katex .fontsize-ensurer.reset-size10.size9 {
+  font-size: 0.83317261em;
+}
+.katex .sizing.reset-size10.size10,
+.katex .fontsize-ensurer.reset-size10.size10 {
+  font-size: 1em;
+}
+.katex .sizing.reset-size10.size11,
+.katex .fontsize-ensurer.reset-size10.size11 {
+  font-size: 1.19961427em;
+}
+.katex .sizing.reset-size11.size1,
+.katex .fontsize-ensurer.reset-size11.size1 {
+  font-size: 0.20096463em;
+}
+.katex .sizing.reset-size11.size2,
+.katex .fontsize-ensurer.reset-size11.size2 {
+  font-size: 0.24115756em;
+}
+.katex .sizing.reset-size11.size3,
+.katex .fontsize-ensurer.reset-size11.size3 {
+  font-size: 0.28135048em;
+}
+.katex .sizing.reset-size11.size4,
+.katex .fontsize-ensurer.reset-size11.size4 {
+  font-size: 0.32154341em;
+}
+.katex .sizing.reset-size11.size5,
+.katex .fontsize-ensurer.reset-size11.size5 {
+  font-size: 0.36173633em;
+}
+.katex .sizing.reset-size11.size6,
+.katex .fontsize-ensurer.reset-size11.size6 {
+  font-size: 0.40192926em;
+}
+.katex .sizing.reset-size11.size7,
+.katex .fontsize-ensurer.reset-size11.size7 {
+  font-size: 0.48231511em;
+}
+.katex .sizing.reset-size11.size8,
+.katex .fontsize-ensurer.reset-size11.size8 {
+  font-size: 0.57877814em;
+}
+.katex .sizing.reset-size11.size9,
+.katex .fontsize-ensurer.reset-size11.size9 {
+  font-size: 0.69453376em;
+}
+.katex .sizing.reset-size11.size10,
+.katex .fontsize-ensurer.reset-size11.size10 {
+  font-size: 0.83360129em;
+}
+.katex .sizing.reset-size11.size11,
+.katex .fontsize-ensurer.reset-size11.size11 {
+  font-size: 1em;
+}
+.katex .delimsizing.size1 {
+  font-family: KaTeX_Size1;
+}
+.katex .delimsizing.size2 {
+  font-family: KaTeX_Size2;
+}
+.katex .delimsizing.size3 {
+  font-family: KaTeX_Size3;
+}
+.katex .delimsizing.size4 {
+  font-family: KaTeX_Size4;
+}
+.katex .delimsizing.mult .delim-size1 > span {
+  font-family: KaTeX_Size1;
+}
+.katex .delimsizing.mult .delim-size4 > span {
+  font-family: KaTeX_Size4;
+}
+.katex .nulldelimiter {
+  display: inline-block;
+  width: 0.12em;
+}
+.katex .delimcenter {
+  position: relative;
+}
+.katex .op-symbol {
+  position: relative;
+}
+.katex .op-symbol.small-op {
+  font-family: KaTeX_Size1;
+}
+.katex .op-symbol.large-op {
+  font-family: KaTeX_Size2;
+}
+.katex .op-limits > .vlist-t {
+  text-align: center;
+}
+.katex .accent > .vlist-t {
+  text-align: center;
+}
+.katex .accent .accent-body {
+  position: relative;
+}
+.katex .accent .accent-body:not(.accent-full) {
+  width: 0;
+}
+.katex .overlay {
+  display: block;
+}
+.katex .mtable .vertical-separator {
+  display: inline-block;
+  min-width: 1px;
+}
+.katex .mtable .arraycolsep {
+  display: inline-block;
+}
+.katex .mtable .col-align-c > .vlist-t {
+  text-align: center;
+}
+.katex .mtable .col-align-l > .vlist-t {
+  text-align: left;
+}
+.katex .mtable .col-align-r > .vlist-t {
+  text-align: right;
+}
+.katex .svg-align {
+  text-align: left;
+}
+.katex svg {
+  display: block;
+  position: absolute;
+  width: 100%;
+  height: inherit;
+  fill: currentColor;
+  stroke: currentColor;
+  fill-rule: nonzero;
+  fill-opacity: 1;
+  stroke-width: 1;
+  stroke-linecap: butt;
+  stroke-linejoin: miter;
+  stroke-miterlimit: 4;
+  stroke-dasharray: none;
+  stroke-dashoffset: 0;
+  stroke-opacity: 1;
+}
+.katex svg path {
+  stroke: none;
+}
+.katex img {
+  border-style: none;
+  min-width: 0;
+  min-height: 0;
+  max-width: none;
+  max-height: none;
+}
+.katex .stretchy {
+  width: 100%;
+  display: block;
+  position: relative;
+  overflow: hidden;
+}
+.katex .stretchy::before,
+.katex .stretchy::after {
+  content: "";
+}
+.katex .hide-tail {
+  width: 100%;
+  position: relative;
+  overflow: hidden;
+}
+.katex .halfarrow-left {
+  position: absolute;
+  left: 0;
+  width: 50.2%;
+  overflow: hidden;
+}
+.katex .halfarrow-right {
+  position: absolute;
+  right: 0;
+  width: 50.2%;
+  overflow: hidden;
+}
+.katex .brace-left {
+  position: absolute;
+  left: 0;
+  width: 25.1%;
+  overflow: hidden;
+}
+.katex .brace-center {
+  position: absolute;
+  left: 25%;
+  width: 50%;
+  overflow: hidden;
+}
+.katex .brace-right {
+  position: absolute;
+  right: 0;
+  width: 25.1%;
+  overflow: hidden;
+}
+.katex .x-arrow-pad {
+  padding: 0 0.5em;
+}
+.katex .x-arrow,
+.katex .mover,
+.katex .munder {
+  text-align: center;
+}
+.katex .boxpad {
+  padding: 0 0.3em 0 0.3em;
+}
+.katex .fbox,
+.katex .fcolorbox {
+  box-sizing: border-box;
+  border: 0.04em solid;
+}
+.katex .cancel-pad {
+  padding: 0 0.2em 0 0.2em;
+}
+.katex .cancel-lap {
+  margin-left: -0.2em;
+  margin-right: -0.2em;
+}
+.katex .sout {
+  border-bottom-style: solid;
+  border-bottom-width: 0.08em;
+}
+.katex-display {
+  display: block;
+  margin: 1em 0;
+  text-align: center;
+}
+.katex-display > .katex {
+  display: block;
+  text-align: center;
+  white-space: nowrap;
+}
+.katex-display > .katex > .katex-html {
+  display: block;
+  position: relative;
+}
+.katex-display > .katex > .katex-html > .tag {
+  position: absolute;
+  right: 0;
+}
+.katex-display.leqno > .katex > .katex-html > .tag {
+  left: 0;
+  right: auto;
+}
+.katex-display.fleqn > .katex {
+  text-align: left;
+  padding-left: 2em;
+}
+
diff --git a/docs/pt/katex/katex.js b/docs/pt/katex/katex.js
new file mode 100644
index 0000000..37fb0fa
--- /dev/null
+++ b/docs/pt/katex/katex.js
@@ -0,0 +1,17308 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory();
+	else if(typeof define === 'function' && define.amd)
+		define([], factory);
+	else if(typeof exports === 'object')
+		exports["katex"] = factory();
+	else
+		root["katex"] = factory();
+})((typeof self !== 'undefined' ? self : this), function() {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
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+/******/ 		// Create a new module (and put it into the cache)
+/******/ 		var module = installedModules[moduleId] = {
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+/******/ 			l: false,
+/******/ 			exports: {}
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+/******/
+/******/ 		// Execute the module function
+/******/ 		modules[moduleId].call(module.exports, module, module.exports, __webpack_require__);
+/******/
+/******/ 		// Flag the module as loaded
+/******/ 		module.l = true;
+/******/
+/******/ 		// Return the exports of the module
+/******/ 		return module.exports;
+/******/ 	}
+/******/
+/******/
+/******/ 	// expose the modules object (__webpack_modules__)
+/******/ 	__webpack_require__.m = modules;
+/******/
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+/******/
+/******/ 	// define getter function for harmony exports
+/******/ 	__webpack_require__.d = function(exports, name, getter) {
+/******/ 		if(!__webpack_require__.o(exports, name)) {
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+/******/ 	};
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+/******/ 	__webpack_require__.r = function(exports) {
+/******/ 		if(typeof Symbol !== 'undefined' && Symbol.toStringTag) {
+/******/ 			Object.defineProperty(exports, Symbol.toStringTag, { value: 'Module' });
+/******/ 		}
+/******/ 		Object.defineProperty(exports, '__esModule', { value: true });
+/******/ 	};
+/******/
+/******/ 	// create a fake namespace object
+/******/ 	// mode & 1: value is a module id, require it
+/******/ 	// mode & 2: merge all properties of value into the ns
+/******/ 	// mode & 4: return value when already ns object
+/******/ 	// mode & 8|1: behave like require
+/******/ 	__webpack_require__.t = function(value, mode) {
+/******/ 		if(mode & 1) value = __webpack_require__(value);
+/******/ 		if(mode & 8) return value;
+/******/ 		if((mode & 4) && typeof value === 'object' && value && value.__esModule) return value;
+/******/ 		var ns = Object.create(null);
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+/******/ 		Object.defineProperty(ns, 'default', { enumerable: true, value: value });
+/******/ 		if(mode & 2 && typeof value != 'string') for(var key in value) __webpack_require__.d(ns, key, function(key) { return value[key]; }.bind(null, key));
+/******/ 		return ns;
+/******/ 	};
+/******/
+/******/ 	// getDefaultExport function for compatibility with non-harmony modules
+/******/ 	__webpack_require__.n = function(module) {
+/******/ 		var getter = module && module.__esModule ?
+/******/ 			function getDefault() { return module['default']; } :
+/******/ 			function getModuleExports() { return module; };
+/******/ 		__webpack_require__.d(getter, 'a', getter);
+/******/ 		return getter;
+/******/ 	};
+/******/
+/******/ 	// Object.prototype.hasOwnProperty.call
+/******/ 	__webpack_require__.o = function(object, property) { return Object.prototype.hasOwnProperty.call(object, property); };
+/******/
+/******/ 	// __webpack_public_path__
+/******/ 	__webpack_require__.p = "";
+/******/
+/******/
+/******/ 	// Load entry module and return exports
+/******/ 	return __webpack_require__(__webpack_require__.s = 1);
+/******/ })
+/************************************************************************/
+/******/ ([
+/* 0 */
+/***/ (function(module, exports, __webpack_require__) {
+
+// extracted by mini-css-extract-plugin
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+
+// EXTERNAL MODULE: ./src/katex.less
+var katex = __webpack_require__(0);
+
+// CONCATENATED MODULE: ./src/SourceLocation.js
+/**
+ * Lexing or parsing positional information for error reporting.
+ * This object is immutable.
+ */
+var SourceLocation =
+/*#__PURE__*/
+function () {
+  // The + prefix indicates that these fields aren't writeable
+  // Lexer holding the input string.
+  // Start offset, zero-based inclusive.
+  // End offset, zero-based exclusive.
+  function SourceLocation(lexer, start, end) {
+    this.lexer = void 0;
+    this.start = void 0;
+    this.end = void 0;
+    this.lexer = lexer;
+    this.start = start;
+    this.end = end;
+  }
+  /**
+   * Merges two `SourceLocation`s from location providers, given they are
+   * provided in order of appearance.
+   * - Returns the first one's location if only the first is provided.
+   * - Returns a merged range of the first and the last if both are provided
+   *   and their lexers match.
+   * - Otherwise, returns null.
+   */
+
+
+  SourceLocation.range = function range(first, second) {
+    if (!second) {
+      return first && first.loc;
+    } else if (!first || !first.loc || !second.loc || first.loc.lexer !== second.loc.lexer) {
+      return null;
+    } else {
+      return new SourceLocation(first.loc.lexer, first.loc.start, second.loc.end);
+    }
+  };
+
+  return SourceLocation;
+}();
+
+
+// CONCATENATED MODULE: ./src/Token.js
+
+/**
+ * Interface required to break circular dependency between Token, Lexer, and
+ * ParseError.
+ */
+
+/**
+ * The resulting token returned from `lex`.
+ *
+ * It consists of the token text plus some position information.
+ * The position information is essentially a range in an input string,
+ * but instead of referencing the bare input string, we refer to the lexer.
+ * That way it is possible to attach extra metadata to the input string,
+ * like for example a file name or similar.
+ *
+ * The position information is optional, so it is OK to construct synthetic
+ * tokens if appropriate. Not providing available position information may
+ * lead to degraded error reporting, though.
+ */
+var Token_Token =
+/*#__PURE__*/
+function () {
+  // don't expand the token
+  // used in \noexpand
+  function Token(text, // the text of this token
+  loc) {
+    this.text = void 0;
+    this.loc = void 0;
+    this.noexpand = void 0;
+    this.treatAsRelax = void 0;
+    this.text = text;
+    this.loc = loc;
+  }
+  /**
+   * Given a pair of tokens (this and endToken), compute a `Token` encompassing
+   * the whole input range enclosed by these two.
+   */
+
+
+  var _proto = Token.prototype;
+
+  _proto.range = function range(endToken, // last token of the range, inclusive
+  text) // the text of the newly constructed token
+  {
+    return new Token(text, SourceLocation.range(this, endToken));
+  };
+
+  return Token;
+}();
+// CONCATENATED MODULE: ./src/ParseError.js
+
+
+/**
+ * This is the ParseError class, which is the main error thrown by KaTeX
+ * functions when something has gone wrong. This is used to distinguish internal
+ * errors from errors in the expression that the user provided.
+ *
+ * If possible, a caller should provide a Token or ParseNode with information
+ * about where in the source string the problem occurred.
+ */
+var ParseError = // Error position based on passed-in Token or ParseNode.
+function ParseError(message, // The error message
+token) // An object providing position information
+{
+  this.position = void 0;
+  var error = "KaTeX parse error: " + message;
+  var start;
+  var loc = token && token.loc;
+
+  if (loc && loc.start <= loc.end) {
+    // If we have the input and a position, make the error a bit fancier
+    // Get the input
+    var input = loc.lexer.input; // Prepend some information
+
+    start = loc.start;
+    var end = loc.end;
+
+    if (start === input.length) {
+      error += " at end of input: ";
+    } else {
+      error += " at position " + (start + 1) + ": ";
+    } // Underline token in question using combining underscores
+
+
+    var underlined = input.slice(start, end).replace(/[^]/g, "$&\u0332"); // Extract some context from the input and add it to the error
+
+    var left;
+
+    if (start > 15) {
+      left = "…" + input.slice(start - 15, start);
+    } else {
+      left = input.slice(0, start);
+    }
+
+    var right;
+
+    if (end + 15 < input.length) {
+      right = input.slice(end, end + 15) + "…";
+    } else {
+      right = input.slice(end);
+    }
+
+    error += left + underlined + right;
+  } // Some hackery to make ParseError a prototype of Error
+  // See http://stackoverflow.com/a/8460753
+
+
+  var self = new Error(error);
+  self.name = "ParseError"; // $FlowFixMe
+
+  self.__proto__ = ParseError.prototype; // $FlowFixMe
+
+  self.position = start;
+  return self;
+}; // $FlowFixMe More hackery
+
+
+ParseError.prototype.__proto__ = Error.prototype;
+/* harmony default export */ var src_ParseError = (ParseError);
+// CONCATENATED MODULE: ./src/utils.js
+/**
+ * This file contains a list of utility functions which are useful in other
+ * files.
+ */
+
+/**
+ * Return whether an element is contained in a list
+ */
+var contains = function contains(list, elem) {
+  return list.indexOf(elem) !== -1;
+};
+/**
+ * Provide a default value if a setting is undefined
+ * NOTE: Couldn't use `T` as the output type due to facebook/flow#5022.
+ */
+
+
+var deflt = function deflt(setting, defaultIfUndefined) {
+  return setting === undefined ? defaultIfUndefined : setting;
+}; // hyphenate and escape adapted from Facebook's React under Apache 2 license
+
+
+var uppercase = /([A-Z])/g;
+
+var hyphenate = function hyphenate(str) {
+  return str.replace(uppercase, "-$1").toLowerCase();
+};
+
+var ESCAPE_LOOKUP = {
+  "&": "&amp;",
+  ">": "&gt;",
+  "<": "&lt;",
+  "\"": "&quot;",
+  "'": "&#x27;"
+};
+var ESCAPE_REGEX = /[&><"']/g;
+/**
+ * Escapes text to prevent scripting attacks.
+ */
+
+function utils_escape(text) {
+  return String(text).replace(ESCAPE_REGEX, function (match) {
+    return ESCAPE_LOOKUP[match];
+  });
+}
+/**
+ * Sometimes we want to pull out the innermost element of a group. In most
+ * cases, this will just be the group itself, but when ordgroups and colors have
+ * a single element, we want to pull that out.
+ */
+
+
+var getBaseElem = function getBaseElem(group) {
+  if (group.type === "ordgroup") {
+    if (group.body.length === 1) {
+      return getBaseElem(group.body[0]);
+    } else {
+      return group;
+    }
+  } else if (group.type === "color") {
+    if (group.body.length === 1) {
+      return getBaseElem(group.body[0]);
+    } else {
+      return group;
+    }
+  } else if (group.type === "font") {
+    return getBaseElem(group.body);
+  } else {
+    return group;
+  }
+};
+/**
+ * TeXbook algorithms often reference "character boxes", which are simply groups
+ * with a single character in them. To decide if something is a character box,
+ * we find its innermost group, and see if it is a single character.
+ */
+
+
+var utils_isCharacterBox = function isCharacterBox(group) {
+  var baseElem = getBaseElem(group); // These are all they types of groups which hold single characters
+
+  return baseElem.type === "mathord" || baseElem.type === "textord" || baseElem.type === "atom";
+};
+
+var assert = function assert(value) {
+  if (!value) {
+    throw new Error('Expected non-null, but got ' + String(value));
+  }
+
+  return value;
+};
+/**
+ * Return the protocol of a URL, or "_relative" if the URL does not specify a
+ * protocol (and thus is relative).
+ */
+
+var protocolFromUrl = function protocolFromUrl(url) {
+  var protocol = /^\s*([^\\/#]*?)(?::|&#0*58|&#x0*3a)/i.exec(url);
+  return protocol != null ? protocol[1] : "_relative";
+};
+/* harmony default export */ var utils = ({
+  contains: contains,
+  deflt: deflt,
+  escape: utils_escape,
+  hyphenate: hyphenate,
+  getBaseElem: getBaseElem,
+  isCharacterBox: utils_isCharacterBox,
+  protocolFromUrl: protocolFromUrl
+});
+// CONCATENATED MODULE: ./src/Settings.js
+/* eslint no-console:0 */
+
+/**
+ * This is a module for storing settings passed into KaTeX. It correctly handles
+ * default settings.
+ */
+
+
+
+
+/**
+ * The main Settings object
+ *
+ * The current options stored are:
+ *  - displayMode: Whether the expression should be typeset as inline math
+ *                 (false, the default), meaning that the math starts in
+ *                 \textstyle and is placed in an inline-block); or as display
+ *                 math (true), meaning that the math starts in \displaystyle
+ *                 and is placed in a block with vertical margin.
+ */
+var Settings_Settings =
+/*#__PURE__*/
+function () {
+  function Settings(options) {
+    this.displayMode = void 0;
+    this.output = void 0;
+    this.leqno = void 0;
+    this.fleqn = void 0;
+    this.throwOnError = void 0;
+    this.errorColor = void 0;
+    this.macros = void 0;
+    this.minRuleThickness = void 0;
+    this.colorIsTextColor = void 0;
+    this.strict = void 0;
+    this.trust = void 0;
+    this.maxSize = void 0;
+    this.maxExpand = void 0;
+    this.globalGroup = void 0;
+    // allow null options
+    options = options || {};
+    this.displayMode = utils.deflt(options.displayMode, false);
+    this.output = utils.deflt(options.output, "htmlAndMathml");
+    this.leqno = utils.deflt(options.leqno, false);
+    this.fleqn = utils.deflt(options.fleqn, false);
+    this.throwOnError = utils.deflt(options.throwOnError, true);
+    this.errorColor = utils.deflt(options.errorColor, "#cc0000");
+    this.macros = options.macros || {};
+    this.minRuleThickness = Math.max(0, utils.deflt(options.minRuleThickness, 0));
+    this.colorIsTextColor = utils.deflt(options.colorIsTextColor, false);
+    this.strict = utils.deflt(options.strict, "warn");
+    this.trust = utils.deflt(options.trust, false);
+    this.maxSize = Math.max(0, utils.deflt(options.maxSize, Infinity));
+    this.maxExpand = Math.max(0, utils.deflt(options.maxExpand, 1000));
+    this.globalGroup = utils.deflt(options.globalGroup, false);
+  }
+  /**
+   * Report nonstrict (non-LaTeX-compatible) input.
+   * Can safely not be called if `this.strict` is false in JavaScript.
+   */
+
+
+  var _proto = Settings.prototype;
+
+  _proto.reportNonstrict = function reportNonstrict(errorCode, errorMsg, token) {
+    var strict = this.strict;
+
+    if (typeof strict === "function") {
+      // Allow return value of strict function to be boolean or string
+      // (or null/undefined, meaning no further processing).
+      strict = strict(errorCode, errorMsg, token);
+    }
+
+    if (!strict || strict === "ignore") {
+      return;
+    } else if (strict === true || strict === "error") {
+      throw new src_ParseError("LaTeX-incompatible input and strict mode is set to 'error': " + (errorMsg + " [" + errorCode + "]"), token);
+    } else if (strict === "warn") {
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to 'warn': " + (errorMsg + " [" + errorCode + "]"));
+    } else {
+      // won't happen in type-safe code
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to " + ("unrecognized '" + strict + "': " + errorMsg + " [" + errorCode + "]"));
+    }
+  }
+  /**
+   * Check whether to apply strict (LaTeX-adhering) behavior for unusual
+   * input (like `\\`).  Unlike `nonstrict`, will not throw an error;
+   * instead, "error" translates to a return value of `true`, while "ignore"
+   * translates to a return value of `false`.  May still print a warning:
+   * "warn" prints a warning and returns `false`.
+   * This is for the second category of `errorCode`s listed in the README.
+   */
+  ;
+
+  _proto.useStrictBehavior = function useStrictBehavior(errorCode, errorMsg, token) {
+    var strict = this.strict;
+
+    if (typeof strict === "function") {
+      // Allow return value of strict function to be boolean or string
+      // (or null/undefined, meaning no further processing).
+      // But catch any exceptions thrown by function, treating them
+      // like "error".
+      try {
+        strict = strict(errorCode, errorMsg, token);
+      } catch (error) {
+        strict = "error";
+      }
+    }
+
+    if (!strict || strict === "ignore") {
+      return false;
+    } else if (strict === true || strict === "error") {
+      return true;
+    } else if (strict === "warn") {
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to 'warn': " + (errorMsg + " [" + errorCode + "]"));
+      return false;
+    } else {
+      // won't happen in type-safe code
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to " + ("unrecognized '" + strict + "': " + errorMsg + " [" + errorCode + "]"));
+      return false;
+    }
+  }
+  /**
+   * Check whether to test potentially dangerous input, and return
+   * `true` (trusted) or `false` (untrusted).  The sole argument `context`
+   * should be an object with `command` field specifying the relevant LaTeX
+   * command (as a string starting with `\`), and any other arguments, etc.
+   * If `context` has a `url` field, a `protocol` field will automatically
+   * get added by this function (changing the specified object).
+   */
+  ;
+
+  _proto.isTrusted = function isTrusted(context) {
+    if (context.url && !context.protocol) {
+      context.protocol = utils.protocolFromUrl(context.url);
+    }
+
+    var trust = typeof this.trust === "function" ? this.trust(context) : this.trust;
+    return Boolean(trust);
+  };
+
+  return Settings;
+}();
+
+
+// CONCATENATED MODULE: ./src/Style.js
+/**
+ * This file contains information and classes for the various kinds of styles
+ * used in TeX. It provides a generic `Style` class, which holds information
+ * about a specific style. It then provides instances of all the different kinds
+ * of styles possible, and provides functions to move between them and get
+ * information about them.
+ */
+
+/**
+ * The main style class. Contains a unique id for the style, a size (which is
+ * the same for cramped and uncramped version of a style), and a cramped flag.
+ */
+var Style =
+/*#__PURE__*/
+function () {
+  function Style(id, size, cramped) {
+    this.id = void 0;
+    this.size = void 0;
+    this.cramped = void 0;
+    this.id = id;
+    this.size = size;
+    this.cramped = cramped;
+  }
+  /**
+   * Get the style of a superscript given a base in the current style.
+   */
+
+
+  var _proto = Style.prototype;
+
+  _proto.sup = function sup() {
+    return Style_styles[_sup[this.id]];
+  }
+  /**
+   * Get the style of a subscript given a base in the current style.
+   */
+  ;
+
+  _proto.sub = function sub() {
+    return Style_styles[_sub[this.id]];
+  }
+  /**
+   * Get the style of a fraction numerator given the fraction in the current
+   * style.
+   */
+  ;
+
+  _proto.fracNum = function fracNum() {
+    return Style_styles[_fracNum[this.id]];
+  }
+  /**
+   * Get the style of a fraction denominator given the fraction in the current
+   * style.
+   */
+  ;
+
+  _proto.fracDen = function fracDen() {
+    return Style_styles[_fracDen[this.id]];
+  }
+  /**
+   * Get the cramped version of a style (in particular, cramping a cramped style
+   * doesn't change the style).
+   */
+  ;
+
+  _proto.cramp = function cramp() {
+    return Style_styles[_cramp[this.id]];
+  }
+  /**
+   * Get a text or display version of this style.
+   */
+  ;
+
+  _proto.text = function text() {
+    return Style_styles[_text[this.id]];
+  }
+  /**
+   * Return true if this style is tightly spaced (scriptstyle/scriptscriptstyle)
+   */
+  ;
+
+  _proto.isTight = function isTight() {
+    return this.size >= 2;
+  };
+
+  return Style;
+}(); // Export an interface for type checking, but don't expose the implementation.
+// This way, no more styles can be generated.
+
+
+// IDs of the different styles
+var D = 0;
+var Dc = 1;
+var T = 2;
+var Tc = 3;
+var S = 4;
+var Sc = 5;
+var SS = 6;
+var SSc = 7; // Instances of the different styles
+
+var Style_styles = [new Style(D, 0, false), new Style(Dc, 0, true), new Style(T, 1, false), new Style(Tc, 1, true), new Style(S, 2, false), new Style(Sc, 2, true), new Style(SS, 3, false), new Style(SSc, 3, true)]; // Lookup tables for switching from one style to another
+
+var _sup = [S, Sc, S, Sc, SS, SSc, SS, SSc];
+var _sub = [Sc, Sc, Sc, Sc, SSc, SSc, SSc, SSc];
+var _fracNum = [T, Tc, S, Sc, SS, SSc, SS, SSc];
+var _fracDen = [Tc, Tc, Sc, Sc, SSc, SSc, SSc, SSc];
+var _cramp = [Dc, Dc, Tc, Tc, Sc, Sc, SSc, SSc];
+var _text = [D, Dc, T, Tc, T, Tc, T, Tc]; // We only export some of the styles.
+
+/* harmony default export */ var src_Style = ({
+  DISPLAY: Style_styles[D],
+  TEXT: Style_styles[T],
+  SCRIPT: Style_styles[S],
+  SCRIPTSCRIPT: Style_styles[SS]
+});
+// CONCATENATED MODULE: ./src/unicodeScripts.js
+/*
+ * This file defines the Unicode scripts and script families that we
+ * support. To add new scripts or families, just add a new entry to the
+ * scriptData array below. Adding scripts to the scriptData array allows
+ * characters from that script to appear in \text{} environments.
+ */
+
+/**
+ * Each script or script family has a name and an array of blocks.
+ * Each block is an array of two numbers which specify the start and
+ * end points (inclusive) of a block of Unicode codepoints.
+ */
+
+/**
+ * Unicode block data for the families of scripts we support in \text{}.
+ * Scripts only need to appear here if they do not have font metrics.
+ */
+var scriptData = [{
+  // Latin characters beyond the Latin-1 characters we have metrics for.
+  // Needed for Czech, Hungarian and Turkish text, for example.
+  name: 'latin',
+  blocks: [[0x0100, 0x024f], // Latin Extended-A and Latin Extended-B
+  [0x0300, 0x036f]]
+}, {
+  // The Cyrillic script used by Russian and related languages.
+  // A Cyrillic subset used to be supported as explicitly defined
+  // symbols in symbols.js
+  name: 'cyrillic',
+  blocks: [[0x0400, 0x04ff]]
+}, {
+  // The Brahmic scripts of South and Southeast Asia
+  // Devanagari (0900–097F)
+  // Bengali (0980–09FF)
+  // Gurmukhi (0A00–0A7F)
+  // Gujarati (0A80–0AFF)
+  // Oriya (0B00–0B7F)
+  // Tamil (0B80–0BFF)
+  // Telugu (0C00–0C7F)
+  // Kannada (0C80–0CFF)
+  // Malayalam (0D00–0D7F)
+  // Sinhala (0D80–0DFF)
+  // Thai (0E00–0E7F)
+  // Lao (0E80–0EFF)
+  // Tibetan (0F00–0FFF)
+  // Myanmar (1000–109F)
+  name: 'brahmic',
+  blocks: [[0x0900, 0x109F]]
+}, {
+  name: 'georgian',
+  blocks: [[0x10A0, 0x10ff]]
+}, {
+  // Chinese and Japanese.
+  // The "k" in cjk is for Korean, but we've separated Korean out
+  name: "cjk",
+  blocks: [[0x3000, 0x30FF], // CJK symbols and punctuation, Hiragana, Katakana
+  [0x4E00, 0x9FAF], // CJK ideograms
+  [0xFF00, 0xFF60]]
+}, {
+  // Korean
+  name: 'hangul',
+  blocks: [[0xAC00, 0xD7AF]]
+}];
+/**
+ * Given a codepoint, return the name of the script or script family
+ * it is from, or null if it is not part of a known block
+ */
+
+function scriptFromCodepoint(codepoint) {
+  for (var i = 0; i < scriptData.length; i++) {
+    var script = scriptData[i];
+
+    for (var _i = 0; _i < script.blocks.length; _i++) {
+      var block = script.blocks[_i];
+
+      if (codepoint >= block[0] && codepoint <= block[1]) {
+        return script.name;
+      }
+    }
+  }
+
+  return null;
+}
+/**
+ * A flattened version of all the supported blocks in a single array.
+ * This is an optimization to make supportedCodepoint() fast.
+ */
+
+var allBlocks = [];
+scriptData.forEach(function (s) {
+  return s.blocks.forEach(function (b) {
+    return allBlocks.push.apply(allBlocks, b);
+  });
+});
+/**
+ * Given a codepoint, return true if it falls within one of the
+ * scripts or script families defined above and false otherwise.
+ *
+ * Micro benchmarks shows that this is faster than
+ * /[\u3000-\u30FF\u4E00-\u9FAF\uFF00-\uFF60\uAC00-\uD7AF\u0900-\u109F]/.test()
+ * in Firefox, Chrome and Node.
+ */
+
+function supportedCodepoint(codepoint) {
+  for (var i = 0; i < allBlocks.length; i += 2) {
+    if (codepoint >= allBlocks[i] && codepoint <= allBlocks[i + 1]) {
+      return true;
+    }
+  }
+
+  return false;
+}
+// CONCATENATED MODULE: ./src/svgGeometry.js
+/**
+ * This file provides support to domTree.js and delimiter.js.
+ * It's a storehouse of path geometry for SVG images.
+ */
+// In all paths below, the viewBox-to-em scale is 1000:1.
+var hLinePad = 80; // padding above a sqrt viniculum. Prevents image cropping.
+// The viniculum of a \sqrt can be made thicker by a KaTeX rendering option.
+// Think of variable extraViniculum as two detours in the SVG path.
+// The detour begins at the lower left of the area labeled extraViniculum below.
+// The detour proceeds one extraViniculum distance up and slightly to the right,
+// displacing the radiused corner between surd and viniculum. The radius is
+// traversed as usual, then the detour resumes. It goes right, to the end of
+// the very long viniculumn, then down one extraViniculum distance,
+// after which it resumes regular path geometry for the radical.
+
+/*                                                  viniculum
+                                                   /
+         /▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒←extraViniculum
+        / █████████████████████←0.04em (40 unit) std viniculum thickness
+       / /
+      / /
+     / /\
+    / / surd
+*/
+
+var sqrtMain = function sqrtMain(extraViniculum, hLinePad) {
+  // sqrtMain path geometry is from glyph U221A in the font KaTeX Main
+  return "M95," + (622 + extraViniculum + hLinePad) + "\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl" + extraViniculum / 2.075 + " -" + extraViniculum + "\nc5.3,-9.3,12,-14,20,-14\nH400000v" + (40 + extraViniculum) + "H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM" + (834 + extraViniculum) + " " + hLinePad + "h400000v" + (40 + extraViniculum) + "h-400000z";
+};
+
+var sqrtSize1 = function sqrtSize1(extraViniculum, hLinePad) {
+  // size1 is from glyph U221A in the font KaTeX_Size1-Regular
+  return "M263," + (601 + extraViniculum + hLinePad) + "c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl" + extraViniculum / 2.084 + " -" + extraViniculum + "\nc4.7,-7.3,11,-11,19,-11\nH40000v" + (40 + extraViniculum) + "H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM" + (1001 + extraViniculum) + " " + hLinePad + "h400000v" + (40 + extraViniculum) + "h-400000z";
+};
+
+var sqrtSize2 = function sqrtSize2(extraViniculum, hLinePad) {
+  // size2 is from glyph U221A in the font KaTeX_Size2-Regular
+  return "M983 " + (10 + extraViniculum + hLinePad) + "\nl" + extraViniculum / 3.13 + " -" + extraViniculum + "\nc4,-6.7,10,-10,18,-10 H400000v" + (40 + extraViniculum) + "\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM" + (1001 + extraViniculum) + " " + hLinePad + "h400000v" + (40 + extraViniculum) + "h-400000z";
+};
+
+var sqrtSize3 = function sqrtSize3(extraViniculum, hLinePad) {
+  // size3 is from glyph U221A in the font KaTeX_Size3-Regular
+  return "M424," + (2398 + extraViniculum + hLinePad) + "\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl" + extraViniculum / 4.223 + " -" + extraViniculum + "c4,-6.7,10,-10,18,-10 H400000\nv" + (40 + extraViniculum) + "H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M" + (1001 + extraViniculum) + " " + hLinePad + "\nh400000v" + (40 + extraViniculum) + "h-400000z";
+};
+
+var sqrtSize4 = function sqrtSize4(extraViniculum, hLinePad) {
+  // size4 is from glyph U221A in the font KaTeX_Size4-Regular
+  return "M473," + (2713 + extraViniculum + hLinePad) + "\nc339.3,-1799.3,509.3,-2700,510,-2702 l" + extraViniculum / 5.298 + " -" + extraViniculum + "\nc3.3,-7.3,9.3,-11,18,-11 H400000v" + (40 + extraViniculum) + "H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM" + (1001 + extraViniculum) + " " + hLinePad + "h400000v" + (40 + extraViniculum) + "H1017.7z";
+};
+
+var sqrtTall = function sqrtTall(extraViniculum, hLinePad, viewBoxHeight) {
+  // sqrtTall is from glyph U23B7 in the font KaTeX_Size4-Regular
+  // One path edge has a variable length. It runs vertically from the viniculumn
+  // to a point near (14 units) the bottom of the surd. The viniculum
+  // is normally 40 units thick. So the length of the line in question is:
+  var vertSegment = viewBoxHeight - 54 - hLinePad - extraViniculum;
+  return "M702 " + (extraViniculum + hLinePad) + "H400000" + (40 + extraViniculum) + "\nH742v" + vertSegment + "l-4 4-4 4c-.667.7 -2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1\nh-12l-28-84c-16.667-52-96.667 -294.333-240-727l-212 -643 -85 170\nc-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667\n219 661 l218 661zM702 " + hLinePad + "H400000v" + (40 + extraViniculum) + "H742z";
+};
+
+var sqrtPath = function sqrtPath(size, extraViniculum, viewBoxHeight) {
+  extraViniculum = 1000 * extraViniculum; // Convert from document ems to viewBox.
+
+  var path = "";
+
+  switch (size) {
+    case "sqrtMain":
+      path = sqrtMain(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize1":
+      path = sqrtSize1(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize2":
+      path = sqrtSize2(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize3":
+      path = sqrtSize3(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize4":
+      path = sqrtSize4(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtTall":
+      path = sqrtTall(extraViniculum, hLinePad, viewBoxHeight);
+  }
+
+  return path;
+};
+var svgGeometry_path = {
+  // Two paths that cover gaps in built-up parentheses.
+  leftParenInner: "M291 0 H417 V300 H291 z",
+  rightParenInner: "M457 0 H583 V300 H457 z",
+  // The doubleleftarrow geometry is from glyph U+21D0 in the font KaTeX Main
+  doubleleftarrow: "M262 157\nl10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3\n 0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28\n 14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5\nc2 1.7 6.3 3.5 13 5.5 68 17.3 128.2 47.8 180.5 91.5 52.3 43.7 93.8 96.2 124.5\n 157.5 9.3 8 15.3 12.3 18 13h6c12-.7 18-4 18-10 0-2-1.7-7-5-15-23.3-46-52-87\n-86-123l-10-10h399738v-40H218c328 0 0 0 0 0l-10-8c-26.7-20-65.7-43-117-69 2.7\n-2 6-3.7 10-5 36.7-16 72.3-37.3 107-64l10-8h399782v-40z\nm8 0v40h399730v-40zm0 194v40h399730v-40z",
+  // doublerightarrow is from glyph U+21D2 in font KaTeX Main
+  doublerightarrow: "M399738 392l\n-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5\n 14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88\n-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68\n-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18\n-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782\nc-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3\n-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z",
+  // leftarrow is from glyph U+2190 in font KaTeX Main
+  leftarrow: "M400000 241H110l3-3c68.7-52.7 113.7-120\n 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8\n-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247\nc-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208\n 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3\n 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202\n l-3-3h399890zM100 241v40h399900v-40z",
+  // overbrace is from glyphs U+23A9/23A8/23A7 in font KaTeX_Size4-Regular
+  leftbrace: "M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117\n-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7\n 5-6 9-10 13-.7 1-7.3 1-20 1H6z",
+  leftbraceunder: "M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",
+  // overgroup is from the MnSymbol package (public domain)
+  leftgroup: "M400000 80\nH435C64 80 168.3 229.4 21 260c-5.9 1.2-18 0-18 0-2 0-3-1-3-3v-38C76 61 257 0\n 435 0h399565z",
+  leftgroupunder: "M400000 262\nH435C64 262 168.3 112.6 21 82c-5.9-1.2-18 0-18 0-2 0-3 1-3 3v38c76 158 257 219\n 435 219h399565z",
+  // Harpoons are from glyph U+21BD in font KaTeX Main
+  leftharpoon: "M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3\n-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5\n-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7\n-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z",
+  leftharpoonplus: "M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3-3.3 10.2-9.5\n 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5-18.3 3-21-1.3\n-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7-196 228-6.7 4.7\n-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40zM0 435v40h400000v-40z\nm0 0v40h400000v-40z",
+  leftharpoondown: "M7 241c-4 4-6.333 8.667-7 14 0 5.333.667 9 2 11s5.333\n 5.333 12 10c90.667 54 156 130 196 228 3.333 10.667 6.333 16.333 9 17 2 .667 5\n 1 9 1h5c10.667 0 16.667-2 18-6 2-2.667 1-9.667-3-21-32-87.333-82.667-157.667\n-152-211l-3-3h399907v-40zM93 281 H400000 v-40L7 241z",
+  leftharpoondownplus: "M7 435c-4 4-6.3 8.7-7 14 0 5.3.7 9 2 11s5.3 5.3 12\n 10c90.7 54 156 130 196 228 3.3 10.7 6.3 16.3 9 17 2 .7 5 1 9 1h5c10.7 0 16.7\n-2 18-6 2-2.7 1-9.7-3-21-32-87.3-82.7-157.7-152-211l-3-3h399907v-40H7zm93 0\nv40h399900v-40zM0 241v40h399900v-40zm0 0v40h399900v-40z",
+  // hook is from glyph U+21A9 in font KaTeX Main
+  lefthook: "M400000 281 H103s-33-11.2-61-33.5S0 197.3 0 164s14.2-61.2 42.5\n-83.5C70.8 58.2 104 47 142 47 c16.7 0 25 6.7 25 20 0 12-8.7 18.7-26 20-40 3.3\n-68.7 15.7-86 37-10 12-15 25.3-15 40 0 22.7 9.8 40.7 29.5 54 19.7 13.3 43.5 21\n 71.5 23h399859zM103 281v-40h399897v40z",
+  leftlinesegment: "M40 281 V428 H0 V94 H40 V241 H400000 v40z\nM40 281 V428 H0 V94 H40 V241 H400000 v40z",
+  leftmapsto: "M40 281 V448H0V74H40V241H400000v40z\nM40 281 V448H0V74H40V241H400000v40z",
+  // tofrom is from glyph U+21C4 in font KaTeX AMS Regular
+  leftToFrom: "M0 147h400000v40H0zm0 214c68 40 115.7 95.7 143 167h22c15.3 0 23\n-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69-70-101l-7-8h399905v-40H95l7-8\nc28.7-32 52-65.7 70-101 10.7-23.3 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 265.3\n 68 321 0 361zm0-174v-40h399900v40zm100 154v40h399900v-40z",
+  longequal: "M0 50 h400000 v40H0z m0 194h40000v40H0z\nM0 50 h400000 v40H0z m0 194h40000v40H0z",
+  midbrace: "M200428 334\nc-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14\n-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7\n 311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11\n 12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z",
+  midbraceunder: "M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",
+  oiintSize1: "M512.6 71.6c272.6 0 320.3 106.8 320.3 178.2 0 70.8-47.7 177.6\n-320.3 177.6S193.1 320.6 193.1 249.8c0-71.4 46.9-178.2 319.5-178.2z\nm368.1 178.2c0-86.4-60.9-215.4-368.1-215.4-306.4 0-367.3 129-367.3 215.4 0 85.8\n60.9 214.8 367.3 214.8 307.2 0 368.1-129 368.1-214.8z",
+  oiintSize2: "M757.8 100.1c384.7 0 451.1 137.6 451.1 230 0 91.3-66.4 228.8\n-451.1 228.8-386.3 0-452.7-137.5-452.7-228.8 0-92.4 66.4-230 452.7-230z\nm502.4 230c0-111.2-82.4-277.2-502.4-277.2s-504 166-504 277.2\nc0 110 84 276 504 276s502.4-166 502.4-276z",
+  oiiintSize1: "M681.4 71.6c408.9 0 480.5 106.8 480.5 178.2 0 70.8-71.6 177.6\n-480.5 177.6S202.1 320.6 202.1 249.8c0-71.4 70.5-178.2 479.3-178.2z\nm525.8 178.2c0-86.4-86.8-215.4-525.7-215.4-437.9 0-524.7 129-524.7 215.4 0\n85.8 86.8 214.8 524.7 214.8 438.9 0 525.7-129 525.7-214.8z",
+  oiiintSize2: "M1021.2 53c603.6 0 707.8 165.8 707.8 277.2 0 110-104.2 275.8\n-707.8 275.8-606 0-710.2-165.8-710.2-275.8C311 218.8 415.2 53 1021.2 53z\nm770.4 277.1c0-131.2-126.4-327.6-770.5-327.6S248.4 198.9 248.4 330.1\nc0 130 128.8 326.4 772.7 326.4s770.5-196.4 770.5-326.4z",
+  rightarrow: "M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z",
+  rightbrace: "M400000 542l\n-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5\ns-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1\nc124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z",
+  rightbraceunder: "M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",
+  rightgroup: "M0 80h399565c371 0 266.7 149.4 414 180 5.9 1.2 18 0 18 0 2 0\n 3-1 3-3v-38c-76-158-257-219-435-219H0z",
+  rightgroupunder: "M0 262h399565c371 0 266.7-149.4 414-180 5.9-1.2 18 0 18\n 0 2 0 3 1 3 3v38c-76 158-257 219-435 219H0z",
+  rightharpoon: "M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3\n-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2\n-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58\n 69.2 92 94.5zm0 0v40h399900v-40z",
+  rightharpoonplus: "M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3-3.7-15.3-11\n-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2-10.7 0-16.7\n 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58 69.2 92 94.5z\nm0 0v40h399900v-40z m100 194v40h399900v-40zm0 0v40h399900v-40z",
+  rightharpoondown: "M399747 511c0 7.3 6.7 11 20 11 8 0 13-.8 15-2.5s4.7-6.8\n 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3 8.5-5.8 9.5\n-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3-64.7 57-92 95\n-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 241v40h399900v-40z",
+  rightharpoondownplus: "M399747 705c0 7.3 6.7 11 20 11 8 0 13-.8\n 15-2.5s4.7-6.8 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3\n 8.5-5.8 9.5-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3\n-64.7 57-92 95-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 435v40h399900v-40z\nm0-194v40h400000v-40zm0 0v40h400000v-40z",
+  righthook: "M399859 241c-764 0 0 0 0 0 40-3.3 68.7-15.7 86-37 10-12 15-25.3\n 15-40 0-22.7-9.8-40.7-29.5-54-19.7-13.3-43.5-21-71.5-23-17.3-1.3-26-8-26-20 0\n-13.3 8.7-20 26-20 38 0 71 11.2 99 33.5 0 0 7 5.6 21 16.7 14 11.2 21 33.5 21\n 66.8s-14 61.2-42 83.5c-28 22.3-61 33.5-99 33.5L0 241z M0 281v-40h399859v40z",
+  rightlinesegment: "M399960 241 V94 h40 V428 h-40 V281 H0 v-40z\nM399960 241 V94 h40 V428 h-40 V281 H0 v-40z",
+  rightToFrom: "M400000 167c-70.7-42-118-97.7-142-167h-23c-15.3 0-23 .3-23\n 1 0 1.3 5.3 13.7 16 37 18 35.3 41.3 69 70 101l7 8H0v40h399905l-7 8c-28.7 32\n-52 65.7-70 101-10.7 23.3-16 35.7-16 37 0 .7 7.7 1 23 1h23c24-69.3 71.3-125 142\n-167z M100 147v40h399900v-40zM0 341v40h399900v-40z",
+  // twoheadleftarrow is from glyph U+219E in font KaTeX AMS Regular
+  twoheadleftarrow: "M0 167c68 40\n 115.7 95.7 143 167h22c15.3 0 23-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69\n-70-101l-7-8h125l9 7c50.7 39.3 85 86 103 140h46c0-4.7-6.3-18.7-19-42-18-35.3\n-40-67.3-66-96l-9-9h399716v-40H284l9-9c26-28.7 48-60.7 66-96 12.7-23.333 19\n-37.333 19-42h-46c-18 54-52.3 100.7-103 140l-9 7H95l7-8c28.7-32 52-65.7 70-101\n 10.7-23.333 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 71.3 68 127 0 167z",
+  twoheadrightarrow: "M400000 167\nc-68-40-115.7-95.7-143-167h-22c-15.3 0-23 .3-23 1 0 1.3 5.3 13.7 16 37 18 35.3\n 41.3 69 70 101l7 8h-125l-9-7c-50.7-39.3-85-86-103-140h-46c0 4.7 6.3 18.7 19 42\n 18 35.3 40 67.3 66 96l9 9H0v40h399716l-9 9c-26 28.7-48 60.7-66 96-12.7 23.333\n-19 37.333-19 42h46c18-54 52.3-100.7 103-140l9-7h125l-7 8c-28.7 32-52 65.7-70\n 101-10.7 23.333-16 35.7-16 37 0 .7 7.7 1 23 1h22c27.3-71.3 75-127 143-167z",
+  // tilde1 is a modified version of a glyph from the MnSymbol package
+  tilde1: "M200 55.538c-77 0-168 73.953-177 73.953-3 0-7\n-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0\n 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0\n 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128\n-68.267.847-113-73.952-191-73.952z",
+  // ditto tilde2, tilde3, & tilde4
+  tilde2: "M344 55.266c-142 0-300.638 81.316-311.5 86.418\n-8.01 3.762-22.5 10.91-23.5 5.562L1 120c-1-2-1-3-1-4 0-5 3-9 8-10l18.4-9C160.9\n 31.9 283 0 358 0c148 0 188 122 331 122s314-97 326-97c4 0 8 2 10 7l7 21.114\nc1 2.14 1 3.21 1 4.28 0 5.347-3 9.626-7 10.696l-22.3 12.622C852.6 158.372 751\n 181.476 676 181.476c-149 0-189-126.21-332-126.21z",
+  tilde3: "M786 59C457 59 32 175.242 13 175.242c-6 0-10-3.457\n-11-10.37L.15 138c-1-7 3-12 10-13l19.2-6.4C378.4 40.7 634.3 0 804.3 0c337 0\n 411.8 157 746.8 157 328 0 754-112 773-112 5 0 10 3 11 9l1 14.075c1 8.066-.697\n 16.595-6.697 17.492l-21.052 7.31c-367.9 98.146-609.15 122.696-778.15 122.696\n -338 0-409-156.573-744-156.573z",
+  tilde4: "M786 58C457 58 32 177.487 13 177.487c-6 0-10-3.345\n-11-10.035L.15 143c-1-7 3-12 10-13l22-6.7C381.2 35 637.15 0 807.15 0c337 0 409\n 177 744 177 328 0 754-127 773-127 5 0 10 3 11 9l1 14.794c1 7.805-3 13.38-9\n 14.495l-20.7 5.574c-366.85 99.79-607.3 139.372-776.3 139.372-338 0-409\n -175.236-744-175.236z",
+  // vec is from glyph U+20D7 in font KaTeX Main
+  vec: "M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z",
+  // widehat1 is a modified version of a glyph from the MnSymbol package
+  widehat1: "M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z",
+  // ditto widehat2, widehat3, & widehat4
+  widehat2: "M1181 0h2l1171 176c6 0 10 5 10 11l-2 23c-1 6-5 10\n-11 10h-1L1182 67 15 220h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z",
+  widehat3: "M1181 0h2l1171 236c6 0 10 5 10 11l-2 23c-1 6-5 10\n-11 10h-1L1182 67 15 280h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z",
+  widehat4: "M1181 0h2l1171 296c6 0 10 5 10 11l-2 23c-1 6-5 10\n-11 10h-1L1182 67 15 340h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z",
+  // widecheck paths are all inverted versions of widehat
+  widecheck1: "M529,159h5l519,-115c5,-1,9,-5,9,-10c0,-1,-1,-2,-1,-3l-4,-22c-1,\n-5,-5,-9,-11,-9h-2l-512,92l-513,-92h-2c-5,0,-9,4,-11,9l-5,22c-1,6,2,12,8,13z",
+  widecheck2: "M1181,220h2l1171,-176c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,\n-11,-10h-1l-1168,153l-1167,-153h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z",
+  widecheck3: "M1181,280h2l1171,-236c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,\n-11,-10h-1l-1168,213l-1167,-213h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z",
+  widecheck4: "M1181,340h2l1171,-296c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,\n-11,-10h-1l-1168,273l-1167,-273h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z",
+  // The next ten paths support reaction arrows from the mhchem package.
+  // Arrows for \ce{<-->} are offset from xAxis by 0.22ex, per mhchem in LaTeX
+  // baraboveleftarrow is mostly from from glyph U+2190 in font KaTeX Main
+  baraboveleftarrow: "M400000 620h-399890l3 -3c68.7 -52.7 113.7 -120 135 -202\nc4 -14.7 6 -23 6 -25c0 -7.3 -7 -11 -21 -11c-8 0 -13.2 0.8 -15.5 2.5\nc-2.3 1.7 -4.2 5.8 -5.5 12.5c-1.3 4.7 -2.7 10.3 -4 17c-12 48.7 -34.8 92 -68.5 130\ns-74.2 66.3 -121.5 85c-10 4 -16 7.7 -18 11c0 8.7 6 14.3 18 17c47.3 18.7 87.8 47\n121.5 85s56.5 81.3 68.5 130c0.7 2 1.3 5 2 9s1.2 6.7 1.5 8c0.3 1.3 1 3.3 2 6\ns2.2 4.5 3.5 5.5c1.3 1 3.3 1.8 6 2.5s6 1 10 1c14 0 21 -3.7 21 -11\nc0 -2 -2 -10.3 -6 -25c-20 -79.3 -65 -146.7 -135 -202l-3 -3h399890z\nM100 620v40h399900v-40z M0 241v40h399900v-40zM0 241v40h399900v-40z",
+  // rightarrowabovebar is mostly from glyph U+2192, KaTeX Main
+  rightarrowabovebar: "M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32\n-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0\n13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39\n-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5\n-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n151.7 139 205zm96 379h399894v40H0zm0 0h399904v40H0z",
+  // The short left harpoon has 0.5em (i.e. 500 units) kern on the left end.
+  // Ref from mhchem.sty: \rlap{\raisebox{-.22ex}{$\kern0.5em
+  baraboveshortleftharpoon: "M507,435c-4,4,-6.3,8.7,-7,14c0,5.3,0.7,9,2,11\nc1.3,2,5.3,5.3,12,10c90.7,54,156,130,196,228c3.3,10.7,6.3,16.3,9,17\nc2,0.7,5,1,9,1c0,0,5,0,5,0c10.7,0,16.7,-2,18,-6c2,-2.7,1,-9.7,-3,-21\nc-32,-87.3,-82.7,-157.7,-152,-211c0,0,-3,-3,-3,-3l399351,0l0,-40\nc-398570,0,-399437,0,-399437,0z M593 435 v40 H399500 v-40z\nM0 281 v-40 H399908 v40z M0 281 v-40 H399908 v40z",
+  rightharpoonaboveshortbar: "M0,241 l0,40c399126,0,399993,0,399993,0\nc4.7,-4.7,7,-9.3,7,-14c0,-9.3,-3.7,-15.3,-11,-18c-92.7,-56.7,-159,-133.7,-199,\n-231c-3.3,-9.3,-6,-14.7,-8,-16c-2,-1.3,-7,-2,-15,-2c-10.7,0,-16.7,2,-18,6\nc-2,2.7,-1,9.7,3,21c15.3,42,36.7,81.8,64,119.5c27.3,37.7,58,69.2,92,94.5z\nM0 241 v40 H399908 v-40z M0 475 v-40 H399500 v40z M0 475 v-40 H399500 v40z",
+  shortbaraboveleftharpoon: "M7,435c-4,4,-6.3,8.7,-7,14c0,5.3,0.7,9,2,11\nc1.3,2,5.3,5.3,12,10c90.7,54,156,130,196,228c3.3,10.7,6.3,16.3,9,17c2,0.7,5,1,9,\n1c0,0,5,0,5,0c10.7,0,16.7,-2,18,-6c2,-2.7,1,-9.7,-3,-21c-32,-87.3,-82.7,-157.7,\n-152,-211c0,0,-3,-3,-3,-3l399907,0l0,-40c-399126,0,-399993,0,-399993,0z\nM93 435 v40 H400000 v-40z M500 241 v40 H400000 v-40z M500 241 v40 H400000 v-40z",
+  shortrightharpoonabovebar: "M53,241l0,40c398570,0,399437,0,399437,0\nc4.7,-4.7,7,-9.3,7,-14c0,-9.3,-3.7,-15.3,-11,-18c-92.7,-56.7,-159,-133.7,-199,\n-231c-3.3,-9.3,-6,-14.7,-8,-16c-2,-1.3,-7,-2,-15,-2c-10.7,0,-16.7,2,-18,6\nc-2,2.7,-1,9.7,3,21c15.3,42,36.7,81.8,64,119.5c27.3,37.7,58,69.2,92,94.5z\nM500 241 v40 H399408 v-40z M500 435 v40 H400000 v-40z"
+};
+// CONCATENATED MODULE: ./src/tree.js
+
+
+/**
+ * This node represents a document fragment, which contains elements, but when
+ * placed into the DOM doesn't have any representation itself. It only contains
+ * children and doesn't have any DOM node properties.
+ */
+var tree_DocumentFragment =
+/*#__PURE__*/
+function () {
+  // HtmlDomNode
+  // Never used; needed for satisfying interface.
+  function DocumentFragment(children) {
+    this.children = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    this.children = children;
+    this.classes = [];
+    this.height = 0;
+    this.depth = 0;
+    this.maxFontSize = 0;
+    this.style = {};
+  }
+
+  var _proto = DocumentFragment.prototype;
+
+  _proto.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+  /** Convert the fragment into a node. */
+  ;
+
+  _proto.toNode = function toNode() {
+    var frag = document.createDocumentFragment();
+
+    for (var i = 0; i < this.children.length; i++) {
+      frag.appendChild(this.children[i].toNode());
+    }
+
+    return frag;
+  }
+  /** Convert the fragment into HTML markup. */
+  ;
+
+  _proto.toMarkup = function toMarkup() {
+    var markup = ""; // Simply concatenate the markup for the children together.
+
+    for (var i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    return markup;
+  }
+  /**
+   * Converts the math node into a string, similar to innerText. Applies to
+   * MathDomNode's only.
+   */
+  ;
+
+  _proto.toText = function toText() {
+    // To avoid this, we would subclass documentFragment separately for
+    // MathML, but polyfills for subclassing is expensive per PR 1469.
+    // $FlowFixMe: Only works for ChildType = MathDomNode.
+    var toText = function toText(child) {
+      return child.toText();
+    };
+
+    return this.children.map(toText).join("");
+  };
+
+  return DocumentFragment;
+}();
+// CONCATENATED MODULE: ./src/domTree.js
+/**
+ * These objects store the data about the DOM nodes we create, as well as some
+ * extra data. They can then be transformed into real DOM nodes with the
+ * `toNode` function or HTML markup using `toMarkup`. They are useful for both
+ * storing extra properties on the nodes, as well as providing a way to easily
+ * work with the DOM.
+ *
+ * Similar functions for working with MathML nodes exist in mathMLTree.js.
+ *
+ * TODO: refactor `span` and `anchor` into common superclass when
+ * target environments support class inheritance
+ */
+
+
+
+
+
+/**
+ * Create an HTML className based on a list of classes. In addition to joining
+ * with spaces, we also remove empty classes.
+ */
+var createClass = function createClass(classes) {
+  return classes.filter(function (cls) {
+    return cls;
+  }).join(" ");
+};
+
+var initNode = function initNode(classes, options, style) {
+  this.classes = classes || [];
+  this.attributes = {};
+  this.height = 0;
+  this.depth = 0;
+  this.maxFontSize = 0;
+  this.style = style || {};
+
+  if (options) {
+    if (options.style.isTight()) {
+      this.classes.push("mtight");
+    }
+
+    var color = options.getColor();
+
+    if (color) {
+      this.style.color = color;
+    }
+  }
+};
+/**
+ * Convert into an HTML node
+ */
+
+
+var _toNode = function toNode(tagName) {
+  var node = document.createElement(tagName); // Apply the class
+
+  node.className = createClass(this.classes); // Apply inline styles
+
+  for (var style in this.style) {
+    if (this.style.hasOwnProperty(style)) {
+      // $FlowFixMe Flow doesn't seem to understand span.style's type.
+      node.style[style] = this.style[style];
+    }
+  } // Apply attributes
+
+
+  for (var attr in this.attributes) {
+    if (this.attributes.hasOwnProperty(attr)) {
+      node.setAttribute(attr, this.attributes[attr]);
+    }
+  } // Append the children, also as HTML nodes
+
+
+  for (var i = 0; i < this.children.length; i++) {
+    node.appendChild(this.children[i].toNode());
+  }
+
+  return node;
+};
+/**
+ * Convert into an HTML markup string
+ */
+
+
+var _toMarkup = function toMarkup(tagName) {
+  var markup = "<" + tagName; // Add the class
+
+  if (this.classes.length) {
+    markup += " class=\"" + utils.escape(createClass(this.classes)) + "\"";
+  }
+
+  var styles = ""; // Add the styles, after hyphenation
+
+  for (var style in this.style) {
+    if (this.style.hasOwnProperty(style)) {
+      styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+    }
+  }
+
+  if (styles) {
+    markup += " style=\"" + utils.escape(styles) + "\"";
+  } // Add the attributes
+
+
+  for (var attr in this.attributes) {
+    if (this.attributes.hasOwnProperty(attr)) {
+      markup += " " + attr + "=\"" + utils.escape(this.attributes[attr]) + "\"";
+    }
+  }
+
+  markup += ">"; // Add the markup of the children, also as markup
+
+  for (var i = 0; i < this.children.length; i++) {
+    markup += this.children[i].toMarkup();
+  }
+
+  markup += "</" + tagName + ">";
+  return markup;
+}; // Making the type below exact with all optional fields doesn't work due to
+// - https://github.com/facebook/flow/issues/4582
+// - https://github.com/facebook/flow/issues/5688
+// However, since *all* fields are optional, $Shape<> works as suggested in 5688
+// above.
+// This type does not include all CSS properties. Additional properties should
+// be added as needed.
+
+
+/**
+ * This node represents a span node, with a className, a list of children, and
+ * an inline style. It also contains information about its height, depth, and
+ * maxFontSize.
+ *
+ * Represents two types with different uses: SvgSpan to wrap an SVG and DomSpan
+ * otherwise. This typesafety is important when HTML builders access a span's
+ * children.
+ */
+var domTree_Span =
+/*#__PURE__*/
+function () {
+  function Span(classes, children, options, style) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.width = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    initNode.call(this, classes, options, style);
+    this.children = children || [];
+  }
+  /**
+   * Sets an arbitrary attribute on the span. Warning: use this wisely. Not
+   * all browsers support attributes the same, and having too many custom
+   * attributes is probably bad.
+   */
+
+
+  var _proto = Span.prototype;
+
+  _proto.setAttribute = function setAttribute(attribute, value) {
+    this.attributes[attribute] = value;
+  };
+
+  _proto.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  };
+
+  _proto.toNode = function toNode() {
+    return _toNode.call(this, "span");
+  };
+
+  _proto.toMarkup = function toMarkup() {
+    return _toMarkup.call(this, "span");
+  };
+
+  return Span;
+}();
+/**
+ * This node represents an anchor (<a>) element with a hyperlink.  See `span`
+ * for further details.
+ */
+
+var domTree_Anchor =
+/*#__PURE__*/
+function () {
+  function Anchor(href, classes, children, options) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    initNode.call(this, classes, options);
+    this.children = children || [];
+    this.setAttribute('href', href);
+  }
+
+  var _proto2 = Anchor.prototype;
+
+  _proto2.setAttribute = function setAttribute(attribute, value) {
+    this.attributes[attribute] = value;
+  };
+
+  _proto2.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  };
+
+  _proto2.toNode = function toNode() {
+    return _toNode.call(this, "a");
+  };
+
+  _proto2.toMarkup = function toMarkup() {
+    return _toMarkup.call(this, "a");
+  };
+
+  return Anchor;
+}();
+/**
+ * This node represents an image embed (<img>) element.
+ */
+
+var domTree_Img =
+/*#__PURE__*/
+function () {
+  function Img(src, alt, style) {
+    this.src = void 0;
+    this.alt = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    this.alt = alt;
+    this.src = src;
+    this.classes = ["mord"];
+    this.style = style;
+  }
+
+  var _proto3 = Img.prototype;
+
+  _proto3.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  };
+
+  _proto3.toNode = function toNode() {
+    var node = document.createElement("img");
+    node.src = this.src;
+    node.alt = this.alt;
+    node.className = "mord"; // Apply inline styles
+
+    for (var style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        // $FlowFixMe
+        node.style[style] = this.style[style];
+      }
+    }
+
+    return node;
+  };
+
+  _proto3.toMarkup = function toMarkup() {
+    var markup = "<img  src='" + this.src + " 'alt='" + this.alt + "' "; // Add the styles, after hyphenation
+
+    var styles = "";
+
+    for (var style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+      }
+    }
+
+    if (styles) {
+      markup += " style=\"" + utils.escape(styles) + "\"";
+    }
+
+    markup += "'/>";
+    return markup;
+  };
+
+  return Img;
+}();
+var iCombinations = {
+  'î': "\u0131\u0302",
+  'ï': "\u0131\u0308",
+  'í': "\u0131\u0301",
+  // 'ī': '\u0131\u0304', // enable when we add Extended Latin
+  'ì': "\u0131\u0300"
+};
+/**
+ * A symbol node contains information about a single symbol. It either renders
+ * to a single text node, or a span with a single text node in it, depending on
+ * whether it has CSS classes, styles, or needs italic correction.
+ */
+
+var domTree_SymbolNode =
+/*#__PURE__*/
+function () {
+  function SymbolNode(text, height, depth, italic, skew, width, classes, style) {
+    this.text = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.italic = void 0;
+    this.skew = void 0;
+    this.width = void 0;
+    this.maxFontSize = void 0;
+    this.classes = void 0;
+    this.style = void 0;
+    this.text = text;
+    this.height = height || 0;
+    this.depth = depth || 0;
+    this.italic = italic || 0;
+    this.skew = skew || 0;
+    this.width = width || 0;
+    this.classes = classes || [];
+    this.style = style || {};
+    this.maxFontSize = 0; // Mark text from non-Latin scripts with specific classes so that we
+    // can specify which fonts to use.  This allows us to render these
+    // characters with a serif font in situations where the browser would
+    // either default to a sans serif or render a placeholder character.
+    // We use CSS class names like cjk_fallback, hangul_fallback and
+    // brahmic_fallback. See ./unicodeScripts.js for the set of possible
+    // script names
+
+    var script = scriptFromCodepoint(this.text.charCodeAt(0));
+
+    if (script) {
+      this.classes.push(script + "_fallback");
+    }
+
+    if (/[îïíì]/.test(this.text)) {
+      // add ī when we add Extended Latin
+      this.text = iCombinations[this.text];
+    }
+  }
+
+  var _proto4 = SymbolNode.prototype;
+
+  _proto4.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+  /**
+   * Creates a text node or span from a symbol node. Note that a span is only
+   * created if it is needed.
+   */
+  ;
+
+  _proto4.toNode = function toNode() {
+    var node = document.createTextNode(this.text);
+    var span = null;
+
+    if (this.italic > 0) {
+      span = document.createElement("span");
+      span.style.marginRight = this.italic + "em";
+    }
+
+    if (this.classes.length > 0) {
+      span = span || document.createElement("span");
+      span.className = createClass(this.classes);
+    }
+
+    for (var style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        span = span || document.createElement("span"); // $FlowFixMe Flow doesn't seem to understand span.style's type.
+
+        span.style[style] = this.style[style];
+      }
+    }
+
+    if (span) {
+      span.appendChild(node);
+      return span;
+    } else {
+      return node;
+    }
+  }
+  /**
+   * Creates markup for a symbol node.
+   */
+  ;
+
+  _proto4.toMarkup = function toMarkup() {
+    // TODO(alpert): More duplication than I'd like from
+    // span.prototype.toMarkup and symbolNode.prototype.toNode...
+    var needsSpan = false;
+    var markup = "<span";
+
+    if (this.classes.length) {
+      needsSpan = true;
+      markup += " class=\"";
+      markup += utils.escape(createClass(this.classes));
+      markup += "\"";
+    }
+
+    var styles = "";
+
+    if (this.italic > 0) {
+      styles += "margin-right:" + this.italic + "em;";
+    }
+
+    for (var style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+      }
+    }
+
+    if (styles) {
+      needsSpan = true;
+      markup += " style=\"" + utils.escape(styles) + "\"";
+    }
+
+    var escaped = utils.escape(this.text);
+
+    if (needsSpan) {
+      markup += ">";
+      markup += escaped;
+      markup += "</span>";
+      return markup;
+    } else {
+      return escaped;
+    }
+  };
+
+  return SymbolNode;
+}();
+/**
+ * SVG nodes are used to render stretchy wide elements.
+ */
+
+var SvgNode =
+/*#__PURE__*/
+function () {
+  function SvgNode(children, attributes) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.children = children || [];
+    this.attributes = attributes || {};
+  }
+
+  var _proto5 = SvgNode.prototype;
+
+  _proto5.toNode = function toNode() {
+    var svgNS = "http://www.w3.org/2000/svg";
+    var node = document.createElementNS(svgNS, "svg"); // Apply attributes
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    for (var i = 0; i < this.children.length; i++) {
+      node.appendChild(this.children[i].toNode());
+    }
+
+    return node;
+  };
+
+  _proto5.toMarkup = function toMarkup() {
+    var markup = "<svg"; // Apply attributes
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += " " + attr + "='" + this.attributes[attr] + "'";
+      }
+    }
+
+    markup += ">";
+
+    for (var i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    markup += "</svg>";
+    return markup;
+  };
+
+  return SvgNode;
+}();
+var domTree_PathNode =
+/*#__PURE__*/
+function () {
+  function PathNode(pathName, alternate) {
+    this.pathName = void 0;
+    this.alternate = void 0;
+    this.pathName = pathName;
+    this.alternate = alternate; // Used only for \sqrt
+  }
+
+  var _proto6 = PathNode.prototype;
+
+  _proto6.toNode = function toNode() {
+    var svgNS = "http://www.w3.org/2000/svg";
+    var node = document.createElementNS(svgNS, "path");
+
+    if (this.alternate) {
+      node.setAttribute("d", this.alternate);
+    } else {
+      node.setAttribute("d", svgGeometry_path[this.pathName]);
+    }
+
+    return node;
+  };
+
+  _proto6.toMarkup = function toMarkup() {
+    if (this.alternate) {
+      return "<path d='" + this.alternate + "'/>";
+    } else {
+      return "<path d='" + svgGeometry_path[this.pathName] + "'/>";
+    }
+  };
+
+  return PathNode;
+}();
+var LineNode =
+/*#__PURE__*/
+function () {
+  function LineNode(attributes) {
+    this.attributes = void 0;
+    this.attributes = attributes || {};
+  }
+
+  var _proto7 = LineNode.prototype;
+
+  _proto7.toNode = function toNode() {
+    var svgNS = "http://www.w3.org/2000/svg";
+    var node = document.createElementNS(svgNS, "line"); // Apply attributes
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    return node;
+  };
+
+  _proto7.toMarkup = function toMarkup() {
+    var markup = "<line";
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += " " + attr + "='" + this.attributes[attr] + "'";
+      }
+    }
+
+    markup += "/>";
+    return markup;
+  };
+
+  return LineNode;
+}();
+function assertSymbolDomNode(group) {
+  if (group instanceof domTree_SymbolNode) {
+    return group;
+  } else {
+    throw new Error("Expected symbolNode but got " + String(group) + ".");
+  }
+}
+function assertSpan(group) {
+  if (group instanceof domTree_Span) {
+    return group;
+  } else {
+    throw new Error("Expected span<HtmlDomNode> but got " + String(group) + ".");
+  }
+}
+// CONCATENATED MODULE: ./submodules/katex-fonts/fontMetricsData.js
+// This file is GENERATED by buildMetrics.sh. DO NOT MODIFY.
+/* harmony default export */ var fontMetricsData = ({
+  "AMS-Regular": {
+    "32": [0, 0, 0, 0, 0.25],
+    "65": [0, 0.68889, 0, 0, 0.72222],
+    "66": [0, 0.68889, 0, 0, 0.66667],
+    "67": [0, 0.68889, 0, 0, 0.72222],
+    "68": [0, 0.68889, 0, 0, 0.72222],
+    "69": [0, 0.68889, 0, 0, 0.66667],
+    "70": [0, 0.68889, 0, 0, 0.61111],
+    "71": [0, 0.68889, 0, 0, 0.77778],
+    "72": [0, 0.68889, 0, 0, 0.77778],
+    "73": [0, 0.68889, 0, 0, 0.38889],
+    "74": [0.16667, 0.68889, 0, 0, 0.5],
+    "75": [0, 0.68889, 0, 0, 0.77778],
+    "76": [0, 0.68889, 0, 0, 0.66667],
+    "77": [0, 0.68889, 0, 0, 0.94445],
+    "78": [0, 0.68889, 0, 0, 0.72222],
+    "79": [0.16667, 0.68889, 0, 0, 0.77778],
+    "80": [0, 0.68889, 0, 0, 0.61111],
+    "81": [0.16667, 0.68889, 0, 0, 0.77778],
+    "82": [0, 0.68889, 0, 0, 0.72222],
+    "83": [0, 0.68889, 0, 0, 0.55556],
+    "84": [0, 0.68889, 0, 0, 0.66667],
+    "85": [0, 0.68889, 0, 0, 0.72222],
+    "86": [0, 0.68889, 0, 0, 0.72222],
+    "87": [0, 0.68889, 0, 0, 1.0],
+    "88": [0, 0.68889, 0, 0, 0.72222],
+    "89": [0, 0.68889, 0, 0, 0.72222],
+    "90": [0, 0.68889, 0, 0, 0.66667],
+    "107": [0, 0.68889, 0, 0, 0.55556],
+    "160": [0, 0, 0, 0, 0.25],
+    "165": [0, 0.675, 0.025, 0, 0.75],
+    "174": [0.15559, 0.69224, 0, 0, 0.94666],
+    "240": [0, 0.68889, 0, 0, 0.55556],
+    "295": [0, 0.68889, 0, 0, 0.54028],
+    "710": [0, 0.825, 0, 0, 2.33334],
+    "732": [0, 0.9, 0, 0, 2.33334],
+    "770": [0, 0.825, 0, 0, 2.33334],
+    "771": [0, 0.9, 0, 0, 2.33334],
+    "989": [0.08167, 0.58167, 0, 0, 0.77778],
+    "1008": [0, 0.43056, 0.04028, 0, 0.66667],
+    "8245": [0, 0.54986, 0, 0, 0.275],
+    "8463": [0, 0.68889, 0, 0, 0.54028],
+    "8487": [0, 0.68889, 0, 0, 0.72222],
+    "8498": [0, 0.68889, 0, 0, 0.55556],
+    "8502": [0, 0.68889, 0, 0, 0.66667],
+    "8503": [0, 0.68889, 0, 0, 0.44445],
+    "8504": [0, 0.68889, 0, 0, 0.66667],
+    "8513": [0, 0.68889, 0, 0, 0.63889],
+    "8592": [-0.03598, 0.46402, 0, 0, 0.5],
+    "8594": [-0.03598, 0.46402, 0, 0, 0.5],
+    "8602": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8603": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8606": [0.01354, 0.52239, 0, 0, 1.0],
+    "8608": [0.01354, 0.52239, 0, 0, 1.0],
+    "8610": [0.01354, 0.52239, 0, 0, 1.11111],
+    "8611": [0.01354, 0.52239, 0, 0, 1.11111],
+    "8619": [0, 0.54986, 0, 0, 1.0],
+    "8620": [0, 0.54986, 0, 0, 1.0],
+    "8621": [-0.13313, 0.37788, 0, 0, 1.38889],
+    "8622": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8624": [0, 0.69224, 0, 0, 0.5],
+    "8625": [0, 0.69224, 0, 0, 0.5],
+    "8630": [0, 0.43056, 0, 0, 1.0],
+    "8631": [0, 0.43056, 0, 0, 1.0],
+    "8634": [0.08198, 0.58198, 0, 0, 0.77778],
+    "8635": [0.08198, 0.58198, 0, 0, 0.77778],
+    "8638": [0.19444, 0.69224, 0, 0, 0.41667],
+    "8639": [0.19444, 0.69224, 0, 0, 0.41667],
+    "8642": [0.19444, 0.69224, 0, 0, 0.41667],
+    "8643": [0.19444, 0.69224, 0, 0, 0.41667],
+    "8644": [0.1808, 0.675, 0, 0, 1.0],
+    "8646": [0.1808, 0.675, 0, 0, 1.0],
+    "8647": [0.1808, 0.675, 0, 0, 1.0],
+    "8648": [0.19444, 0.69224, 0, 0, 0.83334],
+    "8649": [0.1808, 0.675, 0, 0, 1.0],
+    "8650": [0.19444, 0.69224, 0, 0, 0.83334],
+    "8651": [0.01354, 0.52239, 0, 0, 1.0],
+    "8652": [0.01354, 0.52239, 0, 0, 1.0],
+    "8653": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8654": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8655": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8666": [0.13667, 0.63667, 0, 0, 1.0],
+    "8667": [0.13667, 0.63667, 0, 0, 1.0],
+    "8669": [-0.13313, 0.37788, 0, 0, 1.0],
+    "8672": [-0.064, 0.437, 0, 0, 1.334],
+    "8674": [-0.064, 0.437, 0, 0, 1.334],
+    "8705": [0, 0.825, 0, 0, 0.5],
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+    "8741": [0.08167, 0.58167, 0, 0, 0.38889],
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+    "8757": [0, 0.69224, 0, 0, 0.66667],
+    "8764": [-0.13313, 0.36687, 0, 0, 0.77778],
+    "8765": [-0.13313, 0.37788, 0, 0, 0.77778],
+    "8769": [-0.13313, 0.36687, 0, 0, 0.77778],
+    "8770": [-0.03625, 0.46375, 0, 0, 0.77778],
+    "8774": [0.30274, 0.79383, 0, 0, 0.77778],
+    "8776": [-0.01688, 0.48312, 0, 0, 0.77778],
+    "8778": [0.08167, 0.58167, 0, 0, 0.77778],
+    "8782": [0.06062, 0.54986, 0, 0, 0.77778],
+    "8783": [0.06062, 0.54986, 0, 0, 0.77778],
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+    "8595": [1e-05, 0.6, 0, 0, 0.66667],
+    "8657": [1e-05, 0.6, 0, 0, 0.77778],
+    "8659": [1e-05, 0.6, 0, 0, 0.77778],
+    "8719": [0.25001, 0.75, 0, 0, 0.94445],
+    "8720": [0.25001, 0.75, 0, 0, 0.94445],
+    "8721": [0.25001, 0.75, 0, 0, 1.05556],
+    "8730": [0.35001, 0.85, 0, 0, 1.0],
+    "8739": [-0.00599, 0.606, 0, 0, 0.33333],
+    "8741": [-0.00599, 0.606, 0, 0, 0.55556],
+    "8747": [0.30612, 0.805, 0.19445, 0, 0.47222],
+    "8748": [0.306, 0.805, 0.19445, 0, 0.47222],
+    "8749": [0.306, 0.805, 0.19445, 0, 0.47222],
+    "8750": [0.30612, 0.805, 0.19445, 0, 0.47222],
+    "8896": [0.25001, 0.75, 0, 0, 0.83334],
+    "8897": [0.25001, 0.75, 0, 0, 0.83334],
+    "8898": [0.25001, 0.75, 0, 0, 0.83334],
+    "8899": [0.25001, 0.75, 0, 0, 0.83334],
+    "8968": [0.35001, 0.85, 0, 0, 0.47222],
+    "8969": [0.35001, 0.85, 0, 0, 0.47222],
+    "8970": [0.35001, 0.85, 0, 0, 0.47222],
+    "8971": [0.35001, 0.85, 0, 0, 0.47222],
+    "9168": [-0.00099, 0.601, 0, 0, 0.66667],
+    "10216": [0.35001, 0.85, 0, 0, 0.47222],
+    "10217": [0.35001, 0.85, 0, 0, 0.47222],
+    "10752": [0.25001, 0.75, 0, 0, 1.11111],
+    "10753": [0.25001, 0.75, 0, 0, 1.11111],
+    "10754": [0.25001, 0.75, 0, 0, 1.11111],
+    "10756": [0.25001, 0.75, 0, 0, 0.83334],
+    "10758": [0.25001, 0.75, 0, 0, 0.83334]
+  },
+  "Size2-Regular": {
+    "32": [0, 0, 0, 0, 0.25],
+    "40": [0.65002, 1.15, 0, 0, 0.59722],
+    "41": [0.65002, 1.15, 0, 0, 0.59722],
+    "47": [0.65002, 1.15, 0, 0, 0.81111],
+    "91": [0.65002, 1.15, 0, 0, 0.47222],
+    "92": [0.65002, 1.15, 0, 0, 0.81111],
+    "93": [0.65002, 1.15, 0, 0, 0.47222],
+    "123": [0.65002, 1.15, 0, 0, 0.66667],
+    "125": [0.65002, 1.15, 0, 0, 0.66667],
+    "160": [0, 0, 0, 0, 0.25],
+    "710": [0, 0.75, 0, 0, 1.0],
+    "732": [0, 0.75, 0, 0, 1.0],
+    "770": [0, 0.75, 0, 0, 1.0],
+    "771": [0, 0.75, 0, 0, 1.0],
+    "8719": [0.55001, 1.05, 0, 0, 1.27778],
+    "8720": [0.55001, 1.05, 0, 0, 1.27778],
+    "8721": [0.55001, 1.05, 0, 0, 1.44445],
+    "8730": [0.65002, 1.15, 0, 0, 1.0],
+    "8747": [0.86225, 1.36, 0.44445, 0, 0.55556],
+    "8748": [0.862, 1.36, 0.44445, 0, 0.55556],
+    "8749": [0.862, 1.36, 0.44445, 0, 0.55556],
+    "8750": [0.86225, 1.36, 0.44445, 0, 0.55556],
+    "8896": [0.55001, 1.05, 0, 0, 1.11111],
+    "8897": [0.55001, 1.05, 0, 0, 1.11111],
+    "8898": [0.55001, 1.05, 0, 0, 1.11111],
+    "8899": [0.55001, 1.05, 0, 0, 1.11111],
+    "8968": [0.65002, 1.15, 0, 0, 0.52778],
+    "8969": [0.65002, 1.15, 0, 0, 0.52778],
+    "8970": [0.65002, 1.15, 0, 0, 0.52778],
+    "8971": [0.65002, 1.15, 0, 0, 0.52778],
+    "10216": [0.65002, 1.15, 0, 0, 0.61111],
+    "10217": [0.65002, 1.15, 0, 0, 0.61111],
+    "10752": [0.55001, 1.05, 0, 0, 1.51112],
+    "10753": [0.55001, 1.05, 0, 0, 1.51112],
+    "10754": [0.55001, 1.05, 0, 0, 1.51112],
+    "10756": [0.55001, 1.05, 0, 0, 1.11111],
+    "10758": [0.55001, 1.05, 0, 0, 1.11111]
+  },
+  "Size3-Regular": {
+    "32": [0, 0, 0, 0, 0.25],
+    "40": [0.95003, 1.45, 0, 0, 0.73611],
+    "41": [0.95003, 1.45, 0, 0, 0.73611],
+    "47": [0.95003, 1.45, 0, 0, 1.04445],
+    "91": [0.95003, 1.45, 0, 0, 0.52778],
+    "92": [0.95003, 1.45, 0, 0, 1.04445],
+    "93": [0.95003, 1.45, 0, 0, 0.52778],
+    "123": [0.95003, 1.45, 0, 0, 0.75],
+    "125": [0.95003, 1.45, 0, 0, 0.75],
+    "160": [0, 0, 0, 0, 0.25],
+    "710": [0, 0.75, 0, 0, 1.44445],
+    "732": [0, 0.75, 0, 0, 1.44445],
+    "770": [0, 0.75, 0, 0, 1.44445],
+    "771": [0, 0.75, 0, 0, 1.44445],
+    "8730": [0.95003, 1.45, 0, 0, 1.0],
+    "8968": [0.95003, 1.45, 0, 0, 0.58334],
+    "8969": [0.95003, 1.45, 0, 0, 0.58334],
+    "8970": [0.95003, 1.45, 0, 0, 0.58334],
+    "8971": [0.95003, 1.45, 0, 0, 0.58334],
+    "10216": [0.95003, 1.45, 0, 0, 0.75],
+    "10217": [0.95003, 1.45, 0, 0, 0.75]
+  },
+  "Size4-Regular": {
+    "32": [0, 0, 0, 0, 0.25],
+    "40": [1.25003, 1.75, 0, 0, 0.79167],
+    "41": [1.25003, 1.75, 0, 0, 0.79167],
+    "47": [1.25003, 1.75, 0, 0, 1.27778],
+    "91": [1.25003, 1.75, 0, 0, 0.58334],
+    "92": [1.25003, 1.75, 0, 0, 1.27778],
+    "93": [1.25003, 1.75, 0, 0, 0.58334],
+    "123": [1.25003, 1.75, 0, 0, 0.80556],
+    "125": [1.25003, 1.75, 0, 0, 0.80556],
+    "160": [0, 0, 0, 0, 0.25],
+    "710": [0, 0.825, 0, 0, 1.8889],
+    "732": [0, 0.825, 0, 0, 1.8889],
+    "770": [0, 0.825, 0, 0, 1.8889],
+    "771": [0, 0.825, 0, 0, 1.8889],
+    "8730": [1.25003, 1.75, 0, 0, 1.0],
+    "8968": [1.25003, 1.75, 0, 0, 0.63889],
+    "8969": [1.25003, 1.75, 0, 0, 0.63889],
+    "8970": [1.25003, 1.75, 0, 0, 0.63889],
+    "8971": [1.25003, 1.75, 0, 0, 0.63889],
+    "9115": [0.64502, 1.155, 0, 0, 0.875],
+    "9116": [1e-05, 0.6, 0, 0, 0.875],
+    "9117": [0.64502, 1.155, 0, 0, 0.875],
+    "9118": [0.64502, 1.155, 0, 0, 0.875],
+    "9119": [1e-05, 0.6, 0, 0, 0.875],
+    "9120": [0.64502, 1.155, 0, 0, 0.875],
+    "9121": [0.64502, 1.155, 0, 0, 0.66667],
+    "9122": [-0.00099, 0.601, 0, 0, 0.66667],
+    "9123": [0.64502, 1.155, 0, 0, 0.66667],
+    "9124": [0.64502, 1.155, 0, 0, 0.66667],
+    "9125": [-0.00099, 0.601, 0, 0, 0.66667],
+    "9126": [0.64502, 1.155, 0, 0, 0.66667],
+    "9127": [1e-05, 0.9, 0, 0, 0.88889],
+    "9128": [0.65002, 1.15, 0, 0, 0.88889],
+    "9129": [0.90001, 0, 0, 0, 0.88889],
+    "9130": [0, 0.3, 0, 0, 0.88889],
+    "9131": [1e-05, 0.9, 0, 0, 0.88889],
+    "9132": [0.65002, 1.15, 0, 0, 0.88889],
+    "9133": [0.90001, 0, 0, 0, 0.88889],
+    "9143": [0.88502, 0.915, 0, 0, 1.05556],
+    "10216": [1.25003, 1.75, 0, 0, 0.80556],
+    "10217": [1.25003, 1.75, 0, 0, 0.80556],
+    "57344": [-0.00499, 0.605, 0, 0, 1.05556],
+    "57345": [-0.00499, 0.605, 0, 0, 1.05556],
+    "57680": [0, 0.12, 0, 0, 0.45],
+    "57681": [0, 0.12, 0, 0, 0.45],
+    "57682": [0, 0.12, 0, 0, 0.45],
+    "57683": [0, 0.12, 0, 0, 0.45]
+  },
+  "Typewriter-Regular": {
+    "32": [0, 0, 0, 0, 0.525],
+    "33": [0, 0.61111, 0, 0, 0.525],
+    "34": [0, 0.61111, 0, 0, 0.525],
+    "35": [0, 0.61111, 0, 0, 0.525],
+    "36": [0.08333, 0.69444, 0, 0, 0.525],
+    "37": [0.08333, 0.69444, 0, 0, 0.525],
+    "38": [0, 0.61111, 0, 0, 0.525],
+    "39": [0, 0.61111, 0, 0, 0.525],
+    "40": [0.08333, 0.69444, 0, 0, 0.525],
+    "41": [0.08333, 0.69444, 0, 0, 0.525],
+    "42": [0, 0.52083, 0, 0, 0.525],
+    "43": [-0.08056, 0.53055, 0, 0, 0.525],
+    "44": [0.13889, 0.125, 0, 0, 0.525],
+    "45": [-0.08056, 0.53055, 0, 0, 0.525],
+    "46": [0, 0.125, 0, 0, 0.525],
+    "47": [0.08333, 0.69444, 0, 0, 0.525],
+    "48": [0, 0.61111, 0, 0, 0.525],
+    "49": [0, 0.61111, 0, 0, 0.525],
+    "50": [0, 0.61111, 0, 0, 0.525],
+    "51": [0, 0.61111, 0, 0, 0.525],
+    "52": [0, 0.61111, 0, 0, 0.525],
+    "53": [0, 0.61111, 0, 0, 0.525],
+    "54": [0, 0.61111, 0, 0, 0.525],
+    "55": [0, 0.61111, 0, 0, 0.525],
+    "56": [0, 0.61111, 0, 0, 0.525],
+    "57": [0, 0.61111, 0, 0, 0.525],
+    "58": [0, 0.43056, 0, 0, 0.525],
+    "59": [0.13889, 0.43056, 0, 0, 0.525],
+    "60": [-0.05556, 0.55556, 0, 0, 0.525],
+    "61": [-0.19549, 0.41562, 0, 0, 0.525],
+    "62": [-0.05556, 0.55556, 0, 0, 0.525],
+    "63": [0, 0.61111, 0, 0, 0.525],
+    "64": [0, 0.61111, 0, 0, 0.525],
+    "65": [0, 0.61111, 0, 0, 0.525],
+    "66": [0, 0.61111, 0, 0, 0.525],
+    "67": [0, 0.61111, 0, 0, 0.525],
+    "68": [0, 0.61111, 0, 0, 0.525],
+    "69": [0, 0.61111, 0, 0, 0.525],
+    "70": [0, 0.61111, 0, 0, 0.525],
+    "71": [0, 0.61111, 0, 0, 0.525],
+    "72": [0, 0.61111, 0, 0, 0.525],
+    "73": [0, 0.61111, 0, 0, 0.525],
+    "74": [0, 0.61111, 0, 0, 0.525],
+    "75": [0, 0.61111, 0, 0, 0.525],
+    "76": [0, 0.61111, 0, 0, 0.525],
+    "77": [0, 0.61111, 0, 0, 0.525],
+    "78": [0, 0.61111, 0, 0, 0.525],
+    "79": [0, 0.61111, 0, 0, 0.525],
+    "80": [0, 0.61111, 0, 0, 0.525],
+    "81": [0.13889, 0.61111, 0, 0, 0.525],
+    "82": [0, 0.61111, 0, 0, 0.525],
+    "83": [0, 0.61111, 0, 0, 0.525],
+    "84": [0, 0.61111, 0, 0, 0.525],
+    "85": [0, 0.61111, 0, 0, 0.525],
+    "86": [0, 0.61111, 0, 0, 0.525],
+    "87": [0, 0.61111, 0, 0, 0.525],
+    "88": [0, 0.61111, 0, 0, 0.525],
+    "89": [0, 0.61111, 0, 0, 0.525],
+    "90": [0, 0.61111, 0, 0, 0.525],
+    "91": [0.08333, 0.69444, 0, 0, 0.525],
+    "92": [0.08333, 0.69444, 0, 0, 0.525],
+    "93": [0.08333, 0.69444, 0, 0, 0.525],
+    "94": [0, 0.61111, 0, 0, 0.525],
+    "95": [0.09514, 0, 0, 0, 0.525],
+    "96": [0, 0.61111, 0, 0, 0.525],
+    "97": [0, 0.43056, 0, 0, 0.525],
+    "98": [0, 0.61111, 0, 0, 0.525],
+    "99": [0, 0.43056, 0, 0, 0.525],
+    "100": [0, 0.61111, 0, 0, 0.525],
+    "101": [0, 0.43056, 0, 0, 0.525],
+    "102": [0, 0.61111, 0, 0, 0.525],
+    "103": [0.22222, 0.43056, 0, 0, 0.525],
+    "104": [0, 0.61111, 0, 0, 0.525],
+    "105": [0, 0.61111, 0, 0, 0.525],
+    "106": [0.22222, 0.61111, 0, 0, 0.525],
+    "107": [0, 0.61111, 0, 0, 0.525],
+    "108": [0, 0.61111, 0, 0, 0.525],
+    "109": [0, 0.43056, 0, 0, 0.525],
+    "110": [0, 0.43056, 0, 0, 0.525],
+    "111": [0, 0.43056, 0, 0, 0.525],
+    "112": [0.22222, 0.43056, 0, 0, 0.525],
+    "113": [0.22222, 0.43056, 0, 0, 0.525],
+    "114": [0, 0.43056, 0, 0, 0.525],
+    "115": [0, 0.43056, 0, 0, 0.525],
+    "116": [0, 0.55358, 0, 0, 0.525],
+    "117": [0, 0.43056, 0, 0, 0.525],
+    "118": [0, 0.43056, 0, 0, 0.525],
+    "119": [0, 0.43056, 0, 0, 0.525],
+    "120": [0, 0.43056, 0, 0, 0.525],
+    "121": [0.22222, 0.43056, 0, 0, 0.525],
+    "122": [0, 0.43056, 0, 0, 0.525],
+    "123": [0.08333, 0.69444, 0, 0, 0.525],
+    "124": [0.08333, 0.69444, 0, 0, 0.525],
+    "125": [0.08333, 0.69444, 0, 0, 0.525],
+    "126": [0, 0.61111, 0, 0, 0.525],
+    "127": [0, 0.61111, 0, 0, 0.525],
+    "160": [0, 0, 0, 0, 0.525],
+    "176": [0, 0.61111, 0, 0, 0.525],
+    "184": [0.19445, 0, 0, 0, 0.525],
+    "305": [0, 0.43056, 0, 0, 0.525],
+    "567": [0.22222, 0.43056, 0, 0, 0.525],
+    "711": [0, 0.56597, 0, 0, 0.525],
+    "713": [0, 0.56555, 0, 0, 0.525],
+    "714": [0, 0.61111, 0, 0, 0.525],
+    "715": [0, 0.61111, 0, 0, 0.525],
+    "728": [0, 0.61111, 0, 0, 0.525],
+    "730": [0, 0.61111, 0, 0, 0.525],
+    "770": [0, 0.61111, 0, 0, 0.525],
+    "771": [0, 0.61111, 0, 0, 0.525],
+    "776": [0, 0.61111, 0, 0, 0.525],
+    "915": [0, 0.61111, 0, 0, 0.525],
+    "916": [0, 0.61111, 0, 0, 0.525],
+    "920": [0, 0.61111, 0, 0, 0.525],
+    "923": [0, 0.61111, 0, 0, 0.525],
+    "926": [0, 0.61111, 0, 0, 0.525],
+    "928": [0, 0.61111, 0, 0, 0.525],
+    "931": [0, 0.61111, 0, 0, 0.525],
+    "933": [0, 0.61111, 0, 0, 0.525],
+    "934": [0, 0.61111, 0, 0, 0.525],
+    "936": [0, 0.61111, 0, 0, 0.525],
+    "937": [0, 0.61111, 0, 0, 0.525],
+    "8216": [0, 0.61111, 0, 0, 0.525],
+    "8217": [0, 0.61111, 0, 0, 0.525],
+    "8242": [0, 0.61111, 0, 0, 0.525],
+    "9251": [0.11111, 0.21944, 0, 0, 0.525]
+  }
+});
+// CONCATENATED MODULE: ./src/fontMetrics.js
+
+
+/**
+ * This file contains metrics regarding fonts and individual symbols. The sigma
+ * and xi variables, as well as the metricMap map contain data extracted from
+ * TeX, TeX font metrics, and the TTF files. These data are then exposed via the
+ * `metrics` variable and the getCharacterMetrics function.
+ */
+// In TeX, there are actually three sets of dimensions, one for each of
+// textstyle (size index 5 and higher: >=9pt), scriptstyle (size index 3 and 4:
+// 7-8pt), and scriptscriptstyle (size index 1 and 2: 5-6pt).  These are
+// provided in the the arrays below, in that order.
+//
+// The font metrics are stored in fonts cmsy10, cmsy7, and cmsy5 respsectively.
+// This was determined by running the following script:
+//
+//     latex -interaction=nonstopmode \
+//     '\documentclass{article}\usepackage{amsmath}\begin{document}' \
+//     '$a$ \expandafter\show\the\textfont2' \
+//     '\expandafter\show\the\scriptfont2' \
+//     '\expandafter\show\the\scriptscriptfont2' \
+//     '\stop'
+//
+// The metrics themselves were retreived using the following commands:
+//
+//     tftopl cmsy10
+//     tftopl cmsy7
+//     tftopl cmsy5
+//
+// The output of each of these commands is quite lengthy.  The only part we
+// care about is the FONTDIMEN section. Each value is measured in EMs.
+var sigmasAndXis = {
+  slant: [0.250, 0.250, 0.250],
+  // sigma1
+  space: [0.000, 0.000, 0.000],
+  // sigma2
+  stretch: [0.000, 0.000, 0.000],
+  // sigma3
+  shrink: [0.000, 0.000, 0.000],
+  // sigma4
+  xHeight: [0.431, 0.431, 0.431],
+  // sigma5
+  quad: [1.000, 1.171, 1.472],
+  // sigma6
+  extraSpace: [0.000, 0.000, 0.000],
+  // sigma7
+  num1: [0.677, 0.732, 0.925],
+  // sigma8
+  num2: [0.394, 0.384, 0.387],
+  // sigma9
+  num3: [0.444, 0.471, 0.504],
+  // sigma10
+  denom1: [0.686, 0.752, 1.025],
+  // sigma11
+  denom2: [0.345, 0.344, 0.532],
+  // sigma12
+  sup1: [0.413, 0.503, 0.504],
+  // sigma13
+  sup2: [0.363, 0.431, 0.404],
+  // sigma14
+  sup3: [0.289, 0.286, 0.294],
+  // sigma15
+  sub1: [0.150, 0.143, 0.200],
+  // sigma16
+  sub2: [0.247, 0.286, 0.400],
+  // sigma17
+  supDrop: [0.386, 0.353, 0.494],
+  // sigma18
+  subDrop: [0.050, 0.071, 0.100],
+  // sigma19
+  delim1: [2.390, 1.700, 1.980],
+  // sigma20
+  delim2: [1.010, 1.157, 1.420],
+  // sigma21
+  axisHeight: [0.250, 0.250, 0.250],
+  // sigma22
+  // These font metrics are extracted from TeX by using tftopl on cmex10.tfm;
+  // they correspond to the font parameters of the extension fonts (family 3).
+  // See the TeXbook, page 441. In AMSTeX, the extension fonts scale; to
+  // match cmex7, we'd use cmex7.tfm values for script and scriptscript
+  // values.
+  defaultRuleThickness: [0.04, 0.049, 0.049],
+  // xi8; cmex7: 0.049
+  bigOpSpacing1: [0.111, 0.111, 0.111],
+  // xi9
+  bigOpSpacing2: [0.166, 0.166, 0.166],
+  // xi10
+  bigOpSpacing3: [0.2, 0.2, 0.2],
+  // xi11
+  bigOpSpacing4: [0.6, 0.611, 0.611],
+  // xi12; cmex7: 0.611
+  bigOpSpacing5: [0.1, 0.143, 0.143],
+  // xi13; cmex7: 0.143
+  // The \sqrt rule width is taken from the height of the surd character.
+  // Since we use the same font at all sizes, this thickness doesn't scale.
+  sqrtRuleThickness: [0.04, 0.04, 0.04],
+  // This value determines how large a pt is, for metrics which are defined
+  // in terms of pts.
+  // This value is also used in katex.less; if you change it make sure the
+  // values match.
+  ptPerEm: [10.0, 10.0, 10.0],
+  // The space between adjacent `|` columns in an array definition. From
+  // `\showthe\doublerulesep` in LaTeX. Equals 2.0 / ptPerEm.
+  doubleRuleSep: [0.2, 0.2, 0.2],
+  // The width of separator lines in {array} environments. From
+  // `\showthe\arrayrulewidth` in LaTeX. Equals 0.4 / ptPerEm.
+  arrayRuleWidth: [0.04, 0.04, 0.04],
+  // Two values from LaTeX source2e:
+  fboxsep: [0.3, 0.3, 0.3],
+  //        3 pt / ptPerEm
+  fboxrule: [0.04, 0.04, 0.04] // 0.4 pt / ptPerEm
+
+}; // This map contains a mapping from font name and character code to character
+// metrics, including height, depth, italic correction, and skew (kern from the
+// character to the corresponding \skewchar)
+// This map is generated via `make metrics`. It should not be changed manually.
+
+ // These are very rough approximations.  We default to Times New Roman which
+// should have Latin-1 and Cyrillic characters, but may not depending on the
+// operating system.  The metrics do not account for extra height from the
+// accents.  In the case of Cyrillic characters which have both ascenders and
+// descenders we prefer approximations with ascenders, primarily to prevent
+// the fraction bar or root line from intersecting the glyph.
+// TODO(kevinb) allow union of multiple glyph metrics for better accuracy.
+
+var extraCharacterMap = {
+  // Latin-1
+  'Å': 'A',
+  'Ç': 'C',
+  'Ð': 'D',
+  'Þ': 'o',
+  'å': 'a',
+  'ç': 'c',
+  'ð': 'd',
+  'þ': 'o',
+  // Cyrillic
+  'А': 'A',
+  'Б': 'B',
+  'В': 'B',
+  'Г': 'F',
+  'Д': 'A',
+  'Е': 'E',
+  'Ж': 'K',
+  'З': '3',
+  'И': 'N',
+  'Й': 'N',
+  'К': 'K',
+  'Л': 'N',
+  'М': 'M',
+  'Н': 'H',
+  'О': 'O',
+  'П': 'N',
+  'Р': 'P',
+  'С': 'C',
+  'Т': 'T',
+  'У': 'y',
+  'Ф': 'O',
+  'Х': 'X',
+  'Ц': 'U',
+  'Ч': 'h',
+  'Ш': 'W',
+  'Щ': 'W',
+  'Ъ': 'B',
+  'Ы': 'X',
+  'Ь': 'B',
+  'Э': '3',
+  'Ю': 'X',
+  'Я': 'R',
+  'а': 'a',
+  'б': 'b',
+  'в': 'a',
+  'г': 'r',
+  'д': 'y',
+  'е': 'e',
+  'ж': 'm',
+  'з': 'e',
+  'и': 'n',
+  'й': 'n',
+  'к': 'n',
+  'л': 'n',
+  'м': 'm',
+  'н': 'n',
+  'о': 'o',
+  'п': 'n',
+  'р': 'p',
+  'с': 'c',
+  'т': 'o',
+  'у': 'y',
+  'ф': 'b',
+  'х': 'x',
+  'ц': 'n',
+  'ч': 'n',
+  'ш': 'w',
+  'щ': 'w',
+  'ъ': 'a',
+  'ы': 'm',
+  'ь': 'a',
+  'э': 'e',
+  'ю': 'm',
+  'я': 'r'
+};
+
+/**
+ * This function adds new font metrics to default metricMap
+ * It can also override existing metrics
+ */
+function setFontMetrics(fontName, metrics) {
+  fontMetricsData[fontName] = metrics;
+}
+/**
+ * This function is a convenience function for looking up information in the
+ * metricMap table. It takes a character as a string, and a font.
+ *
+ * Note: the `width` property may be undefined if fontMetricsData.js wasn't
+ * built using `Make extended_metrics`.
+ */
+
+function getCharacterMetrics(character, font, mode) {
+  if (!fontMetricsData[font]) {
+    throw new Error("Font metrics not found for font: " + font + ".");
+  }
+
+  var ch = character.charCodeAt(0);
+  var metrics = fontMetricsData[font][ch];
+
+  if (!metrics && character[0] in extraCharacterMap) {
+    ch = extraCharacterMap[character[0]].charCodeAt(0);
+    metrics = fontMetricsData[font][ch];
+  }
+
+  if (!metrics && mode === 'text') {
+    // We don't typically have font metrics for Asian scripts.
+    // But since we support them in text mode, we need to return
+    // some sort of metrics.
+    // So if the character is in a script we support but we
+    // don't have metrics for it, just use the metrics for
+    // the Latin capital letter M. This is close enough because
+    // we (currently) only care about the height of the glpyh
+    // not its width.
+    if (supportedCodepoint(ch)) {
+      metrics = fontMetricsData[font][77]; // 77 is the charcode for 'M'
+    }
+  }
+
+  if (metrics) {
+    return {
+      depth: metrics[0],
+      height: metrics[1],
+      italic: metrics[2],
+      skew: metrics[3],
+      width: metrics[4]
+    };
+  }
+}
+var fontMetricsBySizeIndex = {};
+/**
+ * Get the font metrics for a given size.
+ */
+
+function getGlobalMetrics(size) {
+  var sizeIndex;
+
+  if (size >= 5) {
+    sizeIndex = 0;
+  } else if (size >= 3) {
+    sizeIndex = 1;
+  } else {
+    sizeIndex = 2;
+  }
+
+  if (!fontMetricsBySizeIndex[sizeIndex]) {
+    var metrics = fontMetricsBySizeIndex[sizeIndex] = {
+      cssEmPerMu: sigmasAndXis.quad[sizeIndex] / 18
+    };
+
+    for (var key in sigmasAndXis) {
+      if (sigmasAndXis.hasOwnProperty(key)) {
+        metrics[key] = sigmasAndXis[key][sizeIndex];
+      }
+    }
+  }
+
+  return fontMetricsBySizeIndex[sizeIndex];
+}
+// CONCATENATED MODULE: ./src/symbols.js
+/**
+ * This file holds a list of all no-argument functions and single-character
+ * symbols (like 'a' or ';').
+ *
+ * For each of the symbols, there are three properties they can have:
+ * - font (required): the font to be used for this symbol. Either "main" (the
+     normal font), or "ams" (the ams fonts).
+ * - group (required): the ParseNode group type the symbol should have (i.e.
+     "textord", "mathord", etc).
+     See https://github.com/KaTeX/KaTeX/wiki/Examining-TeX#group-types
+ * - replace: the character that this symbol or function should be
+ *   replaced with (i.e. "\phi" has a replace value of "\u03d5", the phi
+ *   character in the main font).
+ *
+ * The outermost map in the table indicates what mode the symbols should be
+ * accepted in (e.g. "math" or "text").
+ */
+// Some of these have a "-token" suffix since these are also used as `ParseNode`
+// types for raw text tokens, and we want to avoid conflicts with higher-level
+// `ParseNode` types. These `ParseNode`s are constructed within `Parser` by
+// looking up the `symbols` map.
+var ATOMS = {
+  "bin": 1,
+  "close": 1,
+  "inner": 1,
+  "open": 1,
+  "punct": 1,
+  "rel": 1
+};
+var NON_ATOMS = {
+  "accent-token": 1,
+  "mathord": 1,
+  "op-token": 1,
+  "spacing": 1,
+  "textord": 1
+};
+var symbols = {
+  "math": {},
+  "text": {}
+};
+/* harmony default export */ var src_symbols = (symbols);
+/** `acceptUnicodeChar = true` is only applicable if `replace` is set. */
+
+function defineSymbol(mode, font, group, replace, name, acceptUnicodeChar) {
+  symbols[mode][name] = {
+    font: font,
+    group: group,
+    replace: replace
+  };
+
+  if (acceptUnicodeChar && replace) {
+    symbols[mode][replace] = symbols[mode][name];
+  }
+} // Some abbreviations for commonly used strings.
+// This helps minify the code, and also spotting typos using jshint.
+// modes:
+
+var symbols_math = "math";
+var symbols_text = "text"; // fonts:
+
+var main = "main";
+var ams = "ams"; // groups:
+
+var symbols_accent = "accent-token";
+var bin = "bin";
+var symbols_close = "close";
+var symbols_inner = "inner";
+var mathord = "mathord";
+var op = "op-token";
+var symbols_open = "open";
+var punct = "punct";
+var rel = "rel";
+var symbols_spacing = "spacing";
+var symbols_textord = "textord"; // Now comes the symbol table
+// Relation Symbols
+
+defineSymbol(symbols_math, main, rel, "\u2261", "\\equiv", true);
+defineSymbol(symbols_math, main, rel, "\u227A", "\\prec", true);
+defineSymbol(symbols_math, main, rel, "\u227B", "\\succ", true);
+defineSymbol(symbols_math, main, rel, "\u223C", "\\sim", true);
+defineSymbol(symbols_math, main, rel, "\u22A5", "\\perp");
+defineSymbol(symbols_math, main, rel, "\u2AAF", "\\preceq", true);
+defineSymbol(symbols_math, main, rel, "\u2AB0", "\\succeq", true);
+defineSymbol(symbols_math, main, rel, "\u2243", "\\simeq", true);
+defineSymbol(symbols_math, main, rel, "\u2223", "\\mid", true);
+defineSymbol(symbols_math, main, rel, "\u226A", "\\ll", true);
+defineSymbol(symbols_math, main, rel, "\u226B", "\\gg", true);
+defineSymbol(symbols_math, main, rel, "\u224D", "\\asymp", true);
+defineSymbol(symbols_math, main, rel, "\u2225", "\\parallel");
+defineSymbol(symbols_math, main, rel, "\u22C8", "\\bowtie", true);
+defineSymbol(symbols_math, main, rel, "\u2323", "\\smile", true);
+defineSymbol(symbols_math, main, rel, "\u2291", "\\sqsubseteq", true);
+defineSymbol(symbols_math, main, rel, "\u2292", "\\sqsupseteq", true);
+defineSymbol(symbols_math, main, rel, "\u2250", "\\doteq", true);
+defineSymbol(symbols_math, main, rel, "\u2322", "\\frown", true);
+defineSymbol(symbols_math, main, rel, "\u220B", "\\ni", true);
+defineSymbol(symbols_math, main, rel, "\u221D", "\\propto", true);
+defineSymbol(symbols_math, main, rel, "\u22A2", "\\vdash", true);
+defineSymbol(symbols_math, main, rel, "\u22A3", "\\dashv", true);
+defineSymbol(symbols_math, main, rel, "\u220B", "\\owns"); // Punctuation
+
+defineSymbol(symbols_math, main, punct, ".", "\\ldotp");
+defineSymbol(symbols_math, main, punct, "\u22C5", "\\cdotp"); // Misc Symbols
+
+defineSymbol(symbols_math, main, symbols_textord, "#", "\\#");
+defineSymbol(symbols_text, main, symbols_textord, "#", "\\#");
+defineSymbol(symbols_math, main, symbols_textord, "&", "\\&");
+defineSymbol(symbols_text, main, symbols_textord, "&", "\\&");
+defineSymbol(symbols_math, main, symbols_textord, "\u2135", "\\aleph", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2200", "\\forall", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u210F", "\\hbar", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2203", "\\exists", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2207", "\\nabla", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u266D", "\\flat", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2113", "\\ell", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u266E", "\\natural", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2663", "\\clubsuit", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2118", "\\wp", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u266F", "\\sharp", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2662", "\\diamondsuit", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u211C", "\\Re", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2661", "\\heartsuit", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2111", "\\Im", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2660", "\\spadesuit", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xA7", "\\S", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xB6", "\\P", true); // Math and Text
+
+defineSymbol(symbols_math, main, symbols_textord, "\u2020", "\\dag");
+defineSymbol(symbols_text, main, symbols_textord, "\u2020", "\\dag");
+defineSymbol(symbols_text, main, symbols_textord, "\u2020", "\\textdagger");
+defineSymbol(symbols_math, main, symbols_textord, "\u2021", "\\ddag");
+defineSymbol(symbols_text, main, symbols_textord, "\u2021", "\\ddag");
+defineSymbol(symbols_text, main, symbols_textord, "\u2021", "\\textdaggerdbl"); // Large Delimiters
+
+defineSymbol(symbols_math, main, symbols_close, "\u23B1", "\\rmoustache", true);
+defineSymbol(symbols_math, main, symbols_open, "\u23B0", "\\lmoustache", true);
+defineSymbol(symbols_math, main, symbols_close, "\u27EF", "\\rgroup", true);
+defineSymbol(symbols_math, main, symbols_open, "\u27EE", "\\lgroup", true); // Binary Operators
+
+defineSymbol(symbols_math, main, bin, "\u2213", "\\mp", true);
+defineSymbol(symbols_math, main, bin, "\u2296", "\\ominus", true);
+defineSymbol(symbols_math, main, bin, "\u228E", "\\uplus", true);
+defineSymbol(symbols_math, main, bin, "\u2293", "\\sqcap", true);
+defineSymbol(symbols_math, main, bin, "\u2217", "\\ast");
+defineSymbol(symbols_math, main, bin, "\u2294", "\\sqcup", true);
+defineSymbol(symbols_math, main, bin, "\u25EF", "\\bigcirc");
+defineSymbol(symbols_math, main, bin, "\u2219", "\\bullet");
+defineSymbol(symbols_math, main, bin, "\u2021", "\\ddagger");
+defineSymbol(symbols_math, main, bin, "\u2240", "\\wr", true);
+defineSymbol(symbols_math, main, bin, "\u2A3F", "\\amalg");
+defineSymbol(symbols_math, main, bin, "&", "\\And"); // from amsmath
+// Arrow Symbols
+
+defineSymbol(symbols_math, main, rel, "\u27F5", "\\longleftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D0", "\\Leftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27F8", "\\Longleftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27F6", "\\longrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D2", "\\Rightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27F9", "\\Longrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2194", "\\leftrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27F7", "\\longleftrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D4", "\\Leftrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27FA", "\\Longleftrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21A6", "\\mapsto", true);
+defineSymbol(symbols_math, main, rel, "\u27FC", "\\longmapsto", true);
+defineSymbol(symbols_math, main, rel, "\u2197", "\\nearrow", true);
+defineSymbol(symbols_math, main, rel, "\u21A9", "\\hookleftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21AA", "\\hookrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2198", "\\searrow", true);
+defineSymbol(symbols_math, main, rel, "\u21BC", "\\leftharpoonup", true);
+defineSymbol(symbols_math, main, rel, "\u21C0", "\\rightharpoonup", true);
+defineSymbol(symbols_math, main, rel, "\u2199", "\\swarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21BD", "\\leftharpoondown", true);
+defineSymbol(symbols_math, main, rel, "\u21C1", "\\rightharpoondown", true);
+defineSymbol(symbols_math, main, rel, "\u2196", "\\nwarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21CC", "\\rightleftharpoons", true); // AMS Negated Binary Relations
+
+defineSymbol(symbols_math, ams, rel, "\u226E", "\\nless", true); // Symbol names preceeded by "@" each have a corresponding macro.
+
+defineSymbol(symbols_math, ams, rel, "\uE010", "\\@nleqslant");
+defineSymbol(symbols_math, ams, rel, "\uE011", "\\@nleqq");
+defineSymbol(symbols_math, ams, rel, "\u2A87", "\\lneq", true);
+defineSymbol(symbols_math, ams, rel, "\u2268", "\\lneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE00C", "\\@lvertneqq");
+defineSymbol(symbols_math, ams, rel, "\u22E6", "\\lnsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2A89", "\\lnapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u2280", "\\nprec", true); // unicode-math maps \u22e0 to \npreccurlyeq. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u22E0", "\\npreceq", true);
+defineSymbol(symbols_math, ams, rel, "\u22E8", "\\precnsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB9", "\\precnapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u2241", "\\nsim", true);
+defineSymbol(symbols_math, ams, rel, "\uE006", "\\@nshortmid");
+defineSymbol(symbols_math, ams, rel, "\u2224", "\\nmid", true);
+defineSymbol(symbols_math, ams, rel, "\u22AC", "\\nvdash", true);
+defineSymbol(symbols_math, ams, rel, "\u22AD", "\\nvDash", true);
+defineSymbol(symbols_math, ams, rel, "\u22EA", "\\ntriangleleft");
+defineSymbol(symbols_math, ams, rel, "\u22EC", "\\ntrianglelefteq", true);
+defineSymbol(symbols_math, ams, rel, "\u228A", "\\subsetneq", true);
+defineSymbol(symbols_math, ams, rel, "\uE01A", "\\@varsubsetneq");
+defineSymbol(symbols_math, ams, rel, "\u2ACB", "\\subsetneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE017", "\\@varsubsetneqq");
+defineSymbol(symbols_math, ams, rel, "\u226F", "\\ngtr", true);
+defineSymbol(symbols_math, ams, rel, "\uE00F", "\\@ngeqslant");
+defineSymbol(symbols_math, ams, rel, "\uE00E", "\\@ngeqq");
+defineSymbol(symbols_math, ams, rel, "\u2A88", "\\gneq", true);
+defineSymbol(symbols_math, ams, rel, "\u2269", "\\gneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE00D", "\\@gvertneqq");
+defineSymbol(symbols_math, ams, rel, "\u22E7", "\\gnsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2A8A", "\\gnapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u2281", "\\nsucc", true); // unicode-math maps \u22e1 to \nsucccurlyeq. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u22E1", "\\nsucceq", true);
+defineSymbol(symbols_math, ams, rel, "\u22E9", "\\succnsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2ABA", "\\succnapprox", true); // unicode-math maps \u2246 to \simneqq. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u2246", "\\ncong", true);
+defineSymbol(symbols_math, ams, rel, "\uE007", "\\@nshortparallel");
+defineSymbol(symbols_math, ams, rel, "\u2226", "\\nparallel", true);
+defineSymbol(symbols_math, ams, rel, "\u22AF", "\\nVDash", true);
+defineSymbol(symbols_math, ams, rel, "\u22EB", "\\ntriangleright");
+defineSymbol(symbols_math, ams, rel, "\u22ED", "\\ntrianglerighteq", true);
+defineSymbol(symbols_math, ams, rel, "\uE018", "\\@nsupseteqq");
+defineSymbol(symbols_math, ams, rel, "\u228B", "\\supsetneq", true);
+defineSymbol(symbols_math, ams, rel, "\uE01B", "\\@varsupsetneq");
+defineSymbol(symbols_math, ams, rel, "\u2ACC", "\\supsetneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE019", "\\@varsupsetneqq");
+defineSymbol(symbols_math, ams, rel, "\u22AE", "\\nVdash", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB5", "\\precneqq", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB6", "\\succneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE016", "\\@nsubseteqq");
+defineSymbol(symbols_math, ams, bin, "\u22B4", "\\unlhd");
+defineSymbol(symbols_math, ams, bin, "\u22B5", "\\unrhd"); // AMS Negated Arrows
+
+defineSymbol(symbols_math, ams, rel, "\u219A", "\\nleftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u219B", "\\nrightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21CD", "\\nLeftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21CF", "\\nRightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21AE", "\\nleftrightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21CE", "\\nLeftrightarrow", true); // AMS Misc
+
+defineSymbol(symbols_math, ams, rel, "\u25B3", "\\vartriangle");
+defineSymbol(symbols_math, ams, symbols_textord, "\u210F", "\\hslash");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25BD", "\\triangledown");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25CA", "\\lozenge");
+defineSymbol(symbols_math, ams, symbols_textord, "\u24C8", "\\circledS");
+defineSymbol(symbols_math, ams, symbols_textord, "\xAE", "\\circledR");
+defineSymbol(symbols_text, ams, symbols_textord, "\xAE", "\\circledR");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2221", "\\measuredangle", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2204", "\\nexists");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2127", "\\mho");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2132", "\\Finv", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2141", "\\Game", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2035", "\\backprime");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25B2", "\\blacktriangle");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25BC", "\\blacktriangledown");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25A0", "\\blacksquare");
+defineSymbol(symbols_math, ams, symbols_textord, "\u29EB", "\\blacklozenge");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2605", "\\bigstar");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2222", "\\sphericalangle", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2201", "\\complement", true); // unicode-math maps U+F0 to \matheth. We map to AMS function \eth
+
+defineSymbol(symbols_math, ams, symbols_textord, "\xF0", "\\eth", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xF0", "\xF0");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2571", "\\diagup");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2572", "\\diagdown");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25A1", "\\square");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25A1", "\\Box");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25CA", "\\Diamond"); // unicode-math maps U+A5 to \mathyen. We map to AMS function \yen
+
+defineSymbol(symbols_math, ams, symbols_textord, "\xA5", "\\yen", true);
+defineSymbol(symbols_text, ams, symbols_textord, "\xA5", "\\yen", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2713", "\\checkmark", true);
+defineSymbol(symbols_text, ams, symbols_textord, "\u2713", "\\checkmark"); // AMS Hebrew
+
+defineSymbol(symbols_math, ams, symbols_textord, "\u2136", "\\beth", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2138", "\\daleth", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2137", "\\gimel", true); // AMS Greek
+
+defineSymbol(symbols_math, ams, symbols_textord, "\u03DD", "\\digamma", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u03F0", "\\varkappa"); // AMS Delimiters
+
+defineSymbol(symbols_math, ams, symbols_open, "\u250C", "\\@ulcorner", true);
+defineSymbol(symbols_math, ams, symbols_close, "\u2510", "\\@urcorner", true);
+defineSymbol(symbols_math, ams, symbols_open, "\u2514", "\\@llcorner", true);
+defineSymbol(symbols_math, ams, symbols_close, "\u2518", "\\@lrcorner", true); // AMS Binary Relations
+
+defineSymbol(symbols_math, ams, rel, "\u2266", "\\leqq", true);
+defineSymbol(symbols_math, ams, rel, "\u2A7D", "\\leqslant", true);
+defineSymbol(symbols_math, ams, rel, "\u2A95", "\\eqslantless", true);
+defineSymbol(symbols_math, ams, rel, "\u2272", "\\lesssim", true);
+defineSymbol(symbols_math, ams, rel, "\u2A85", "\\lessapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u224A", "\\approxeq", true);
+defineSymbol(symbols_math, ams, bin, "\u22D6", "\\lessdot");
+defineSymbol(symbols_math, ams, rel, "\u22D8", "\\lll", true);
+defineSymbol(symbols_math, ams, rel, "\u2276", "\\lessgtr", true);
+defineSymbol(symbols_math, ams, rel, "\u22DA", "\\lesseqgtr", true);
+defineSymbol(symbols_math, ams, rel, "\u2A8B", "\\lesseqqgtr", true);
+defineSymbol(symbols_math, ams, rel, "\u2251", "\\doteqdot");
+defineSymbol(symbols_math, ams, rel, "\u2253", "\\risingdotseq", true);
+defineSymbol(symbols_math, ams, rel, "\u2252", "\\fallingdotseq", true);
+defineSymbol(symbols_math, ams, rel, "\u223D", "\\backsim", true);
+defineSymbol(symbols_math, ams, rel, "\u22CD", "\\backsimeq", true);
+defineSymbol(symbols_math, ams, rel, "\u2AC5", "\\subseteqq", true);
+defineSymbol(symbols_math, ams, rel, "\u22D0", "\\Subset", true);
+defineSymbol(symbols_math, ams, rel, "\u228F", "\\sqsubset", true);
+defineSymbol(symbols_math, ams, rel, "\u227C", "\\preccurlyeq", true);
+defineSymbol(symbols_math, ams, rel, "\u22DE", "\\curlyeqprec", true);
+defineSymbol(symbols_math, ams, rel, "\u227E", "\\precsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB7", "\\precapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u22B2", "\\vartriangleleft");
+defineSymbol(symbols_math, ams, rel, "\u22B4", "\\trianglelefteq");
+defineSymbol(symbols_math, ams, rel, "\u22A8", "\\vDash", true);
+defineSymbol(symbols_math, ams, rel, "\u22AA", "\\Vvdash", true);
+defineSymbol(symbols_math, ams, rel, "\u2323", "\\smallsmile");
+defineSymbol(symbols_math, ams, rel, "\u2322", "\\smallfrown");
+defineSymbol(symbols_math, ams, rel, "\u224F", "\\bumpeq", true);
+defineSymbol(symbols_math, ams, rel, "\u224E", "\\Bumpeq", true);
+defineSymbol(symbols_math, ams, rel, "\u2267", "\\geqq", true);
+defineSymbol(symbols_math, ams, rel, "\u2A7E", "\\geqslant", true);
+defineSymbol(symbols_math, ams, rel, "\u2A96", "\\eqslantgtr", true);
+defineSymbol(symbols_math, ams, rel, "\u2273", "\\gtrsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2A86", "\\gtrapprox", true);
+defineSymbol(symbols_math, ams, bin, "\u22D7", "\\gtrdot");
+defineSymbol(symbols_math, ams, rel, "\u22D9", "\\ggg", true);
+defineSymbol(symbols_math, ams, rel, "\u2277", "\\gtrless", true);
+defineSymbol(symbols_math, ams, rel, "\u22DB", "\\gtreqless", true);
+defineSymbol(symbols_math, ams, rel, "\u2A8C", "\\gtreqqless", true);
+defineSymbol(symbols_math, ams, rel, "\u2256", "\\eqcirc", true);
+defineSymbol(symbols_math, ams, rel, "\u2257", "\\circeq", true);
+defineSymbol(symbols_math, ams, rel, "\u225C", "\\triangleq", true);
+defineSymbol(symbols_math, ams, rel, "\u223C", "\\thicksim");
+defineSymbol(symbols_math, ams, rel, "\u2248", "\\thickapprox");
+defineSymbol(symbols_math, ams, rel, "\u2AC6", "\\supseteqq", true);
+defineSymbol(symbols_math, ams, rel, "\u22D1", "\\Supset", true);
+defineSymbol(symbols_math, ams, rel, "\u2290", "\\sqsupset", true);
+defineSymbol(symbols_math, ams, rel, "\u227D", "\\succcurlyeq", true);
+defineSymbol(symbols_math, ams, rel, "\u22DF", "\\curlyeqsucc", true);
+defineSymbol(symbols_math, ams, rel, "\u227F", "\\succsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB8", "\\succapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u22B3", "\\vartriangleright");
+defineSymbol(symbols_math, ams, rel, "\u22B5", "\\trianglerighteq");
+defineSymbol(symbols_math, ams, rel, "\u22A9", "\\Vdash", true);
+defineSymbol(symbols_math, ams, rel, "\u2223", "\\shortmid");
+defineSymbol(symbols_math, ams, rel, "\u2225", "\\shortparallel");
+defineSymbol(symbols_math, ams, rel, "\u226C", "\\between", true);
+defineSymbol(symbols_math, ams, rel, "\u22D4", "\\pitchfork", true);
+defineSymbol(symbols_math, ams, rel, "\u221D", "\\varpropto");
+defineSymbol(symbols_math, ams, rel, "\u25C0", "\\blacktriangleleft"); // unicode-math says that \therefore is a mathord atom.
+// We kept the amssymb atom type, which is rel.
+
+defineSymbol(symbols_math, ams, rel, "\u2234", "\\therefore", true);
+defineSymbol(symbols_math, ams, rel, "\u220D", "\\backepsilon");
+defineSymbol(symbols_math, ams, rel, "\u25B6", "\\blacktriangleright"); // unicode-math says that \because is a mathord atom.
+// We kept the amssymb atom type, which is rel.
+
+defineSymbol(symbols_math, ams, rel, "\u2235", "\\because", true);
+defineSymbol(symbols_math, ams, rel, "\u22D8", "\\llless");
+defineSymbol(symbols_math, ams, rel, "\u22D9", "\\gggtr");
+defineSymbol(symbols_math, ams, bin, "\u22B2", "\\lhd");
+defineSymbol(symbols_math, ams, bin, "\u22B3", "\\rhd");
+defineSymbol(symbols_math, ams, rel, "\u2242", "\\eqsim", true);
+defineSymbol(symbols_math, main, rel, "\u22C8", "\\Join");
+defineSymbol(symbols_math, ams, rel, "\u2251", "\\Doteq", true); // AMS Binary Operators
+
+defineSymbol(symbols_math, ams, bin, "\u2214", "\\dotplus", true);
+defineSymbol(symbols_math, ams, bin, "\u2216", "\\smallsetminus");
+defineSymbol(symbols_math, ams, bin, "\u22D2", "\\Cap", true);
+defineSymbol(symbols_math, ams, bin, "\u22D3", "\\Cup", true);
+defineSymbol(symbols_math, ams, bin, "\u2A5E", "\\doublebarwedge", true);
+defineSymbol(symbols_math, ams, bin, "\u229F", "\\boxminus", true);
+defineSymbol(symbols_math, ams, bin, "\u229E", "\\boxplus", true);
+defineSymbol(symbols_math, ams, bin, "\u22C7", "\\divideontimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22C9", "\\ltimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22CA", "\\rtimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22CB", "\\leftthreetimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22CC", "\\rightthreetimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22CF", "\\curlywedge", true);
+defineSymbol(symbols_math, ams, bin, "\u22CE", "\\curlyvee", true);
+defineSymbol(symbols_math, ams, bin, "\u229D", "\\circleddash", true);
+defineSymbol(symbols_math, ams, bin, "\u229B", "\\circledast", true);
+defineSymbol(symbols_math, ams, bin, "\u22C5", "\\centerdot");
+defineSymbol(symbols_math, ams, bin, "\u22BA", "\\intercal", true);
+defineSymbol(symbols_math, ams, bin, "\u22D2", "\\doublecap");
+defineSymbol(symbols_math, ams, bin, "\u22D3", "\\doublecup");
+defineSymbol(symbols_math, ams, bin, "\u22A0", "\\boxtimes", true); // AMS Arrows
+// Note: unicode-math maps \u21e2 to their own function \rightdasharrow.
+// We'll map it to AMS function \dashrightarrow. It produces the same atom.
+
+defineSymbol(symbols_math, ams, rel, "\u21E2", "\\dashrightarrow", true); // unicode-math maps \u21e0 to \leftdasharrow. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u21E0", "\\dashleftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21C7", "\\leftleftarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21C6", "\\leftrightarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21DA", "\\Lleftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u219E", "\\twoheadleftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21A2", "\\leftarrowtail", true);
+defineSymbol(symbols_math, ams, rel, "\u21AB", "\\looparrowleft", true);
+defineSymbol(symbols_math, ams, rel, "\u21CB", "\\leftrightharpoons", true);
+defineSymbol(symbols_math, ams, rel, "\u21B6", "\\curvearrowleft", true); // unicode-math maps \u21ba to \acwopencirclearrow. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u21BA", "\\circlearrowleft", true);
+defineSymbol(symbols_math, ams, rel, "\u21B0", "\\Lsh", true);
+defineSymbol(symbols_math, ams, rel, "\u21C8", "\\upuparrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21BF", "\\upharpoonleft", true);
+defineSymbol(symbols_math, ams, rel, "\u21C3", "\\downharpoonleft", true);
+defineSymbol(symbols_math, ams, rel, "\u22B8", "\\multimap", true);
+defineSymbol(symbols_math, ams, rel, "\u21AD", "\\leftrightsquigarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21C9", "\\rightrightarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21C4", "\\rightleftarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21A0", "\\twoheadrightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21A3", "\\rightarrowtail", true);
+defineSymbol(symbols_math, ams, rel, "\u21AC", "\\looparrowright", true);
+defineSymbol(symbols_math, ams, rel, "\u21B7", "\\curvearrowright", true); // unicode-math maps \u21bb to \cwopencirclearrow. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u21BB", "\\circlearrowright", true);
+defineSymbol(symbols_math, ams, rel, "\u21B1", "\\Rsh", true);
+defineSymbol(symbols_math, ams, rel, "\u21CA", "\\downdownarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21BE", "\\upharpoonright", true);
+defineSymbol(symbols_math, ams, rel, "\u21C2", "\\downharpoonright", true);
+defineSymbol(symbols_math, ams, rel, "\u21DD", "\\rightsquigarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21DD", "\\leadsto");
+defineSymbol(symbols_math, ams, rel, "\u21DB", "\\Rrightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21BE", "\\restriction");
+defineSymbol(symbols_math, main, symbols_textord, "\u2018", "`");
+defineSymbol(symbols_math, main, symbols_textord, "$", "\\$");
+defineSymbol(symbols_text, main, symbols_textord, "$", "\\$");
+defineSymbol(symbols_text, main, symbols_textord, "$", "\\textdollar");
+defineSymbol(symbols_math, main, symbols_textord, "%", "\\%");
+defineSymbol(symbols_text, main, symbols_textord, "%", "\\%");
+defineSymbol(symbols_math, main, symbols_textord, "_", "\\_");
+defineSymbol(symbols_text, main, symbols_textord, "_", "\\_");
+defineSymbol(symbols_text, main, symbols_textord, "_", "\\textunderscore");
+defineSymbol(symbols_math, main, symbols_textord, "\u2220", "\\angle", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u221E", "\\infty", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2032", "\\prime");
+defineSymbol(symbols_math, main, symbols_textord, "\u25B3", "\\triangle");
+defineSymbol(symbols_math, main, symbols_textord, "\u0393", "\\Gamma", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u0394", "\\Delta", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u0398", "\\Theta", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u039B", "\\Lambda", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u039E", "\\Xi", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A0", "\\Pi", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A3", "\\Sigma", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A5", "\\Upsilon", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A6", "\\Phi", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A8", "\\Psi", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A9", "\\Omega", true);
+defineSymbol(symbols_math, main, symbols_textord, "A", "\u0391");
+defineSymbol(symbols_math, main, symbols_textord, "B", "\u0392");
+defineSymbol(symbols_math, main, symbols_textord, "E", "\u0395");
+defineSymbol(symbols_math, main, symbols_textord, "Z", "\u0396");
+defineSymbol(symbols_math, main, symbols_textord, "H", "\u0397");
+defineSymbol(symbols_math, main, symbols_textord, "I", "\u0399");
+defineSymbol(symbols_math, main, symbols_textord, "K", "\u039A");
+defineSymbol(symbols_math, main, symbols_textord, "M", "\u039C");
+defineSymbol(symbols_math, main, symbols_textord, "N", "\u039D");
+defineSymbol(symbols_math, main, symbols_textord, "O", "\u039F");
+defineSymbol(symbols_math, main, symbols_textord, "P", "\u03A1");
+defineSymbol(symbols_math, main, symbols_textord, "T", "\u03A4");
+defineSymbol(symbols_math, main, symbols_textord, "X", "\u03A7");
+defineSymbol(symbols_math, main, symbols_textord, "\xAC", "\\neg", true);
+defineSymbol(symbols_math, main, symbols_textord, "\xAC", "\\lnot");
+defineSymbol(symbols_math, main, symbols_textord, "\u22A4", "\\top");
+defineSymbol(symbols_math, main, symbols_textord, "\u22A5", "\\bot");
+defineSymbol(symbols_math, main, symbols_textord, "\u2205", "\\emptyset");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2205", "\\varnothing");
+defineSymbol(symbols_math, main, mathord, "\u03B1", "\\alpha", true);
+defineSymbol(symbols_math, main, mathord, "\u03B2", "\\beta", true);
+defineSymbol(symbols_math, main, mathord, "\u03B3", "\\gamma", true);
+defineSymbol(symbols_math, main, mathord, "\u03B4", "\\delta", true);
+defineSymbol(symbols_math, main, mathord, "\u03F5", "\\epsilon", true);
+defineSymbol(symbols_math, main, mathord, "\u03B6", "\\zeta", true);
+defineSymbol(symbols_math, main, mathord, "\u03B7", "\\eta", true);
+defineSymbol(symbols_math, main, mathord, "\u03B8", "\\theta", true);
+defineSymbol(symbols_math, main, mathord, "\u03B9", "\\iota", true);
+defineSymbol(symbols_math, main, mathord, "\u03BA", "\\kappa", true);
+defineSymbol(symbols_math, main, mathord, "\u03BB", "\\lambda", true);
+defineSymbol(symbols_math, main, mathord, "\u03BC", "\\mu", true);
+defineSymbol(symbols_math, main, mathord, "\u03BD", "\\nu", true);
+defineSymbol(symbols_math, main, mathord, "\u03BE", "\\xi", true);
+defineSymbol(symbols_math, main, mathord, "\u03BF", "\\omicron", true);
+defineSymbol(symbols_math, main, mathord, "\u03C0", "\\pi", true);
+defineSymbol(symbols_math, main, mathord, "\u03C1", "\\rho", true);
+defineSymbol(symbols_math, main, mathord, "\u03C3", "\\sigma", true);
+defineSymbol(symbols_math, main, mathord, "\u03C4", "\\tau", true);
+defineSymbol(symbols_math, main, mathord, "\u03C5", "\\upsilon", true);
+defineSymbol(symbols_math, main, mathord, "\u03D5", "\\phi", true);
+defineSymbol(symbols_math, main, mathord, "\u03C7", "\\chi", true);
+defineSymbol(symbols_math, main, mathord, "\u03C8", "\\psi", true);
+defineSymbol(symbols_math, main, mathord, "\u03C9", "\\omega", true);
+defineSymbol(symbols_math, main, mathord, "\u03B5", "\\varepsilon", true);
+defineSymbol(symbols_math, main, mathord, "\u03D1", "\\vartheta", true);
+defineSymbol(symbols_math, main, mathord, "\u03D6", "\\varpi", true);
+defineSymbol(symbols_math, main, mathord, "\u03F1", "\\varrho", true);
+defineSymbol(symbols_math, main, mathord, "\u03C2", "\\varsigma", true);
+defineSymbol(symbols_math, main, mathord, "\u03C6", "\\varphi", true);
+defineSymbol(symbols_math, main, bin, "\u2217", "*");
+defineSymbol(symbols_math, main, bin, "+", "+");
+defineSymbol(symbols_math, main, bin, "\u2212", "-");
+defineSymbol(symbols_math, main, bin, "\u22C5", "\\cdot", true);
+defineSymbol(symbols_math, main, bin, "\u2218", "\\circ");
+defineSymbol(symbols_math, main, bin, "\xF7", "\\div", true);
+defineSymbol(symbols_math, main, bin, "\xB1", "\\pm", true);
+defineSymbol(symbols_math, main, bin, "\xD7", "\\times", true);
+defineSymbol(symbols_math, main, bin, "\u2229", "\\cap", true);
+defineSymbol(symbols_math, main, bin, "\u222A", "\\cup", true);
+defineSymbol(symbols_math, main, bin, "\u2216", "\\setminus");
+defineSymbol(symbols_math, main, bin, "\u2227", "\\land");
+defineSymbol(symbols_math, main, bin, "\u2228", "\\lor");
+defineSymbol(symbols_math, main, bin, "\u2227", "\\wedge", true);
+defineSymbol(symbols_math, main, bin, "\u2228", "\\vee", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u221A", "\\surd");
+defineSymbol(symbols_math, main, symbols_open, "\u27E8", "\\langle", true);
+defineSymbol(symbols_math, main, symbols_open, "\u2223", "\\lvert");
+defineSymbol(symbols_math, main, symbols_open, "\u2225", "\\lVert");
+defineSymbol(symbols_math, main, symbols_close, "?", "?");
+defineSymbol(symbols_math, main, symbols_close, "!", "!");
+defineSymbol(symbols_math, main, symbols_close, "\u27E9", "\\rangle", true);
+defineSymbol(symbols_math, main, symbols_close, "\u2223", "\\rvert");
+defineSymbol(symbols_math, main, symbols_close, "\u2225", "\\rVert");
+defineSymbol(symbols_math, main, rel, "=", "=");
+defineSymbol(symbols_math, main, rel, ":", ":");
+defineSymbol(symbols_math, main, rel, "\u2248", "\\approx", true);
+defineSymbol(symbols_math, main, rel, "\u2245", "\\cong", true);
+defineSymbol(symbols_math, main, rel, "\u2265", "\\ge");
+defineSymbol(symbols_math, main, rel, "\u2265", "\\geq", true);
+defineSymbol(symbols_math, main, rel, "\u2190", "\\gets");
+defineSymbol(symbols_math, main, rel, ">", "\\gt", true);
+defineSymbol(symbols_math, main, rel, "\u2208", "\\in", true);
+defineSymbol(symbols_math, main, rel, "\uE020", "\\@not");
+defineSymbol(symbols_math, main, rel, "\u2282", "\\subset", true);
+defineSymbol(symbols_math, main, rel, "\u2283", "\\supset", true);
+defineSymbol(symbols_math, main, rel, "\u2286", "\\subseteq", true);
+defineSymbol(symbols_math, main, rel, "\u2287", "\\supseteq", true);
+defineSymbol(symbols_math, ams, rel, "\u2288", "\\nsubseteq", true);
+defineSymbol(symbols_math, ams, rel, "\u2289", "\\nsupseteq", true);
+defineSymbol(symbols_math, main, rel, "\u22A8", "\\models");
+defineSymbol(symbols_math, main, rel, "\u2190", "\\leftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2264", "\\le");
+defineSymbol(symbols_math, main, rel, "\u2264", "\\leq", true);
+defineSymbol(symbols_math, main, rel, "<", "\\lt", true);
+defineSymbol(symbols_math, main, rel, "\u2192", "\\rightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2192", "\\to");
+defineSymbol(symbols_math, ams, rel, "\u2271", "\\ngeq", true);
+defineSymbol(symbols_math, ams, rel, "\u2270", "\\nleq", true);
+defineSymbol(symbols_math, main, symbols_spacing, "\xA0", "\\ ");
+defineSymbol(symbols_math, main, symbols_spacing, "\xA0", "~");
+defineSymbol(symbols_math, main, symbols_spacing, "\xA0", "\\space"); // Ref: LaTeX Source 2e: \DeclareRobustCommand{\nobreakspace}{%
+
+defineSymbol(symbols_math, main, symbols_spacing, "\xA0", "\\nobreakspace");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", "\\ ");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", " ");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", "~");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", "\\space");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", "\\nobreakspace");
+defineSymbol(symbols_math, main, symbols_spacing, null, "\\nobreak");
+defineSymbol(symbols_math, main, symbols_spacing, null, "\\allowbreak");
+defineSymbol(symbols_math, main, punct, ",", ",");
+defineSymbol(symbols_math, main, punct, ";", ";");
+defineSymbol(symbols_math, ams, bin, "\u22BC", "\\barwedge", true);
+defineSymbol(symbols_math, ams, bin, "\u22BB", "\\veebar", true);
+defineSymbol(symbols_math, main, bin, "\u2299", "\\odot", true);
+defineSymbol(symbols_math, main, bin, "\u2295", "\\oplus", true);
+defineSymbol(symbols_math, main, bin, "\u2297", "\\otimes", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2202", "\\partial", true);
+defineSymbol(symbols_math, main, bin, "\u2298", "\\oslash", true);
+defineSymbol(symbols_math, ams, bin, "\u229A", "\\circledcirc", true);
+defineSymbol(symbols_math, ams, bin, "\u22A1", "\\boxdot", true);
+defineSymbol(symbols_math, main, bin, "\u25B3", "\\bigtriangleup");
+defineSymbol(symbols_math, main, bin, "\u25BD", "\\bigtriangledown");
+defineSymbol(symbols_math, main, bin, "\u2020", "\\dagger");
+defineSymbol(symbols_math, main, bin, "\u22C4", "\\diamond");
+defineSymbol(symbols_math, main, bin, "\u22C6", "\\star");
+defineSymbol(symbols_math, main, bin, "\u25C3", "\\triangleleft");
+defineSymbol(symbols_math, main, bin, "\u25B9", "\\triangleright");
+defineSymbol(symbols_math, main, symbols_open, "{", "\\{");
+defineSymbol(symbols_text, main, symbols_textord, "{", "\\{");
+defineSymbol(symbols_text, main, symbols_textord, "{", "\\textbraceleft");
+defineSymbol(symbols_math, main, symbols_close, "}", "\\}");
+defineSymbol(symbols_text, main, symbols_textord, "}", "\\}");
+defineSymbol(symbols_text, main, symbols_textord, "}", "\\textbraceright");
+defineSymbol(symbols_math, main, symbols_open, "{", "\\lbrace");
+defineSymbol(symbols_math, main, symbols_close, "}", "\\rbrace");
+defineSymbol(symbols_math, main, symbols_open, "[", "\\lbrack", true);
+defineSymbol(symbols_text, main, symbols_textord, "[", "\\lbrack", true);
+defineSymbol(symbols_math, main, symbols_close, "]", "\\rbrack", true);
+defineSymbol(symbols_text, main, symbols_textord, "]", "\\rbrack", true);
+defineSymbol(symbols_math, main, symbols_open, "(", "\\lparen", true);
+defineSymbol(symbols_math, main, symbols_close, ")", "\\rparen", true);
+defineSymbol(symbols_text, main, symbols_textord, "<", "\\textless", true); // in T1 fontenc
+
+defineSymbol(symbols_text, main, symbols_textord, ">", "\\textgreater", true); // in T1 fontenc
+
+defineSymbol(symbols_math, main, symbols_open, "\u230A", "\\lfloor", true);
+defineSymbol(symbols_math, main, symbols_close, "\u230B", "\\rfloor", true);
+defineSymbol(symbols_math, main, symbols_open, "\u2308", "\\lceil", true);
+defineSymbol(symbols_math, main, symbols_close, "\u2309", "\\rceil", true);
+defineSymbol(symbols_math, main, symbols_textord, "\\", "\\backslash");
+defineSymbol(symbols_math, main, symbols_textord, "\u2223", "|");
+defineSymbol(symbols_math, main, symbols_textord, "\u2223", "\\vert");
+defineSymbol(symbols_text, main, symbols_textord, "|", "\\textbar", true); // in T1 fontenc
+
+defineSymbol(symbols_math, main, symbols_textord, "\u2225", "\\|");
+defineSymbol(symbols_math, main, symbols_textord, "\u2225", "\\Vert");
+defineSymbol(symbols_text, main, symbols_textord, "\u2225", "\\textbardbl");
+defineSymbol(symbols_text, main, symbols_textord, "~", "\\textasciitilde");
+defineSymbol(symbols_text, main, symbols_textord, "\\", "\\textbackslash");
+defineSymbol(symbols_text, main, symbols_textord, "^", "\\textasciicircum");
+defineSymbol(symbols_math, main, rel, "\u2191", "\\uparrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D1", "\\Uparrow", true);
+defineSymbol(symbols_math, main, rel, "\u2193", "\\downarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D3", "\\Downarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2195", "\\updownarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D5", "\\Updownarrow", true);
+defineSymbol(symbols_math, main, op, "\u2210", "\\coprod");
+defineSymbol(symbols_math, main, op, "\u22C1", "\\bigvee");
+defineSymbol(symbols_math, main, op, "\u22C0", "\\bigwedge");
+defineSymbol(symbols_math, main, op, "\u2A04", "\\biguplus");
+defineSymbol(symbols_math, main, op, "\u22C2", "\\bigcap");
+defineSymbol(symbols_math, main, op, "\u22C3", "\\bigcup");
+defineSymbol(symbols_math, main, op, "\u222B", "\\int");
+defineSymbol(symbols_math, main, op, "\u222B", "\\intop");
+defineSymbol(symbols_math, main, op, "\u222C", "\\iint");
+defineSymbol(symbols_math, main, op, "\u222D", "\\iiint");
+defineSymbol(symbols_math, main, op, "\u220F", "\\prod");
+defineSymbol(symbols_math, main, op, "\u2211", "\\sum");
+defineSymbol(symbols_math, main, op, "\u2A02", "\\bigotimes");
+defineSymbol(symbols_math, main, op, "\u2A01", "\\bigoplus");
+defineSymbol(symbols_math, main, op, "\u2A00", "\\bigodot");
+defineSymbol(symbols_math, main, op, "\u222E", "\\oint");
+defineSymbol(symbols_math, main, op, "\u2A06", "\\bigsqcup");
+defineSymbol(symbols_math, main, op, "\u222B", "\\smallint");
+defineSymbol(symbols_text, main, symbols_inner, "\u2026", "\\textellipsis");
+defineSymbol(symbols_math, main, symbols_inner, "\u2026", "\\mathellipsis");
+defineSymbol(symbols_text, main, symbols_inner, "\u2026", "\\ldots", true);
+defineSymbol(symbols_math, main, symbols_inner, "\u2026", "\\ldots", true);
+defineSymbol(symbols_math, main, symbols_inner, "\u22EF", "\\@cdots", true);
+defineSymbol(symbols_math, main, symbols_inner, "\u22F1", "\\ddots", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u22EE", "\\varvdots"); // \vdots is a macro
+
+defineSymbol(symbols_math, main, symbols_accent, "\u02CA", "\\acute");
+defineSymbol(symbols_math, main, symbols_accent, "\u02CB", "\\grave");
+defineSymbol(symbols_math, main, symbols_accent, "\xA8", "\\ddot");
+defineSymbol(symbols_math, main, symbols_accent, "~", "\\tilde");
+defineSymbol(symbols_math, main, symbols_accent, "\u02C9", "\\bar");
+defineSymbol(symbols_math, main, symbols_accent, "\u02D8", "\\breve");
+defineSymbol(symbols_math, main, symbols_accent, "\u02C7", "\\check");
+defineSymbol(symbols_math, main, symbols_accent, "^", "\\hat");
+defineSymbol(symbols_math, main, symbols_accent, "\u20D7", "\\vec");
+defineSymbol(symbols_math, main, symbols_accent, "\u02D9", "\\dot");
+defineSymbol(symbols_math, main, symbols_accent, "\u02DA", "\\mathring"); // \imath and \jmath should be invariant to \mathrm, \mathbf, etc., so use PUA
+
+defineSymbol(symbols_math, main, mathord, "\uE131", "\\@imath");
+defineSymbol(symbols_math, main, mathord, "\uE237", "\\@jmath");
+defineSymbol(symbols_math, main, symbols_textord, "\u0131", "\u0131");
+defineSymbol(symbols_math, main, symbols_textord, "\u0237", "\u0237");
+defineSymbol(symbols_text, main, symbols_textord, "\u0131", "\\i", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u0237", "\\j", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xDF", "\\ss", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xE6", "\\ae", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u0153", "\\oe", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xF8", "\\o", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xC6", "\\AE", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u0152", "\\OE", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xD8", "\\O", true);
+defineSymbol(symbols_text, main, symbols_accent, "\u02CA", "\\'"); // acute
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02CB", "\\`"); // grave
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02C6", "\\^"); // circumflex
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02DC", "\\~"); // tilde
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02C9", "\\="); // macron
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02D8", "\\u"); // breve
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02D9", "\\."); // dot above
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02DA", "\\r"); // ring above
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02C7", "\\v"); // caron
+
+defineSymbol(symbols_text, main, symbols_accent, "\xA8", '\\"'); // diaresis
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02DD", "\\H"); // double acute
+
+defineSymbol(symbols_text, main, symbols_accent, "\u25EF", "\\textcircled"); // \bigcirc glyph
+// These ligatures are detected and created in Parser.js's `formLigatures`.
+
+var ligatures = {
+  "--": true,
+  "---": true,
+  "``": true,
+  "''": true
+};
+defineSymbol(symbols_text, main, symbols_textord, "\u2013", "--", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u2013", "\\textendash");
+defineSymbol(symbols_text, main, symbols_textord, "\u2014", "---", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u2014", "\\textemdash");
+defineSymbol(symbols_text, main, symbols_textord, "\u2018", "`", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u2018", "\\textquoteleft");
+defineSymbol(symbols_text, main, symbols_textord, "\u2019", "'", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u2019", "\\textquoteright");
+defineSymbol(symbols_text, main, symbols_textord, "\u201C", "``", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u201C", "\\textquotedblleft");
+defineSymbol(symbols_text, main, symbols_textord, "\u201D", "''", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u201D", "\\textquotedblright"); //  \degree from gensymb package
+
+defineSymbol(symbols_math, main, symbols_textord, "\xB0", "\\degree", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xB0", "\\degree"); // \textdegree from inputenc package
+
+defineSymbol(symbols_text, main, symbols_textord, "\xB0", "\\textdegree", true); // TODO: In LaTeX, \pounds can generate a different character in text and math
+// mode, but among our fonts, only Main-Regular defines this character "163".
+
+defineSymbol(symbols_math, main, symbols_textord, "\xA3", "\\pounds");
+defineSymbol(symbols_math, main, symbols_textord, "\xA3", "\\mathsterling", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xA3", "\\pounds");
+defineSymbol(symbols_text, main, symbols_textord, "\xA3", "\\textsterling", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2720", "\\maltese");
+defineSymbol(symbols_text, ams, symbols_textord, "\u2720", "\\maltese"); // There are lots of symbols which are the same, so we add them in afterwards.
+// All of these are textords in math mode
+
+var mathTextSymbols = "0123456789/@.\"";
+
+for (var symbols_i = 0; symbols_i < mathTextSymbols.length; symbols_i++) {
+  var symbols_ch = mathTextSymbols.charAt(symbols_i);
+  defineSymbol(symbols_math, main, symbols_textord, symbols_ch, symbols_ch);
+} // All of these are textords in text mode
+
+
+var textSymbols = "0123456789!@*()-=+\";:?/.,";
+
+for (var src_symbols_i = 0; src_symbols_i < textSymbols.length; src_symbols_i++) {
+  var _ch = textSymbols.charAt(src_symbols_i);
+
+  defineSymbol(symbols_text, main, symbols_textord, _ch, _ch);
+} // All of these are textords in text mode, and mathords in math mode
+
+
+var letters = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";
+
+for (var symbols_i2 = 0; symbols_i2 < letters.length; symbols_i2++) {
+  var _ch2 = letters.charAt(symbols_i2);
+
+  defineSymbol(symbols_math, main, mathord, _ch2, _ch2);
+  defineSymbol(symbols_text, main, symbols_textord, _ch2, _ch2);
+} // Blackboard bold and script letters in Unicode range
+
+
+defineSymbol(symbols_math, ams, symbols_textord, "C", "\u2102"); // blackboard bold
+
+defineSymbol(symbols_text, ams, symbols_textord, "C", "\u2102");
+defineSymbol(symbols_math, ams, symbols_textord, "H", "\u210D");
+defineSymbol(symbols_text, ams, symbols_textord, "H", "\u210D");
+defineSymbol(symbols_math, ams, symbols_textord, "N", "\u2115");
+defineSymbol(symbols_text, ams, symbols_textord, "N", "\u2115");
+defineSymbol(symbols_math, ams, symbols_textord, "P", "\u2119");
+defineSymbol(symbols_text, ams, symbols_textord, "P", "\u2119");
+defineSymbol(symbols_math, ams, symbols_textord, "Q", "\u211A");
+defineSymbol(symbols_text, ams, symbols_textord, "Q", "\u211A");
+defineSymbol(symbols_math, ams, symbols_textord, "R", "\u211D");
+defineSymbol(symbols_text, ams, symbols_textord, "R", "\u211D");
+defineSymbol(symbols_math, ams, symbols_textord, "Z", "\u2124");
+defineSymbol(symbols_text, ams, symbols_textord, "Z", "\u2124");
+defineSymbol(symbols_math, main, mathord, "h", "\u210E"); // italic h, Planck constant
+
+defineSymbol(symbols_text, main, mathord, "h", "\u210E"); // The next loop loads wide (surrogate pair) characters.
+// We support some letters in the Unicode range U+1D400 to U+1D7FF,
+// Mathematical Alphanumeric Symbols.
+// Some editors do not deal well with wide characters. So don't write the
+// string into this file. Instead, create the string from the surrogate pair.
+
+var symbols_wideChar = "";
+
+for (var symbols_i3 = 0; symbols_i3 < letters.length; symbols_i3++) {
+  var _ch3 = letters.charAt(symbols_i3); // The hex numbers in the next line are a surrogate pair.
+  // 0xD835 is the high surrogate for all letters in the range we support.
+  // 0xDC00 is the low surrogate for bold A.
+
+
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDC00 + symbols_i3); // A-Z a-z bold
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDC34 + symbols_i3); // A-Z a-z italic
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDC68 + symbols_i3); // A-Z a-z bold italic
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDD04 + symbols_i3); // A-Z a-z Fractur
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDDA0 + symbols_i3); // A-Z a-z sans-serif
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDDD4 + symbols_i3); // A-Z a-z sans bold
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDE08 + symbols_i3); // A-Z a-z sans italic
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDE70 + symbols_i3); // A-Z a-z monospace
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+
+  if (symbols_i3 < 26) {
+    // KaTeX fonts have only capital letters for blackboard bold and script.
+    // See exception for k below.
+    symbols_wideChar = String.fromCharCode(0xD835, 0xDD38 + symbols_i3); // A-Z double struck
+
+    defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+    defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+    symbols_wideChar = String.fromCharCode(0xD835, 0xDC9C + symbols_i3); // A-Z script
+
+    defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+    defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  } // TODO: Add bold script when it is supported by a KaTeX font.
+
+} // "k" is the only double struck lower case letter in the KaTeX fonts.
+
+
+symbols_wideChar = String.fromCharCode(0xD835, 0xDD5C); // k double struck
+
+defineSymbol(symbols_math, main, mathord, "k", symbols_wideChar);
+defineSymbol(symbols_text, main, symbols_textord, "k", symbols_wideChar); // Next, some wide character numerals
+
+for (var symbols_i4 = 0; symbols_i4 < 10; symbols_i4++) {
+  var _ch4 = symbols_i4.toString();
+
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDFCE + symbols_i4); // 0-9 bold
+
+  defineSymbol(symbols_math, main, mathord, _ch4, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch4, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDFE2 + symbols_i4); // 0-9 sans serif
+
+  defineSymbol(symbols_math, main, mathord, _ch4, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch4, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDFEC + symbols_i4); // 0-9 bold sans
+
+  defineSymbol(symbols_math, main, mathord, _ch4, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch4, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDFF6 + symbols_i4); // 0-9 monospace
+
+  defineSymbol(symbols_math, main, mathord, _ch4, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch4, symbols_wideChar);
+} // We add these Latin-1 letters as symbols for backwards-compatibility,
+// but they are not actually in the font, nor are they supported by the
+// Unicode accent mechanism, so they fall back to Times font and look ugly.
+// TODO(edemaine): Fix this.
+
+
+var extraLatin = "\xC7\xD0\xDE\xE7\xFE";
+
+for (var _i5 = 0; _i5 < extraLatin.length; _i5++) {
+  var _ch5 = extraLatin.charAt(_i5);
+
+  defineSymbol(symbols_math, main, mathord, _ch5, _ch5);
+  defineSymbol(symbols_text, main, symbols_textord, _ch5, _ch5);
+}
+// CONCATENATED MODULE: ./src/wide-character.js
+/**
+ * This file provides support for Unicode range U+1D400 to U+1D7FF,
+ * Mathematical Alphanumeric Symbols.
+ *
+ * Function wideCharacterFont takes a wide character as input and returns
+ * the font information necessary to render it properly.
+ */
+
+/**
+ * Data below is from https://www.unicode.org/charts/PDF/U1D400.pdf
+ * That document sorts characters into groups by font type, say bold or italic.
+ *
+ * In the arrays below, each subarray consists three elements:
+ *      * The CSS class of that group when in math mode.
+ *      * The CSS class of that group when in text mode.
+ *      * The font name, so that KaTeX can get font metrics.
+ */
+
+var wideLatinLetterData = [["mathbf", "textbf", "Main-Bold"], // A-Z bold upright
+["mathbf", "textbf", "Main-Bold"], // a-z bold upright
+["mathnormal", "textit", "Math-Italic"], // A-Z italic
+["mathnormal", "textit", "Math-Italic"], // a-z italic
+["boldsymbol", "boldsymbol", "Main-BoldItalic"], // A-Z bold italic
+["boldsymbol", "boldsymbol", "Main-BoldItalic"], // a-z bold italic
+// Map fancy A-Z letters to script, not calligraphic.
+// This aligns with unicode-math and math fonts (except Cambria Math).
+["mathscr", "textscr", "Script-Regular"], // A-Z script
+["", "", ""], // a-z script.  No font
+["", "", ""], // A-Z bold script. No font
+["", "", ""], // a-z bold script. No font
+["mathfrak", "textfrak", "Fraktur-Regular"], // A-Z Fraktur
+["mathfrak", "textfrak", "Fraktur-Regular"], // a-z Fraktur
+["mathbb", "textbb", "AMS-Regular"], // A-Z double-struck
+["mathbb", "textbb", "AMS-Regular"], // k double-struck
+["", "", ""], // A-Z bold Fraktur No font metrics
+["", "", ""], // a-z bold Fraktur.   No font.
+["mathsf", "textsf", "SansSerif-Regular"], // A-Z sans-serif
+["mathsf", "textsf", "SansSerif-Regular"], // a-z sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // A-Z bold sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // a-z bold sans-serif
+["mathitsf", "textitsf", "SansSerif-Italic"], // A-Z italic sans-serif
+["mathitsf", "textitsf", "SansSerif-Italic"], // a-z italic sans-serif
+["", "", ""], // A-Z bold italic sans. No font
+["", "", ""], // a-z bold italic sans. No font
+["mathtt", "texttt", "Typewriter-Regular"], // A-Z monospace
+["mathtt", "texttt", "Typewriter-Regular"]];
+var wideNumeralData = [["mathbf", "textbf", "Main-Bold"], // 0-9 bold
+["", "", ""], // 0-9 double-struck. No KaTeX font.
+["mathsf", "textsf", "SansSerif-Regular"], // 0-9 sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // 0-9 bold sans-serif
+["mathtt", "texttt", "Typewriter-Regular"]];
+var wide_character_wideCharacterFont = function wideCharacterFont(wideChar, mode) {
+  // IE doesn't support codePointAt(). So work with the surrogate pair.
+  var H = wideChar.charCodeAt(0); // high surrogate
+
+  var L = wideChar.charCodeAt(1); // low surrogate
+
+  var codePoint = (H - 0xD800) * 0x400 + (L - 0xDC00) + 0x10000;
+  var j = mode === "math" ? 0 : 1; // column index for CSS class.
+
+  if (0x1D400 <= codePoint && codePoint < 0x1D6A4) {
+    // wideLatinLetterData contains exactly 26 chars on each row.
+    // So we can calculate the relevant row. No traverse necessary.
+    var i = Math.floor((codePoint - 0x1D400) / 26);
+    return [wideLatinLetterData[i][2], wideLatinLetterData[i][j]];
+  } else if (0x1D7CE <= codePoint && codePoint <= 0x1D7FF) {
+    // Numerals, ten per row.
+    var _i = Math.floor((codePoint - 0x1D7CE) / 10);
+
+    return [wideNumeralData[_i][2], wideNumeralData[_i][j]];
+  } else if (codePoint === 0x1D6A5 || codePoint === 0x1D6A6) {
+    // dotless i or j
+    return [wideLatinLetterData[0][2], wideLatinLetterData[0][j]];
+  } else if (0x1D6A6 < codePoint && codePoint < 0x1D7CE) {
+    // Greek letters. Not supported, yet.
+    return ["", ""];
+  } else {
+    // We don't support any wide characters outside 1D400–1D7FF.
+    throw new src_ParseError("Unsupported character: " + wideChar);
+  }
+};
+// CONCATENATED MODULE: ./src/Options.js
+/**
+ * This file contains information about the options that the Parser carries
+ * around with it while parsing. Data is held in an `Options` object, and when
+ * recursing, a new `Options` object can be created with the `.with*` and
+ * `.reset` functions.
+ */
+
+var sizeStyleMap = [// Each element contains [textsize, scriptsize, scriptscriptsize].
+// The size mappings are taken from TeX with \normalsize=10pt.
+[1, 1, 1], // size1: [5, 5, 5]              \tiny
+[2, 1, 1], // size2: [6, 5, 5]
+[3, 1, 1], // size3: [7, 5, 5]              \scriptsize
+[4, 2, 1], // size4: [8, 6, 5]              \footnotesize
+[5, 2, 1], // size5: [9, 6, 5]              \small
+[6, 3, 1], // size6: [10, 7, 5]             \normalsize
+[7, 4, 2], // size7: [12, 8, 6]             \large
+[8, 6, 3], // size8: [14.4, 10, 7]          \Large
+[9, 7, 6], // size9: [17.28, 12, 10]        \LARGE
+[10, 8, 7], // size10: [20.74, 14.4, 12]     \huge
+[11, 10, 9]];
+var sizeMultipliers = [// fontMetrics.js:getGlobalMetrics also uses size indexes, so if
+// you change size indexes, change that function.
+0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.44, 1.728, 2.074, 2.488];
+
+var sizeAtStyle = function sizeAtStyle(size, style) {
+  return style.size < 2 ? size : sizeStyleMap[size - 1][style.size - 1];
+}; // In these types, "" (empty string) means "no change".
+
+
+/**
+ * This is the main options class. It contains the current style, size, color,
+ * and font.
+ *
+ * Options objects should not be modified. To create a new Options with
+ * different properties, call a `.having*` method.
+ */
+var Options_Options =
+/*#__PURE__*/
+function () {
+  // A font family applies to a group of fonts (i.e. SansSerif), while a font
+  // represents a specific font (i.e. SansSerif Bold).
+  // See: https://tex.stackexchange.com/questions/22350/difference-between-textrm-and-mathrm
+
+  /**
+   * The base size index.
+   */
+  function Options(data) {
+    this.style = void 0;
+    this.color = void 0;
+    this.size = void 0;
+    this.textSize = void 0;
+    this.phantom = void 0;
+    this.font = void 0;
+    this.fontFamily = void 0;
+    this.fontWeight = void 0;
+    this.fontShape = void 0;
+    this.sizeMultiplier = void 0;
+    this.maxSize = void 0;
+    this.minRuleThickness = void 0;
+    this._fontMetrics = void 0;
+    this.style = data.style;
+    this.color = data.color;
+    this.size = data.size || Options.BASESIZE;
+    this.textSize = data.textSize || this.size;
+    this.phantom = !!data.phantom;
+    this.font = data.font || "";
+    this.fontFamily = data.fontFamily || "";
+    this.fontWeight = data.fontWeight || '';
+    this.fontShape = data.fontShape || '';
+    this.sizeMultiplier = sizeMultipliers[this.size - 1];
+    this.maxSize = data.maxSize;
+    this.minRuleThickness = data.minRuleThickness;
+    this._fontMetrics = undefined;
+  }
+  /**
+   * Returns a new options object with the same properties as "this".  Properties
+   * from "extension" will be copied to the new options object.
+   */
+
+
+  var _proto = Options.prototype;
+
+  _proto.extend = function extend(extension) {
+    var data = {
+      style: this.style,
+      size: this.size,
+      textSize: this.textSize,
+      color: this.color,
+      phantom: this.phantom,
+      font: this.font,
+      fontFamily: this.fontFamily,
+      fontWeight: this.fontWeight,
+      fontShape: this.fontShape,
+      maxSize: this.maxSize,
+      minRuleThickness: this.minRuleThickness
+    };
+
+    for (var key in extension) {
+      if (extension.hasOwnProperty(key)) {
+        data[key] = extension[key];
+      }
+    }
+
+    return new Options(data);
+  }
+  /**
+   * Return an options object with the given style. If `this.style === style`,
+   * returns `this`.
+   */
+  ;
+
+  _proto.havingStyle = function havingStyle(style) {
+    if (this.style === style) {
+      return this;
+    } else {
+      return this.extend({
+        style: style,
+        size: sizeAtStyle(this.textSize, style)
+      });
+    }
+  }
+  /**
+   * Return an options object with a cramped version of the current style. If
+   * the current style is cramped, returns `this`.
+   */
+  ;
+
+  _proto.havingCrampedStyle = function havingCrampedStyle() {
+    return this.havingStyle(this.style.cramp());
+  }
+  /**
+   * Return an options object with the given size and in at least `\textstyle`.
+   * Returns `this` if appropriate.
+   */
+  ;
+
+  _proto.havingSize = function havingSize(size) {
+    if (this.size === size && this.textSize === size) {
+      return this;
+    } else {
+      return this.extend({
+        style: this.style.text(),
+        size: size,
+        textSize: size,
+        sizeMultiplier: sizeMultipliers[size - 1]
+      });
+    }
+  }
+  /**
+   * Like `this.havingSize(BASESIZE).havingStyle(style)`. If `style` is omitted,
+   * changes to at least `\textstyle`.
+   */
+  ;
+
+  _proto.havingBaseStyle = function havingBaseStyle(style) {
+    style = style || this.style.text();
+    var wantSize = sizeAtStyle(Options.BASESIZE, style);
+
+    if (this.size === wantSize && this.textSize === Options.BASESIZE && this.style === style) {
+      return this;
+    } else {
+      return this.extend({
+        style: style,
+        size: wantSize
+      });
+    }
+  }
+  /**
+   * Remove the effect of sizing changes such as \Huge.
+   * Keep the effect of the current style, such as \scriptstyle.
+   */
+  ;
+
+  _proto.havingBaseSizing = function havingBaseSizing() {
+    var size;
+
+    switch (this.style.id) {
+      case 4:
+      case 5:
+        size = 3; // normalsize in scriptstyle
+
+        break;
+
+      case 6:
+      case 7:
+        size = 1; // normalsize in scriptscriptstyle
+
+        break;
+
+      default:
+        size = 6;
+      // normalsize in textstyle or displaystyle
+    }
+
+    return this.extend({
+      style: this.style.text(),
+      size: size
+    });
+  }
+  /**
+   * Create a new options object with the given color.
+   */
+  ;
+
+  _proto.withColor = function withColor(color) {
+    return this.extend({
+      color: color
+    });
+  }
+  /**
+   * Create a new options object with "phantom" set to true.
+   */
+  ;
+
+  _proto.withPhantom = function withPhantom() {
+    return this.extend({
+      phantom: true
+    });
+  }
+  /**
+   * Creates a new options object with the given math font or old text font.
+   * @type {[type]}
+   */
+  ;
+
+  _proto.withFont = function withFont(font) {
+    return this.extend({
+      font: font
+    });
+  }
+  /**
+   * Create a new options objects with the given fontFamily.
+   */
+  ;
+
+  _proto.withTextFontFamily = function withTextFontFamily(fontFamily) {
+    return this.extend({
+      fontFamily: fontFamily,
+      font: ""
+    });
+  }
+  /**
+   * Creates a new options object with the given font weight
+   */
+  ;
+
+  _proto.withTextFontWeight = function withTextFontWeight(fontWeight) {
+    return this.extend({
+      fontWeight: fontWeight,
+      font: ""
+    });
+  }
+  /**
+   * Creates a new options object with the given font weight
+   */
+  ;
+
+  _proto.withTextFontShape = function withTextFontShape(fontShape) {
+    return this.extend({
+      fontShape: fontShape,
+      font: ""
+    });
+  }
+  /**
+   * Return the CSS sizing classes required to switch from enclosing options
+   * `oldOptions` to `this`. Returns an array of classes.
+   */
+  ;
+
+  _proto.sizingClasses = function sizingClasses(oldOptions) {
+    if (oldOptions.size !== this.size) {
+      return ["sizing", "reset-size" + oldOptions.size, "size" + this.size];
+    } else {
+      return [];
+    }
+  }
+  /**
+   * Return the CSS sizing classes required to switch to the base size. Like
+   * `this.havingSize(BASESIZE).sizingClasses(this)`.
+   */
+  ;
+
+  _proto.baseSizingClasses = function baseSizingClasses() {
+    if (this.size !== Options.BASESIZE) {
+      return ["sizing", "reset-size" + this.size, "size" + Options.BASESIZE];
+    } else {
+      return [];
+    }
+  }
+  /**
+   * Return the font metrics for this size.
+   */
+  ;
+
+  _proto.fontMetrics = function fontMetrics() {
+    if (!this._fontMetrics) {
+      this._fontMetrics = getGlobalMetrics(this.size);
+    }
+
+    return this._fontMetrics;
+  }
+  /**
+   * Gets the CSS color of the current options object
+   */
+  ;
+
+  _proto.getColor = function getColor() {
+    if (this.phantom) {
+      return "transparent";
+    } else {
+      return this.color;
+    }
+  };
+
+  return Options;
+}();
+
+Options_Options.BASESIZE = 6;
+/* harmony default export */ var src_Options = (Options_Options);
+// CONCATENATED MODULE: ./src/units.js
+/**
+ * This file does conversion between units.  In particular, it provides
+ * calculateSize to convert other units into ems.
+ */
+
+ // This table gives the number of TeX pts in one of each *absolute* TeX unit.
+// Thus, multiplying a length by this number converts the length from units
+// into pts.  Dividing the result by ptPerEm gives the number of ems
+// *assuming* a font size of ptPerEm (normal size, normal style).
+
+var ptPerUnit = {
+  // https://en.wikibooks.org/wiki/LaTeX/Lengths and
+  // https://tex.stackexchange.com/a/8263
+  "pt": 1,
+  // TeX point
+  "mm": 7227 / 2540,
+  // millimeter
+  "cm": 7227 / 254,
+  // centimeter
+  "in": 72.27,
+  // inch
+  "bp": 803 / 800,
+  // big (PostScript) points
+  "pc": 12,
+  // pica
+  "dd": 1238 / 1157,
+  // didot
+  "cc": 14856 / 1157,
+  // cicero (12 didot)
+  "nd": 685 / 642,
+  // new didot
+  "nc": 1370 / 107,
+  // new cicero (12 new didot)
+  "sp": 1 / 65536,
+  // scaled point (TeX's internal smallest unit)
+  // https://tex.stackexchange.com/a/41371
+  "px": 803 / 800 // \pdfpxdimen defaults to 1 bp in pdfTeX and LuaTeX
+
+}; // Dictionary of relative units, for fast validity testing.
+
+var relativeUnit = {
+  "ex": true,
+  "em": true,
+  "mu": true
+};
+
+/**
+ * Determine whether the specified unit (either a string defining the unit
+ * or a "size" parse node containing a unit field) is valid.
+ */
+var validUnit = function validUnit(unit) {
+  if (typeof unit !== "string") {
+    unit = unit.unit;
+  }
+
+  return unit in ptPerUnit || unit in relativeUnit || unit === "ex";
+};
+/*
+ * Convert a "size" parse node (with numeric "number" and string "unit" fields,
+ * as parsed by functions.js argType "size") into a CSS em value for the
+ * current style/scale.  `options` gives the current options.
+ */
+
+var units_calculateSize = function calculateSize(sizeValue, options) {
+  var scale;
+
+  if (sizeValue.unit in ptPerUnit) {
+    // Absolute units
+    scale = ptPerUnit[sizeValue.unit] // Convert unit to pt
+    / options.fontMetrics().ptPerEm // Convert pt to CSS em
+    / options.sizeMultiplier; // Unscale to make absolute units
+  } else if (sizeValue.unit === "mu") {
+    // `mu` units scale with scriptstyle/scriptscriptstyle.
+    scale = options.fontMetrics().cssEmPerMu;
+  } else {
+    // Other relative units always refer to the *textstyle* font
+    // in the current size.
+    var unitOptions;
+
+    if (options.style.isTight()) {
+      // isTight() means current style is script/scriptscript.
+      unitOptions = options.havingStyle(options.style.text());
+    } else {
+      unitOptions = options;
+    } // TODO: In TeX these units are relative to the quad of the current
+    // *text* font, e.g. cmr10. KaTeX instead uses values from the
+    // comparably-sized *Computer Modern symbol* font. At 10pt, these
+    // match. At 7pt and 5pt, they differ: cmr7=1.138894, cmsy7=1.170641;
+    // cmr5=1.361133, cmsy5=1.472241. Consider $\scriptsize a\kern1emb$.
+    // TeX \showlists shows a kern of 1.13889 * fontsize;
+    // KaTeX shows a kern of 1.171 * fontsize.
+
+
+    if (sizeValue.unit === "ex") {
+      scale = unitOptions.fontMetrics().xHeight;
+    } else if (sizeValue.unit === "em") {
+      scale = unitOptions.fontMetrics().quad;
+    } else {
+      throw new src_ParseError("Invalid unit: '" + sizeValue.unit + "'");
+    }
+
+    if (unitOptions !== options) {
+      scale *= unitOptions.sizeMultiplier / options.sizeMultiplier;
+    }
+  }
+
+  return Math.min(sizeValue.number * scale, options.maxSize);
+};
+// CONCATENATED MODULE: ./src/buildCommon.js
+/* eslint no-console:0 */
+
+/**
+ * This module contains general functions that can be used for building
+ * different kinds of domTree nodes in a consistent manner.
+ */
+
+
+
+
+
+
+
+/**
+ * Looks up the given symbol in fontMetrics, after applying any symbol
+ * replacements defined in symbol.js
+ */
+var buildCommon_lookupSymbol = function lookupSymbol(value, // TODO(#963): Use a union type for this.
+fontName, mode) {
+  // Replace the value with its replaced value from symbol.js
+  if (src_symbols[mode][value] && src_symbols[mode][value].replace) {
+    value = src_symbols[mode][value].replace;
+  }
+
+  return {
+    value: value,
+    metrics: getCharacterMetrics(value, fontName, mode)
+  };
+};
+/**
+ * Makes a symbolNode after translation via the list of symbols in symbols.js.
+ * Correctly pulls out metrics for the character, and optionally takes a list of
+ * classes to be attached to the node.
+ *
+ * TODO: make argument order closer to makeSpan
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ * TODO(#953): Make `options` mandatory and always pass it in.
+ */
+
+
+var buildCommon_makeSymbol = function makeSymbol(value, fontName, mode, options, classes) {
+  var lookup = buildCommon_lookupSymbol(value, fontName, mode);
+  var metrics = lookup.metrics;
+  value = lookup.value;
+  var symbolNode;
+
+  if (metrics) {
+    var italic = metrics.italic;
+
+    if (mode === "text" || options && options.font === "mathit") {
+      italic = 0;
+    }
+
+    symbolNode = new domTree_SymbolNode(value, metrics.height, metrics.depth, italic, metrics.skew, metrics.width, classes);
+  } else {
+    // TODO(emily): Figure out a good way to only print this in development
+    typeof console !== "undefined" && console.warn("No character metrics " + ("for '" + value + "' in style '" + fontName + "' and mode '" + mode + "'"));
+    symbolNode = new domTree_SymbolNode(value, 0, 0, 0, 0, 0, classes);
+  }
+
+  if (options) {
+    symbolNode.maxFontSize = options.sizeMultiplier;
+
+    if (options.style.isTight()) {
+      symbolNode.classes.push("mtight");
+    }
+
+    var color = options.getColor();
+
+    if (color) {
+      symbolNode.style.color = color;
+    }
+  }
+
+  return symbolNode;
+};
+/**
+ * Makes a symbol in Main-Regular or AMS-Regular.
+ * Used for rel, bin, open, close, inner, and punct.
+ */
+
+
+var buildCommon_mathsym = function mathsym(value, mode, options, classes) {
+  if (classes === void 0) {
+    classes = [];
+  }
+
+  // Decide what font to render the symbol in by its entry in the symbols
+  // table.
+  // Have a special case for when the value = \ because the \ is used as a
+  // textord in unsupported command errors but cannot be parsed as a regular
+  // text ordinal and is therefore not present as a symbol in the symbols
+  // table for text, as well as a special case for boldsymbol because it
+  // can be used for bold + and -
+  if (options.font === "boldsymbol" && buildCommon_lookupSymbol(value, "Main-Bold", mode).metrics) {
+    return buildCommon_makeSymbol(value, "Main-Bold", mode, options, classes.concat(["mathbf"]));
+  } else if (value === "\\" || src_symbols[mode][value].font === "main") {
+    return buildCommon_makeSymbol(value, "Main-Regular", mode, options, classes);
+  } else {
+    return buildCommon_makeSymbol(value, "AMS-Regular", mode, options, classes.concat(["amsrm"]));
+  }
+};
+/**
+ * Determines which of the two font names (Main-Bold and Math-BoldItalic) and
+ * corresponding style tags (mathbf or boldsymbol) to use for font "boldsymbol",
+ * depending on the symbol.  Use this function instead of fontMap for font
+ * "boldsymbol".
+ */
+
+
+var boldsymbol = function boldsymbol(value, mode, options, classes, type) {
+  if (type !== "textord" && buildCommon_lookupSymbol(value, "Math-BoldItalic", mode).metrics) {
+    return {
+      fontName: "Math-BoldItalic",
+      fontClass: "boldsymbol"
+    };
+  } else {
+    // Some glyphs do not exist in Math-BoldItalic so we need to use
+    // Main-Bold instead.
+    return {
+      fontName: "Main-Bold",
+      fontClass: "mathbf"
+    };
+  }
+};
+/**
+ * Makes either a mathord or textord in the correct font and color.
+ */
+
+
+var buildCommon_makeOrd = function makeOrd(group, options, type) {
+  var mode = group.mode;
+  var text = group.text;
+  var classes = ["mord"]; // Math mode or Old font (i.e. \rm)
+
+  var isFont = mode === "math" || mode === "text" && options.font;
+  var fontOrFamily = isFont ? options.font : options.fontFamily;
+
+  if (text.charCodeAt(0) === 0xD835) {
+    // surrogate pairs get special treatment
+    var _wideCharacterFont = wide_character_wideCharacterFont(text, mode),
+        wideFontName = _wideCharacterFont[0],
+        wideFontClass = _wideCharacterFont[1];
+
+    return buildCommon_makeSymbol(text, wideFontName, mode, options, classes.concat(wideFontClass));
+  } else if (fontOrFamily) {
+    var fontName;
+    var fontClasses;
+
+    if (fontOrFamily === "boldsymbol") {
+      var fontData = boldsymbol(text, mode, options, classes, type);
+      fontName = fontData.fontName;
+      fontClasses = [fontData.fontClass];
+    } else if (isFont) {
+      fontName = fontMap[fontOrFamily].fontName;
+      fontClasses = [fontOrFamily];
+    } else {
+      fontName = retrieveTextFontName(fontOrFamily, options.fontWeight, options.fontShape);
+      fontClasses = [fontOrFamily, options.fontWeight, options.fontShape];
+    }
+
+    if (buildCommon_lookupSymbol(text, fontName, mode).metrics) {
+      return buildCommon_makeSymbol(text, fontName, mode, options, classes.concat(fontClasses));
+    } else if (ligatures.hasOwnProperty(text) && fontName.substr(0, 10) === "Typewriter") {
+      // Deconstruct ligatures in monospace fonts (\texttt, \tt).
+      var parts = [];
+
+      for (var i = 0; i < text.length; i++) {
+        parts.push(buildCommon_makeSymbol(text[i], fontName, mode, options, classes.concat(fontClasses)));
+      }
+
+      return buildCommon_makeFragment(parts);
+    }
+  } // Makes a symbol in the default font for mathords and textords.
+
+
+  if (type === "mathord") {
+    return buildCommon_makeSymbol(text, "Math-Italic", mode, options, classes.concat(["mathnormal"]));
+  } else if (type === "textord") {
+    var font = src_symbols[mode][text] && src_symbols[mode][text].font;
+
+    if (font === "ams") {
+      var _fontName = retrieveTextFontName("amsrm", options.fontWeight, options.fontShape);
+
+      return buildCommon_makeSymbol(text, _fontName, mode, options, classes.concat("amsrm", options.fontWeight, options.fontShape));
+    } else if (font === "main" || !font) {
+      var _fontName2 = retrieveTextFontName("textrm", options.fontWeight, options.fontShape);
+
+      return buildCommon_makeSymbol(text, _fontName2, mode, options, classes.concat(options.fontWeight, options.fontShape));
+    } else {
+      // fonts added by plugins
+      var _fontName3 = retrieveTextFontName(font, options.fontWeight, options.fontShape); // We add font name as a css class
+
+
+      return buildCommon_makeSymbol(text, _fontName3, mode, options, classes.concat(_fontName3, options.fontWeight, options.fontShape));
+    }
+  } else {
+    throw new Error("unexpected type: " + type + " in makeOrd");
+  }
+};
+/**
+ * Returns true if subsequent symbolNodes have the same classes, skew, maxFont,
+ * and styles.
+ */
+
+
+var buildCommon_canCombine = function canCombine(prev, next) {
+  if (createClass(prev.classes) !== createClass(next.classes) || prev.skew !== next.skew || prev.maxFontSize !== next.maxFontSize) {
+    return false;
+  }
+
+  for (var style in prev.style) {
+    if (prev.style.hasOwnProperty(style) && prev.style[style] !== next.style[style]) {
+      return false;
+    }
+  }
+
+  for (var _style in next.style) {
+    if (next.style.hasOwnProperty(_style) && prev.style[_style] !== next.style[_style]) {
+      return false;
+    }
+  }
+
+  return true;
+};
+/**
+ * Combine consequetive domTree.symbolNodes into a single symbolNode.
+ * Note: this function mutates the argument.
+ */
+
+
+var buildCommon_tryCombineChars = function tryCombineChars(chars) {
+  for (var i = 0; i < chars.length - 1; i++) {
+    var prev = chars[i];
+    var next = chars[i + 1];
+
+    if (prev instanceof domTree_SymbolNode && next instanceof domTree_SymbolNode && buildCommon_canCombine(prev, next)) {
+      prev.text += next.text;
+      prev.height = Math.max(prev.height, next.height);
+      prev.depth = Math.max(prev.depth, next.depth); // Use the last character's italic correction since we use
+      // it to add padding to the right of the span created from
+      // the combined characters.
+
+      prev.italic = next.italic;
+      chars.splice(i + 1, 1);
+      i--;
+    }
+  }
+
+  return chars;
+};
+/**
+ * Calculate the height, depth, and maxFontSize of an element based on its
+ * children.
+ */
+
+
+var sizeElementFromChildren = function sizeElementFromChildren(elem) {
+  var height = 0;
+  var depth = 0;
+  var maxFontSize = 0;
+
+  for (var i = 0; i < elem.children.length; i++) {
+    var child = elem.children[i];
+
+    if (child.height > height) {
+      height = child.height;
+    }
+
+    if (child.depth > depth) {
+      depth = child.depth;
+    }
+
+    if (child.maxFontSize > maxFontSize) {
+      maxFontSize = child.maxFontSize;
+    }
+  }
+
+  elem.height = height;
+  elem.depth = depth;
+  elem.maxFontSize = maxFontSize;
+};
+/**
+ * Makes a span with the given list of classes, list of children, and options.
+ *
+ * TODO(#953): Ensure that `options` is always provided (currently some call
+ * sites don't pass it) and make the type below mandatory.
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ */
+
+
+var buildCommon_makeSpan = function makeSpan(classes, children, options, style) {
+  var span = new domTree_Span(classes, children, options, style);
+  sizeElementFromChildren(span);
+  return span;
+}; // SVG one is simpler -- doesn't require height, depth, max-font setting.
+// This is also a separate method for typesafety.
+
+
+var buildCommon_makeSvgSpan = function makeSvgSpan(classes, children, options, style) {
+  return new domTree_Span(classes, children, options, style);
+};
+
+var makeLineSpan = function makeLineSpan(className, options, thickness) {
+  var line = buildCommon_makeSpan([className], [], options);
+  line.height = Math.max(thickness || options.fontMetrics().defaultRuleThickness, options.minRuleThickness);
+  line.style.borderBottomWidth = line.height + "em";
+  line.maxFontSize = 1.0;
+  return line;
+};
+/**
+ * Makes an anchor with the given href, list of classes, list of children,
+ * and options.
+ */
+
+
+var buildCommon_makeAnchor = function makeAnchor(href, classes, children, options) {
+  var anchor = new domTree_Anchor(href, classes, children, options);
+  sizeElementFromChildren(anchor);
+  return anchor;
+};
+/**
+ * Makes a document fragment with the given list of children.
+ */
+
+
+var buildCommon_makeFragment = function makeFragment(children) {
+  var fragment = new tree_DocumentFragment(children);
+  sizeElementFromChildren(fragment);
+  return fragment;
+};
+/**
+ * Wraps group in a span if it's a document fragment, allowing to apply classes
+ * and styles
+ */
+
+
+var buildCommon_wrapFragment = function wrapFragment(group, options) {
+  if (group instanceof tree_DocumentFragment) {
+    return buildCommon_makeSpan([], [group], options);
+  }
+
+  return group;
+}; // These are exact object types to catch typos in the names of the optional fields.
+
+
+// Computes the updated `children` list and the overall depth.
+//
+// This helper function for makeVList makes it easier to enforce type safety by
+// allowing early exits (returns) in the logic.
+var getVListChildrenAndDepth = function getVListChildrenAndDepth(params) {
+  if (params.positionType === "individualShift") {
+    var oldChildren = params.children;
+    var children = [oldChildren[0]]; // Add in kerns to the list of params.children to get each element to be
+    // shifted to the correct specified shift
+
+    var _depth = -oldChildren[0].shift - oldChildren[0].elem.depth;
+
+    var currPos = _depth;
+
+    for (var i = 1; i < oldChildren.length; i++) {
+      var diff = -oldChildren[i].shift - currPos - oldChildren[i].elem.depth;
+      var size = diff - (oldChildren[i - 1].elem.height + oldChildren[i - 1].elem.depth);
+      currPos = currPos + diff;
+      children.push({
+        type: "kern",
+        size: size
+      });
+      children.push(oldChildren[i]);
+    }
+
+    return {
+      children: children,
+      depth: _depth
+    };
+  }
+
+  var depth;
+
+  if (params.positionType === "top") {
+    // We always start at the bottom, so calculate the bottom by adding up
+    // all the sizes
+    var bottom = params.positionData;
+
+    for (var _i = 0; _i < params.children.length; _i++) {
+      var child = params.children[_i];
+      bottom -= child.type === "kern" ? child.size : child.elem.height + child.elem.depth;
+    }
+
+    depth = bottom;
+  } else if (params.positionType === "bottom") {
+    depth = -params.positionData;
+  } else {
+    var firstChild = params.children[0];
+
+    if (firstChild.type !== "elem") {
+      throw new Error('First child must have type "elem".');
+    }
+
+    if (params.positionType === "shift") {
+      depth = -firstChild.elem.depth - params.positionData;
+    } else if (params.positionType === "firstBaseline") {
+      depth = -firstChild.elem.depth;
+    } else {
+      throw new Error("Invalid positionType " + params.positionType + ".");
+    }
+  }
+
+  return {
+    children: params.children,
+    depth: depth
+  };
+};
+/**
+ * Makes a vertical list by stacking elements and kerns on top of each other.
+ * Allows for many different ways of specifying the positioning method.
+ *
+ * See VListParam documentation above.
+ */
+
+
+var buildCommon_makeVList = function makeVList(params, options) {
+  var _getVListChildrenAndD = getVListChildrenAndDepth(params),
+      children = _getVListChildrenAndD.children,
+      depth = _getVListChildrenAndD.depth; // Create a strut that is taller than any list item. The strut is added to
+  // each item, where it will determine the item's baseline. Since it has
+  // `overflow:hidden`, the strut's top edge will sit on the item's line box's
+  // top edge and the strut's bottom edge will sit on the item's baseline,
+  // with no additional line-height spacing. This allows the item baseline to
+  // be positioned precisely without worrying about font ascent and
+  // line-height.
+
+
+  var pstrutSize = 0;
+
+  for (var i = 0; i < children.length; i++) {
+    var child = children[i];
+
+    if (child.type === "elem") {
+      var elem = child.elem;
+      pstrutSize = Math.max(pstrutSize, elem.maxFontSize, elem.height);
+    }
+  }
+
+  pstrutSize += 2;
+  var pstrut = buildCommon_makeSpan(["pstrut"], []);
+  pstrut.style.height = pstrutSize + "em"; // Create a new list of actual children at the correct offsets
+
+  var realChildren = [];
+  var minPos = depth;
+  var maxPos = depth;
+  var currPos = depth;
+
+  for (var _i2 = 0; _i2 < children.length; _i2++) {
+    var _child = children[_i2];
+
+    if (_child.type === "kern") {
+      currPos += _child.size;
+    } else {
+      var _elem = _child.elem;
+      var classes = _child.wrapperClasses || [];
+      var style = _child.wrapperStyle || {};
+      var childWrap = buildCommon_makeSpan(classes, [pstrut, _elem], undefined, style);
+      childWrap.style.top = -pstrutSize - currPos - _elem.depth + "em";
+
+      if (_child.marginLeft) {
+        childWrap.style.marginLeft = _child.marginLeft;
+      }
+
+      if (_child.marginRight) {
+        childWrap.style.marginRight = _child.marginRight;
+      }
+
+      realChildren.push(childWrap);
+      currPos += _elem.height + _elem.depth;
+    }
+
+    minPos = Math.min(minPos, currPos);
+    maxPos = Math.max(maxPos, currPos);
+  } // The vlist contents go in a table-cell with `vertical-align:bottom`.
+  // This cell's bottom edge will determine the containing table's baseline
+  // without overly expanding the containing line-box.
+
+
+  var vlist = buildCommon_makeSpan(["vlist"], realChildren);
+  vlist.style.height = maxPos + "em"; // A second row is used if necessary to represent the vlist's depth.
+
+  var rows;
+
+  if (minPos < 0) {
+    // We will define depth in an empty span with display: table-cell.
+    // It should render with the height that we define. But Chrome, in
+    // contenteditable mode only, treats that span as if it contains some
+    // text content. And that min-height over-rides our desired height.
+    // So we put another empty span inside the depth strut span.
+    var emptySpan = buildCommon_makeSpan([], []);
+    var depthStrut = buildCommon_makeSpan(["vlist"], [emptySpan]);
+    depthStrut.style.height = -minPos + "em"; // Safari wants the first row to have inline content; otherwise it
+    // puts the bottom of the *second* row on the baseline.
+
+    var topStrut = buildCommon_makeSpan(["vlist-s"], [new domTree_SymbolNode("\u200B")]);
+    rows = [buildCommon_makeSpan(["vlist-r"], [vlist, topStrut]), buildCommon_makeSpan(["vlist-r"], [depthStrut])];
+  } else {
+    rows = [buildCommon_makeSpan(["vlist-r"], [vlist])];
+  }
+
+  var vtable = buildCommon_makeSpan(["vlist-t"], rows);
+
+  if (rows.length === 2) {
+    vtable.classes.push("vlist-t2");
+  }
+
+  vtable.height = maxPos;
+  vtable.depth = -minPos;
+  return vtable;
+}; // Glue is a concept from TeX which is a flexible space between elements in
+// either a vertical or horizontal list. In KaTeX, at least for now, it's
+// static space between elements in a horizontal layout.
+
+
+var buildCommon_makeGlue = function makeGlue(measurement, options) {
+  // Make an empty span for the space
+  var rule = buildCommon_makeSpan(["mspace"], [], options);
+  var size = units_calculateSize(measurement, options);
+  rule.style.marginRight = size + "em";
+  return rule;
+}; // Takes font options, and returns the appropriate fontLookup name
+
+
+var retrieveTextFontName = function retrieveTextFontName(fontFamily, fontWeight, fontShape) {
+  var baseFontName = "";
+
+  switch (fontFamily) {
+    case "amsrm":
+      baseFontName = "AMS";
+      break;
+
+    case "textrm":
+      baseFontName = "Main";
+      break;
+
+    case "textsf":
+      baseFontName = "SansSerif";
+      break;
+
+    case "texttt":
+      baseFontName = "Typewriter";
+      break;
+
+    default:
+      baseFontName = fontFamily;
+    // use fonts added by a plugin
+  }
+
+  var fontStylesName;
+
+  if (fontWeight === "textbf" && fontShape === "textit") {
+    fontStylesName = "BoldItalic";
+  } else if (fontWeight === "textbf") {
+    fontStylesName = "Bold";
+  } else if (fontWeight === "textit") {
+    fontStylesName = "Italic";
+  } else {
+    fontStylesName = "Regular";
+  }
+
+  return baseFontName + "-" + fontStylesName;
+};
+/**
+ * Maps TeX font commands to objects containing:
+ * - variant: string used for "mathvariant" attribute in buildMathML.js
+ * - fontName: the "style" parameter to fontMetrics.getCharacterMetrics
+ */
+// A map between tex font commands an MathML mathvariant attribute values
+
+
+var fontMap = {
+  // styles
+  "mathbf": {
+    variant: "bold",
+    fontName: "Main-Bold"
+  },
+  "mathrm": {
+    variant: "normal",
+    fontName: "Main-Regular"
+  },
+  "textit": {
+    variant: "italic",
+    fontName: "Main-Italic"
+  },
+  "mathit": {
+    variant: "italic",
+    fontName: "Main-Italic"
+  },
+  "mathnormal": {
+    variant: "italic",
+    fontName: "Math-Italic"
+  },
+  // "boldsymbol" is missing because they require the use of multiple fonts:
+  // Math-BoldItalic and Main-Bold.  This is handled by a special case in
+  // makeOrd which ends up calling boldsymbol.
+  // families
+  "mathbb": {
+    variant: "double-struck",
+    fontName: "AMS-Regular"
+  },
+  "mathcal": {
+    variant: "script",
+    fontName: "Caligraphic-Regular"
+  },
+  "mathfrak": {
+    variant: "fraktur",
+    fontName: "Fraktur-Regular"
+  },
+  "mathscr": {
+    variant: "script",
+    fontName: "Script-Regular"
+  },
+  "mathsf": {
+    variant: "sans-serif",
+    fontName: "SansSerif-Regular"
+  },
+  "mathtt": {
+    variant: "monospace",
+    fontName: "Typewriter-Regular"
+  }
+};
+var svgData = {
+  //   path, width, height
+  vec: ["vec", 0.471, 0.714],
+  // values from the font glyph
+  oiintSize1: ["oiintSize1", 0.957, 0.499],
+  // oval to overlay the integrand
+  oiintSize2: ["oiintSize2", 1.472, 0.659],
+  oiiintSize1: ["oiiintSize1", 1.304, 0.499],
+  oiiintSize2: ["oiiintSize2", 1.98, 0.659],
+  leftParenInner: ["leftParenInner", 0.875, 0.3],
+  rightParenInner: ["rightParenInner", 0.875, 0.3]
+};
+
+var buildCommon_staticSvg = function staticSvg(value, options) {
+  // Create a span with inline SVG for the element.
+  var _svgData$value = svgData[value],
+      pathName = _svgData$value[0],
+      width = _svgData$value[1],
+      height = _svgData$value[2];
+  var path = new domTree_PathNode(pathName);
+  var svgNode = new SvgNode([path], {
+    "width": width + "em",
+    "height": height + "em",
+    // Override CSS rule `.katex svg { width: 100% }`
+    "style": "width:" + width + "em",
+    "viewBox": "0 0 " + 1000 * width + " " + 1000 * height,
+    "preserveAspectRatio": "xMinYMin"
+  });
+  var span = buildCommon_makeSvgSpan(["overlay"], [svgNode], options);
+  span.height = height;
+  span.style.height = height + "em";
+  span.style.width = width + "em";
+  return span;
+};
+
+/* harmony default export */ var buildCommon = ({
+  fontMap: fontMap,
+  makeSymbol: buildCommon_makeSymbol,
+  mathsym: buildCommon_mathsym,
+  makeSpan: buildCommon_makeSpan,
+  makeSvgSpan: buildCommon_makeSvgSpan,
+  makeLineSpan: makeLineSpan,
+  makeAnchor: buildCommon_makeAnchor,
+  makeFragment: buildCommon_makeFragment,
+  wrapFragment: buildCommon_wrapFragment,
+  makeVList: buildCommon_makeVList,
+  makeOrd: buildCommon_makeOrd,
+  makeGlue: buildCommon_makeGlue,
+  staticSvg: buildCommon_staticSvg,
+  svgData: svgData,
+  tryCombineChars: buildCommon_tryCombineChars
+});
+// CONCATENATED MODULE: ./src/spacingData.js
+/**
+ * Describes spaces between different classes of atoms.
+ */
+var thinspace = {
+  number: 3,
+  unit: "mu"
+};
+var mediumspace = {
+  number: 4,
+  unit: "mu"
+};
+var thickspace = {
+  number: 5,
+  unit: "mu"
+}; // Making the type below exact with all optional fields doesn't work due to
+// - https://github.com/facebook/flow/issues/4582
+// - https://github.com/facebook/flow/issues/5688
+// However, since *all* fields are optional, $Shape<> works as suggested in 5688
+// above.
+
+// Spacing relationships for display and text styles
+var spacings = {
+  mord: {
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mop: {
+    mord: thinspace,
+    mop: thinspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mbin: {
+    mord: mediumspace,
+    mop: mediumspace,
+    mopen: mediumspace,
+    minner: mediumspace
+  },
+  mrel: {
+    mord: thickspace,
+    mop: thickspace,
+    mopen: thickspace,
+    minner: thickspace
+  },
+  mopen: {},
+  mclose: {
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mpunct: {
+    mord: thinspace,
+    mop: thinspace,
+    mrel: thickspace,
+    mopen: thinspace,
+    mclose: thinspace,
+    mpunct: thinspace,
+    minner: thinspace
+  },
+  minner: {
+    mord: thinspace,
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    mopen: thinspace,
+    mpunct: thinspace,
+    minner: thinspace
+  }
+}; // Spacing relationships for script and scriptscript styles
+
+var tightSpacings = {
+  mord: {
+    mop: thinspace
+  },
+  mop: {
+    mord: thinspace,
+    mop: thinspace
+  },
+  mbin: {},
+  mrel: {},
+  mopen: {},
+  mclose: {
+    mop: thinspace
+  },
+  mpunct: {},
+  minner: {
+    mop: thinspace
+  }
+};
+// CONCATENATED MODULE: ./src/defineFunction.js
+/** Context provided to function handlers for error messages. */
+// Note: reverse the order of the return type union will cause a flow error.
+// See https://github.com/facebook/flow/issues/3663.
+// More general version of `HtmlBuilder` for nodes (e.g. \sum, accent types)
+// whose presence impacts super/subscripting. In this case, ParseNode<"supsub">
+// delegates its HTML building to the HtmlBuilder corresponding to these nodes.
+
+/**
+ * Final function spec for use at parse time.
+ * This is almost identical to `FunctionPropSpec`, except it
+ * 1. includes the function handler, and
+ * 2. requires all arguments except argTypes.
+ * It is generated by `defineFunction()` below.
+ */
+
+/**
+ * All registered functions.
+ * `functions.js` just exports this same dictionary again and makes it public.
+ * `Parser.js` requires this dictionary.
+ */
+var _functions = {};
+/**
+ * All HTML builders. Should be only used in the `define*` and the `build*ML`
+ * functions.
+ */
+
+var _htmlGroupBuilders = {};
+/**
+ * All MathML builders. Should be only used in the `define*` and the `build*ML`
+ * functions.
+ */
+
+var _mathmlGroupBuilders = {};
+function defineFunction(_ref) {
+  var type = _ref.type,
+      names = _ref.names,
+      props = _ref.props,
+      handler = _ref.handler,
+      htmlBuilder = _ref.htmlBuilder,
+      mathmlBuilder = _ref.mathmlBuilder;
+  // Set default values of functions
+  var data = {
+    type: type,
+    numArgs: props.numArgs,
+    argTypes: props.argTypes,
+    greediness: props.greediness === undefined ? 1 : props.greediness,
+    allowedInText: !!props.allowedInText,
+    allowedInMath: props.allowedInMath === undefined ? true : props.allowedInMath,
+    numOptionalArgs: props.numOptionalArgs || 0,
+    infix: !!props.infix,
+    handler: handler
+  };
+
+  for (var i = 0; i < names.length; ++i) {
+    _functions[names[i]] = data;
+  }
+
+  if (type) {
+    if (htmlBuilder) {
+      _htmlGroupBuilders[type] = htmlBuilder;
+    }
+
+    if (mathmlBuilder) {
+      _mathmlGroupBuilders[type] = mathmlBuilder;
+    }
+  }
+}
+/**
+ * Use this to register only the HTML and MathML builders for a function (e.g.
+ * if the function's ParseNode is generated in Parser.js rather than via a
+ * stand-alone handler provided to `defineFunction`).
+ */
+
+function defineFunctionBuilders(_ref2) {
+  var type = _ref2.type,
+      htmlBuilder = _ref2.htmlBuilder,
+      mathmlBuilder = _ref2.mathmlBuilder;
+  defineFunction({
+    type: type,
+    names: [],
+    props: {
+      numArgs: 0
+    },
+    handler: function handler() {
+      throw new Error('Should never be called.');
+    },
+    htmlBuilder: htmlBuilder,
+    mathmlBuilder: mathmlBuilder
+  });
+} // Since the corresponding buildHTML/buildMathML function expects a
+// list of elements, we normalize for different kinds of arguments
+
+var ordargument = function ordargument(arg) {
+  return arg.type === "ordgroup" ? arg.body : [arg];
+};
+// CONCATENATED MODULE: ./src/buildHTML.js
+/**
+ * This file does the main work of building a domTree structure from a parse
+ * tree. The entry point is the `buildHTML` function, which takes a parse tree.
+ * Then, the buildExpression, buildGroup, and various groupBuilders functions
+ * are called, to produce a final HTML tree.
+ */
+
+
+
+
+
+
+
+
+var buildHTML_makeSpan = buildCommon.makeSpan; // Binary atoms (first class `mbin`) change into ordinary atoms (`mord`)
+// depending on their surroundings. See TeXbook pg. 442-446, Rules 5 and 6,
+// and the text before Rule 19.
+
+var binLeftCanceller = ["leftmost", "mbin", "mopen", "mrel", "mop", "mpunct"];
+var binRightCanceller = ["rightmost", "mrel", "mclose", "mpunct"];
+var styleMap = {
+  "display": src_Style.DISPLAY,
+  "text": src_Style.TEXT,
+  "script": src_Style.SCRIPT,
+  "scriptscript": src_Style.SCRIPTSCRIPT
+};
+var DomEnum = {
+  mord: "mord",
+  mop: "mop",
+  mbin: "mbin",
+  mrel: "mrel",
+  mopen: "mopen",
+  mclose: "mclose",
+  mpunct: "mpunct",
+  minner: "minner"
+};
+
+/**
+ * Take a list of nodes, build them in order, and return a list of the built
+ * nodes. documentFragments are flattened into their contents, so the
+ * returned list contains no fragments. `isRealGroup` is true if `expression`
+ * is a real group (no atoms will be added on either side), as opposed to
+ * a partial group (e.g. one created by \color). `surrounding` is an array
+ * consisting type of nodes that will be added to the left and right.
+ */
+var buildHTML_buildExpression = function buildExpression(expression, options, isRealGroup, surrounding) {
+  if (surrounding === void 0) {
+    surrounding = [null, null];
+  }
+
+  // Parse expressions into `groups`.
+  var groups = [];
+
+  for (var i = 0; i < expression.length; i++) {
+    var output = buildHTML_buildGroup(expression[i], options);
+
+    if (output instanceof tree_DocumentFragment) {
+      var children = output.children;
+      groups.push.apply(groups, children);
+    } else {
+      groups.push(output);
+    }
+  } // If `expression` is a partial group, let the parent handle spacings
+  // to avoid processing groups multiple times.
+
+
+  if (!isRealGroup) {
+    return groups;
+  }
+
+  var glueOptions = options;
+
+  if (expression.length === 1) {
+    var node = expression[0];
+
+    if (node.type === "sizing") {
+      glueOptions = options.havingSize(node.size);
+    } else if (node.type === "styling") {
+      glueOptions = options.havingStyle(styleMap[node.style]);
+    }
+  } // Dummy spans for determining spacings between surrounding atoms.
+  // If `expression` has no atoms on the left or right, class "leftmost"
+  // or "rightmost", respectively, is used to indicate it.
+
+
+  var dummyPrev = buildHTML_makeSpan([surrounding[0] || "leftmost"], [], options);
+  var dummyNext = buildHTML_makeSpan([surrounding[1] || "rightmost"], [], options); // TODO: These code assumes that a node's math class is the first element
+  // of its `classes` array. A later cleanup should ensure this, for
+  // instance by changing the signature of `makeSpan`.
+  // Before determining what spaces to insert, perform bin cancellation.
+  // Binary operators change to ordinary symbols in some contexts.
+
+  var isRoot = isRealGroup === "root";
+  traverseNonSpaceNodes(groups, function (node, prev) {
+    var prevType = prev.classes[0];
+    var type = node.classes[0];
+
+    if (prevType === "mbin" && utils.contains(binRightCanceller, type)) {
+      prev.classes[0] = "mord";
+    } else if (type === "mbin" && utils.contains(binLeftCanceller, prevType)) {
+      node.classes[0] = "mord";
+    }
+  }, {
+    node: dummyPrev
+  }, dummyNext, isRoot);
+  traverseNonSpaceNodes(groups, function (node, prev) {
+    var prevType = getTypeOfDomTree(prev);
+    var type = getTypeOfDomTree(node); // 'mtight' indicates that the node is script or scriptscript style.
+
+    var space = prevType && type ? node.hasClass("mtight") ? tightSpacings[prevType][type] : spacings[prevType][type] : null;
+
+    if (space) {
+      // Insert glue (spacing) after the `prev`.
+      return buildCommon.makeGlue(space, glueOptions);
+    }
+  }, {
+    node: dummyPrev
+  }, dummyNext, isRoot);
+  return groups;
+}; // Depth-first traverse non-space `nodes`, calling `callback` with the current and
+// previous node as arguments, optionally returning a node to insert after the
+// previous node. `prev` is an object with the previous node and `insertAfter`
+// function to insert after it. `next` is a node that will be added to the right.
+// Used for bin cancellation and inserting spacings.
+
+var traverseNonSpaceNodes = function traverseNonSpaceNodes(nodes, callback, prev, next, isRoot) {
+  if (next) {
+    // temporarily append the right node, if exists
+    nodes.push(next);
+  }
+
+  var i = 0;
+
+  for (; i < nodes.length; i++) {
+    var node = nodes[i];
+    var partialGroup = buildHTML_checkPartialGroup(node);
+
+    if (partialGroup) {
+      // Recursive DFS
+      // $FlowFixMe: make nodes a $ReadOnlyArray by returning a new array
+      traverseNonSpaceNodes(partialGroup.children, callback, prev, null, isRoot);
+      continue;
+    } // Ignore explicit spaces (e.g., \;, \,) when determining what implicit
+    // spacing should go between atoms of different classes
+
+
+    var nonspace = !node.hasClass("mspace");
+
+    if (nonspace) {
+      var result = callback(node, prev.node);
+
+      if (result) {
+        if (prev.insertAfter) {
+          prev.insertAfter(result);
+        } else {
+          // insert at front
+          nodes.unshift(result);
+          i++;
+        }
+      }
+    }
+
+    if (nonspace) {
+      prev.node = node;
+    } else if (isRoot && node.hasClass("newline")) {
+      prev.node = buildHTML_makeSpan(["leftmost"]); // treat like beginning of line
+    }
+
+    prev.insertAfter = function (index) {
+      return function (n) {
+        nodes.splice(index + 1, 0, n);
+        i++;
+      };
+    }(i);
+  }
+
+  if (next) {
+    nodes.pop();
+  }
+}; // Check if given node is a partial group, i.e., does not affect spacing around.
+
+
+var buildHTML_checkPartialGroup = function checkPartialGroup(node) {
+  if (node instanceof tree_DocumentFragment || node instanceof domTree_Anchor || node instanceof domTree_Span && node.hasClass("enclosing")) {
+    return node;
+  }
+
+  return null;
+}; // Return the outermost node of a domTree.
+
+
+var getOutermostNode = function getOutermostNode(node, side) {
+  var partialGroup = buildHTML_checkPartialGroup(node);
+
+  if (partialGroup) {
+    var children = partialGroup.children;
+
+    if (children.length) {
+      if (side === "right") {
+        return getOutermostNode(children[children.length - 1], "right");
+      } else if (side === "left") {
+        return getOutermostNode(children[0], "left");
+      }
+    }
+  }
+
+  return node;
+}; // Return math atom class (mclass) of a domTree.
+// If `side` is given, it will get the type of the outermost node at given side.
+
+
+var getTypeOfDomTree = function getTypeOfDomTree(node, side) {
+  if (!node) {
+    return null;
+  }
+
+  if (side) {
+    node = getOutermostNode(node, side);
+  } // This makes a lot of assumptions as to where the type of atom
+  // appears.  We should do a better job of enforcing this.
+
+
+  return DomEnum[node.classes[0]] || null;
+};
+var makeNullDelimiter = function makeNullDelimiter(options, classes) {
+  var moreClasses = ["nulldelimiter"].concat(options.baseSizingClasses());
+  return buildHTML_makeSpan(classes.concat(moreClasses));
+};
+/**
+ * buildGroup is the function that takes a group and calls the correct groupType
+ * function for it. It also handles the interaction of size and style changes
+ * between parents and children.
+ */
+
+var buildHTML_buildGroup = function buildGroup(group, options, baseOptions) {
+  if (!group) {
+    return buildHTML_makeSpan();
+  }
+
+  if (_htmlGroupBuilders[group.type]) {
+    // Call the groupBuilders function
+    var groupNode = _htmlGroupBuilders[group.type](group, options); // If the size changed between the parent and the current group, account
+    // for that size difference.
+
+    if (baseOptions && options.size !== baseOptions.size) {
+      groupNode = buildHTML_makeSpan(options.sizingClasses(baseOptions), [groupNode], options);
+      var multiplier = options.sizeMultiplier / baseOptions.sizeMultiplier;
+      groupNode.height *= multiplier;
+      groupNode.depth *= multiplier;
+    }
+
+    return groupNode;
+  } else {
+    throw new src_ParseError("Got group of unknown type: '" + group.type + "'");
+  }
+};
+/**
+ * Combine an array of HTML DOM nodes (e.g., the output of `buildExpression`)
+ * into an unbreakable HTML node of class .base, with proper struts to
+ * guarantee correct vertical extent.  `buildHTML` calls this repeatedly to
+ * make up the entire expression as a sequence of unbreakable units.
+ */
+
+function buildHTMLUnbreakable(children, options) {
+  // Compute height and depth of this chunk.
+  var body = buildHTML_makeSpan(["base"], children, options); // Add strut, which ensures that the top of the HTML element falls at
+  // the height of the expression, and the bottom of the HTML element
+  // falls at the depth of the expression.
+
+  var strut = buildHTML_makeSpan(["strut"]);
+  strut.style.height = body.height + body.depth + "em";
+  strut.style.verticalAlign = -body.depth + "em";
+  body.children.unshift(strut);
+  return body;
+}
+/**
+ * Take an entire parse tree, and build it into an appropriate set of HTML
+ * nodes.
+ */
+
+
+function buildHTML(tree, options) {
+  // Strip off outer tag wrapper for processing below.
+  var tag = null;
+
+  if (tree.length === 1 && tree[0].type === "tag") {
+    tag = tree[0].tag;
+    tree = tree[0].body;
+  } // Build the expression contained in the tree
+
+
+  var expression = buildHTML_buildExpression(tree, options, "root");
+  var children = []; // Create one base node for each chunk between potential line breaks.
+  // The TeXBook [p.173] says "A formula will be broken only after a
+  // relation symbol like $=$ or $<$ or $\rightarrow$, or after a binary
+  // operation symbol like $+$ or $-$ or $\times$, where the relation or
+  // binary operation is on the ``outer level'' of the formula (i.e., not
+  // enclosed in {...} and not part of an \over construction)."
+
+  var parts = [];
+
+  for (var i = 0; i < expression.length; i++) {
+    parts.push(expression[i]);
+
+    if (expression[i].hasClass("mbin") || expression[i].hasClass("mrel") || expression[i].hasClass("allowbreak")) {
+      // Put any post-operator glue on same line as operator.
+      // Watch for \nobreak along the way, and stop at \newline.
+      var nobreak = false;
+
+      while (i < expression.length - 1 && expression[i + 1].hasClass("mspace") && !expression[i + 1].hasClass("newline")) {
+        i++;
+        parts.push(expression[i]);
+
+        if (expression[i].hasClass("nobreak")) {
+          nobreak = true;
+        }
+      } // Don't allow break if \nobreak among the post-operator glue.
+
+
+      if (!nobreak) {
+        children.push(buildHTMLUnbreakable(parts, options));
+        parts = [];
+      }
+    } else if (expression[i].hasClass("newline")) {
+      // Write the line except the newline
+      parts.pop();
+
+      if (parts.length > 0) {
+        children.push(buildHTMLUnbreakable(parts, options));
+        parts = [];
+      } // Put the newline at the top level
+
+
+      children.push(expression[i]);
+    }
+  }
+
+  if (parts.length > 0) {
+    children.push(buildHTMLUnbreakable(parts, options));
+  } // Now, if there was a tag, build it too and append it as a final child.
+
+
+  var tagChild;
+
+  if (tag) {
+    tagChild = buildHTMLUnbreakable(buildHTML_buildExpression(tag, options, true));
+    tagChild.classes = ["tag"];
+    children.push(tagChild);
+  }
+
+  var htmlNode = buildHTML_makeSpan(["katex-html"], children);
+  htmlNode.setAttribute("aria-hidden", "true"); // Adjust the strut of the tag to be the maximum height of all children
+  // (the height of the enclosing htmlNode) for proper vertical alignment.
+
+  if (tagChild) {
+    var strut = tagChild.children[0];
+    strut.style.height = htmlNode.height + htmlNode.depth + "em";
+    strut.style.verticalAlign = -htmlNode.depth + "em";
+  }
+
+  return htmlNode;
+}
+// CONCATENATED MODULE: ./src/mathMLTree.js
+/**
+ * These objects store data about MathML nodes. This is the MathML equivalent
+ * of the types in domTree.js. Since MathML handles its own rendering, and
+ * since we're mainly using MathML to improve accessibility, we don't manage
+ * any of the styling state that the plain DOM nodes do.
+ *
+ * The `toNode` and `toMarkup` functions work simlarly to how they do in
+ * domTree.js, creating namespaced DOM nodes and HTML text markup respectively.
+ */
+
+
+function newDocumentFragment(children) {
+  return new tree_DocumentFragment(children);
+}
+/**
+ * This node represents a general purpose MathML node of any type. The
+ * constructor requires the type of node to create (for example, `"mo"` or
+ * `"mspace"`, corresponding to `<mo>` and `<mspace>` tags).
+ */
+
+var mathMLTree_MathNode =
+/*#__PURE__*/
+function () {
+  function MathNode(type, children) {
+    this.type = void 0;
+    this.attributes = void 0;
+    this.children = void 0;
+    this.type = type;
+    this.attributes = {};
+    this.children = children || [];
+  }
+  /**
+   * Sets an attribute on a MathML node. MathML depends on attributes to convey a
+   * semantic content, so this is used heavily.
+   */
+
+
+  var _proto = MathNode.prototype;
+
+  _proto.setAttribute = function setAttribute(name, value) {
+    this.attributes[name] = value;
+  }
+  /**
+   * Gets an attribute on a MathML node.
+   */
+  ;
+
+  _proto.getAttribute = function getAttribute(name) {
+    return this.attributes[name];
+  }
+  /**
+   * Converts the math node into a MathML-namespaced DOM element.
+   */
+  ;
+
+  _proto.toNode = function toNode() {
+    var node = document.createElementNS("http://www.w3.org/1998/Math/MathML", this.type);
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    for (var i = 0; i < this.children.length; i++) {
+      node.appendChild(this.children[i].toNode());
+    }
+
+    return node;
+  }
+  /**
+   * Converts the math node into an HTML markup string.
+   */
+  ;
+
+  _proto.toMarkup = function toMarkup() {
+    var markup = "<" + this.type; // Add the attributes
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += " " + attr + "=\"";
+        markup += utils.escape(this.attributes[attr]);
+        markup += "\"";
+      }
+    }
+
+    markup += ">";
+
+    for (var i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    markup += "</" + this.type + ">";
+    return markup;
+  }
+  /**
+   * Converts the math node into a string, similar to innerText, but escaped.
+   */
+  ;
+
+  _proto.toText = function toText() {
+    return this.children.map(function (child) {
+      return child.toText();
+    }).join("");
+  };
+
+  return MathNode;
+}();
+/**
+ * This node represents a piece of text.
+ */
+
+var mathMLTree_TextNode =
+/*#__PURE__*/
+function () {
+  function TextNode(text) {
+    this.text = void 0;
+    this.text = text;
+  }
+  /**
+   * Converts the text node into a DOM text node.
+   */
+
+
+  var _proto2 = TextNode.prototype;
+
+  _proto2.toNode = function toNode() {
+    return document.createTextNode(this.text);
+  }
+  /**
+   * Converts the text node into escaped HTML markup
+   * (representing the text itself).
+   */
+  ;
+
+  _proto2.toMarkup = function toMarkup() {
+    return utils.escape(this.toText());
+  }
+  /**
+   * Converts the text node into a string
+   * (representing the text iteself).
+   */
+  ;
+
+  _proto2.toText = function toText() {
+    return this.text;
+  };
+
+  return TextNode;
+}();
+/**
+ * This node represents a space, but may render as <mspace.../> or as text,
+ * depending on the width.
+ */
+
+var SpaceNode =
+/*#__PURE__*/
+function () {
+  /**
+   * Create a Space node with width given in CSS ems.
+   */
+  function SpaceNode(width) {
+    this.width = void 0;
+    this.character = void 0;
+    this.width = width; // See https://www.w3.org/TR/2000/WD-MathML2-20000328/chapter6.html
+    // for a table of space-like characters.  We use Unicode
+    // representations instead of &LongNames; as it's not clear how to
+    // make the latter via document.createTextNode.
+
+    if (width >= 0.05555 && width <= 0.05556) {
+      this.character = "\u200A"; // &VeryThinSpace;
+    } else if (width >= 0.1666 && width <= 0.1667) {
+      this.character = "\u2009"; // &ThinSpace;
+    } else if (width >= 0.2222 && width <= 0.2223) {
+      this.character = "\u2005"; // &MediumSpace;
+    } else if (width >= 0.2777 && width <= 0.2778) {
+      this.character = "\u2005\u200A"; // &ThickSpace;
+    } else if (width >= -0.05556 && width <= -0.05555) {
+      this.character = "\u200A\u2063"; // &NegativeVeryThinSpace;
+    } else if (width >= -0.1667 && width <= -0.1666) {
+      this.character = "\u2009\u2063"; // &NegativeThinSpace;
+    } else if (width >= -0.2223 && width <= -0.2222) {
+      this.character = "\u205F\u2063"; // &NegativeMediumSpace;
+    } else if (width >= -0.2778 && width <= -0.2777) {
+      this.character = "\u2005\u2063"; // &NegativeThickSpace;
+    } else {
+      this.character = null;
+    }
+  }
+  /**
+   * Converts the math node into a MathML-namespaced DOM element.
+   */
+
+
+  var _proto3 = SpaceNode.prototype;
+
+  _proto3.toNode = function toNode() {
+    if (this.character) {
+      return document.createTextNode(this.character);
+    } else {
+      var node = document.createElementNS("http://www.w3.org/1998/Math/MathML", "mspace");
+      node.setAttribute("width", this.width + "em");
+      return node;
+    }
+  }
+  /**
+   * Converts the math node into an HTML markup string.
+   */
+  ;
+
+  _proto3.toMarkup = function toMarkup() {
+    if (this.character) {
+      return "<mtext>" + this.character + "</mtext>";
+    } else {
+      return "<mspace width=\"" + this.width + "em\"/>";
+    }
+  }
+  /**
+   * Converts the math node into a string, similar to innerText.
+   */
+  ;
+
+  _proto3.toText = function toText() {
+    if (this.character) {
+      return this.character;
+    } else {
+      return " ";
+    }
+  };
+
+  return SpaceNode;
+}();
+
+/* harmony default export */ var mathMLTree = ({
+  MathNode: mathMLTree_MathNode,
+  TextNode: mathMLTree_TextNode,
+  SpaceNode: SpaceNode,
+  newDocumentFragment: newDocumentFragment
+});
+// CONCATENATED MODULE: ./src/buildMathML.js
+/**
+ * This file converts a parse tree into a cooresponding MathML tree. The main
+ * entry point is the `buildMathML` function, which takes a parse tree from the
+ * parser.
+ */
+
+
+
+
+
+
+
+
+
+/**
+ * Takes a symbol and converts it into a MathML text node after performing
+ * optional replacement from symbols.js.
+ */
+var buildMathML_makeText = function makeText(text, mode, options) {
+  if (src_symbols[mode][text] && src_symbols[mode][text].replace && text.charCodeAt(0) !== 0xD835 && !(ligatures.hasOwnProperty(text) && options && (options.fontFamily && options.fontFamily.substr(4, 2) === "tt" || options.font && options.font.substr(4, 2) === "tt"))) {
+    text = src_symbols[mode][text].replace;
+  }
+
+  return new mathMLTree.TextNode(text);
+};
+/**
+ * Wrap the given array of nodes in an <mrow> node if needed, i.e.,
+ * unless the array has length 1.  Always returns a single node.
+ */
+
+var buildMathML_makeRow = function makeRow(body) {
+  if (body.length === 1) {
+    return body[0];
+  } else {
+    return new mathMLTree.MathNode("mrow", body);
+  }
+};
+/**
+ * Returns the math variant as a string or null if none is required.
+ */
+
+var buildMathML_getVariant = function getVariant(group, options) {
+  // Handle \text... font specifiers as best we can.
+  // MathML has a limited list of allowable mathvariant specifiers; see
+  // https://www.w3.org/TR/MathML3/chapter3.html#presm.commatt
+  if (options.fontFamily === "texttt") {
+    return "monospace";
+  } else if (options.fontFamily === "textsf") {
+    if (options.fontShape === "textit" && options.fontWeight === "textbf") {
+      return "sans-serif-bold-italic";
+    } else if (options.fontShape === "textit") {
+      return "sans-serif-italic";
+    } else if (options.fontWeight === "textbf") {
+      return "bold-sans-serif";
+    } else {
+      return "sans-serif";
+    }
+  } else if (options.fontShape === "textit" && options.fontWeight === "textbf") {
+    return "bold-italic";
+  } else if (options.fontShape === "textit") {
+    return "italic";
+  } else if (options.fontWeight === "textbf") {
+    return "bold";
+  }
+
+  var font = options.font;
+
+  if (!font || font === "mathnormal") {
+    return null;
+  }
+
+  var mode = group.mode;
+
+  if (font === "mathit") {
+    return "italic";
+  } else if (font === "boldsymbol") {
+    return group.type === "textord" ? "bold" : "bold-italic";
+  } else if (font === "mathbf") {
+    return "bold";
+  } else if (font === "mathbb") {
+    return "double-struck";
+  } else if (font === "mathfrak") {
+    return "fraktur";
+  } else if (font === "mathscr" || font === "mathcal") {
+    // MathML makes no distinction between script and caligrahpic
+    return "script";
+  } else if (font === "mathsf") {
+    return "sans-serif";
+  } else if (font === "mathtt") {
+    return "monospace";
+  }
+
+  var text = group.text;
+
+  if (utils.contains(["\\imath", "\\jmath"], text)) {
+    return null;
+  }
+
+  if (src_symbols[mode][text] && src_symbols[mode][text].replace) {
+    text = src_symbols[mode][text].replace;
+  }
+
+  var fontName = buildCommon.fontMap[font].fontName;
+
+  if (getCharacterMetrics(text, fontName, mode)) {
+    return buildCommon.fontMap[font].variant;
+  }
+
+  return null;
+};
+/**
+ * Takes a list of nodes, builds them, and returns a list of the generated
+ * MathML nodes.  Also combine consecutive <mtext> outputs into a single
+ * <mtext> tag.
+ */
+
+var buildMathML_buildExpression = function buildExpression(expression, options, isOrdgroup) {
+  if (expression.length === 1) {
+    var group = buildMathML_buildGroup(expression[0], options);
+
+    if (isOrdgroup && group instanceof mathMLTree_MathNode && group.type === "mo") {
+      // When TeX writers want to suppress spacing on an operator,
+      // they often put the operator by itself inside braces.
+      group.setAttribute("lspace", "0em");
+      group.setAttribute("rspace", "0em");
+    }
+
+    return [group];
+  }
+
+  var groups = [];
+  var lastGroup;
+
+  for (var i = 0; i < expression.length; i++) {
+    var _group = buildMathML_buildGroup(expression[i], options);
+
+    if (_group instanceof mathMLTree_MathNode && lastGroup instanceof mathMLTree_MathNode) {
+      // Concatenate adjacent <mtext>s
+      if (_group.type === 'mtext' && lastGroup.type === 'mtext' && _group.getAttribute('mathvariant') === lastGroup.getAttribute('mathvariant')) {
+        var _lastGroup$children;
+
+        (_lastGroup$children = lastGroup.children).push.apply(_lastGroup$children, _group.children);
+
+        continue; // Concatenate adjacent <mn>s
+      } else if (_group.type === 'mn' && lastGroup.type === 'mn') {
+        var _lastGroup$children2;
+
+        (_lastGroup$children2 = lastGroup.children).push.apply(_lastGroup$children2, _group.children);
+
+        continue; // Concatenate <mn>...</mn> followed by <mi>.</mi>
+      } else if (_group.type === 'mi' && _group.children.length === 1 && lastGroup.type === 'mn') {
+        var child = _group.children[0];
+
+        if (child instanceof mathMLTree_TextNode && child.text === '.') {
+          var _lastGroup$children3;
+
+          (_lastGroup$children3 = lastGroup.children).push.apply(_lastGroup$children3, _group.children);
+
+          continue;
+        }
+      } else if (lastGroup.type === 'mi' && lastGroup.children.length === 1) {
+        var lastChild = lastGroup.children[0];
+
+        if (lastChild instanceof mathMLTree_TextNode && lastChild.text === "\u0338" && (_group.type === 'mo' || _group.type === 'mi' || _group.type === 'mn')) {
+          var _child = _group.children[0];
+
+          if (_child instanceof mathMLTree_TextNode && _child.text.length > 0) {
+            // Overlay with combining character long solidus
+            _child.text = _child.text.slice(0, 1) + "\u0338" + _child.text.slice(1);
+            groups.pop();
+          }
+        }
+      }
+    }
+
+    groups.push(_group);
+    lastGroup = _group;
+  }
+
+  return groups;
+};
+/**
+ * Equivalent to buildExpression, but wraps the elements in an <mrow>
+ * if there's more than one.  Returns a single node instead of an array.
+ */
+
+var buildExpressionRow = function buildExpressionRow(expression, options, isOrdgroup) {
+  return buildMathML_makeRow(buildMathML_buildExpression(expression, options, isOrdgroup));
+};
+/**
+ * Takes a group from the parser and calls the appropriate groupBuilders function
+ * on it to produce a MathML node.
+ */
+
+var buildMathML_buildGroup = function buildGroup(group, options) {
+  if (!group) {
+    return new mathMLTree.MathNode("mrow");
+  }
+
+  if (_mathmlGroupBuilders[group.type]) {
+    // Call the groupBuilders function
+    var result = _mathmlGroupBuilders[group.type](group, options);
+    return result;
+  } else {
+    throw new src_ParseError("Got group of unknown type: '" + group.type + "'");
+  }
+};
+/**
+ * Takes a full parse tree and settings and builds a MathML representation of
+ * it. In particular, we put the elements from building the parse tree into a
+ * <semantics> tag so we can also include that TeX source as an annotation.
+ *
+ * Note that we actually return a domTree element with a `<math>` inside it so
+ * we can do appropriate styling.
+ */
+
+function buildMathML(tree, texExpression, options, isDisplayMode, forMathmlOnly) {
+  var expression = buildMathML_buildExpression(tree, options); // Wrap up the expression in an mrow so it is presented in the semantics
+  // tag correctly, unless it's a single <mrow> or <mtable>.
+
+  var wrapper;
+
+  if (expression.length === 1 && expression[0] instanceof mathMLTree_MathNode && utils.contains(["mrow", "mtable"], expression[0].type)) {
+    wrapper = expression[0];
+  } else {
+    wrapper = new mathMLTree.MathNode("mrow", expression);
+  } // Build a TeX annotation of the source
+
+
+  var annotation = new mathMLTree.MathNode("annotation", [new mathMLTree.TextNode(texExpression)]);
+  annotation.setAttribute("encoding", "application/x-tex");
+  var semantics = new mathMLTree.MathNode("semantics", [wrapper, annotation]);
+  var math = new mathMLTree.MathNode("math", [semantics]);
+  math.setAttribute("xmlns", "http://www.w3.org/1998/Math/MathML");
+
+  if (isDisplayMode) {
+    math.setAttribute("display", "block");
+  } // You can't style <math> nodes, so we wrap the node in a span.
+  // NOTE: The span class is not typed to have <math> nodes as children, and
+  // we don't want to make the children type more generic since the children
+  // of span are expected to have more fields in `buildHtml` contexts.
+
+
+  var wrapperClass = forMathmlOnly ? "katex" : "katex-mathml"; // $FlowFixMe
+
+  return buildCommon.makeSpan([wrapperClass], [math]);
+}
+// CONCATENATED MODULE: ./src/buildTree.js
+
+
+
+
+
+
+
+var buildTree_optionsFromSettings = function optionsFromSettings(settings) {
+  return new src_Options({
+    style: settings.displayMode ? src_Style.DISPLAY : src_Style.TEXT,
+    maxSize: settings.maxSize,
+    minRuleThickness: settings.minRuleThickness
+  });
+};
+
+var buildTree_displayWrap = function displayWrap(node, settings) {
+  if (settings.displayMode) {
+    var classes = ["katex-display"];
+
+    if (settings.leqno) {
+      classes.push("leqno");
+    }
+
+    if (settings.fleqn) {
+      classes.push("fleqn");
+    }
+
+    node = buildCommon.makeSpan(classes, [node]);
+  }
+
+  return node;
+};
+
+var buildTree_buildTree = function buildTree(tree, expression, settings) {
+  var options = buildTree_optionsFromSettings(settings);
+  var katexNode;
+
+  if (settings.output === "mathml") {
+    return buildMathML(tree, expression, options, settings.displayMode, true);
+  } else if (settings.output === "html") {
+    var htmlNode = buildHTML(tree, options);
+    katexNode = buildCommon.makeSpan(["katex"], [htmlNode]);
+  } else {
+    var mathMLNode = buildMathML(tree, expression, options, settings.displayMode, false);
+
+    var _htmlNode = buildHTML(tree, options);
+
+    katexNode = buildCommon.makeSpan(["katex"], [mathMLNode, _htmlNode]);
+  }
+
+  return buildTree_displayWrap(katexNode, settings);
+};
+var buildTree_buildHTMLTree = function buildHTMLTree(tree, expression, settings) {
+  var options = buildTree_optionsFromSettings(settings);
+  var htmlNode = buildHTML(tree, options);
+  var katexNode = buildCommon.makeSpan(["katex"], [htmlNode]);
+  return buildTree_displayWrap(katexNode, settings);
+};
+/* harmony default export */ var src_buildTree = (buildTree_buildTree);
+// CONCATENATED MODULE: ./src/stretchy.js
+/**
+ * This file provides support to buildMathML.js and buildHTML.js
+ * for stretchy wide elements rendered from SVG files
+ * and other CSS trickery.
+ */
+
+
+
+
+var stretchyCodePoint = {
+  widehat: "^",
+  widecheck: "ˇ",
+  widetilde: "~",
+  utilde: "~",
+  overleftarrow: "\u2190",
+  underleftarrow: "\u2190",
+  xleftarrow: "\u2190",
+  overrightarrow: "\u2192",
+  underrightarrow: "\u2192",
+  xrightarrow: "\u2192",
+  underbrace: "\u23DF",
+  overbrace: "\u23DE",
+  overgroup: "\u23E0",
+  undergroup: "\u23E1",
+  overleftrightarrow: "\u2194",
+  underleftrightarrow: "\u2194",
+  xleftrightarrow: "\u2194",
+  Overrightarrow: "\u21D2",
+  xRightarrow: "\u21D2",
+  overleftharpoon: "\u21BC",
+  xleftharpoonup: "\u21BC",
+  overrightharpoon: "\u21C0",
+  xrightharpoonup: "\u21C0",
+  xLeftarrow: "\u21D0",
+  xLeftrightarrow: "\u21D4",
+  xhookleftarrow: "\u21A9",
+  xhookrightarrow: "\u21AA",
+  xmapsto: "\u21A6",
+  xrightharpoondown: "\u21C1",
+  xleftharpoondown: "\u21BD",
+  xrightleftharpoons: "\u21CC",
+  xleftrightharpoons: "\u21CB",
+  xtwoheadleftarrow: "\u219E",
+  xtwoheadrightarrow: "\u21A0",
+  xlongequal: "=",
+  xtofrom: "\u21C4",
+  xrightleftarrows: "\u21C4",
+  xrightequilibrium: "\u21CC",
+  // Not a perfect match.
+  xleftequilibrium: "\u21CB" // None better available.
+
+};
+
+var stretchy_mathMLnode = function mathMLnode(label) {
+  var node = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(stretchyCodePoint[label.substr(1)])]);
+  node.setAttribute("stretchy", "true");
+  return node;
+}; // Many of the KaTeX SVG images have been adapted from glyphs in KaTeX fonts.
+// Copyright (c) 2009-2010, Design Science, Inc. (<www.mathjax.org>)
+// Copyright (c) 2014-2017 Khan Academy (<www.khanacademy.org>)
+// Licensed under the SIL Open Font License, Version 1.1.
+// See \nhttp://scripts.sil.org/OFL
+// Very Long SVGs
+//    Many of the KaTeX stretchy wide elements use a long SVG image and an
+//    overflow: hidden tactic to achieve a stretchy image while avoiding
+//    distortion of arrowheads or brace corners.
+//    The SVG typically contains a very long (400 em) arrow.
+//    The SVG is in a container span that has overflow: hidden, so the span
+//    acts like a window that exposes only part of the  SVG.
+//    The SVG always has a longer, thinner aspect ratio than the container span.
+//    After the SVG fills 100% of the height of the container span,
+//    there is a long arrow shaft left over. That left-over shaft is not shown.
+//    Instead, it is sliced off because the span's CSS has overflow: hidden.
+//    Thus, the reader sees an arrow that matches the subject matter width
+//    without distortion.
+//    Some functions, such as \cancel, need to vary their aspect ratio. These
+//    functions do not get the overflow SVG treatment.
+// Second Brush Stroke
+//    Low resolution monitors struggle to display images in fine detail.
+//    So browsers apply anti-aliasing. A long straight arrow shaft therefore
+//    will sometimes appear as if it has a blurred edge.
+//    To mitigate this, these SVG files contain a second "brush-stroke" on the
+//    arrow shafts. That is, a second long thin rectangular SVG path has been
+//    written directly on top of each arrow shaft. This reinforcement causes
+//    some of the screen pixels to display as black instead of the anti-aliased
+//    gray pixel that a  single path would generate. So we get arrow shafts
+//    whose edges appear to be sharper.
+// In the katexImagesData object just below, the dimensions all
+// correspond to path geometry inside the relevant SVG.
+// For example, \overrightarrow uses the same arrowhead as glyph U+2192
+// from the KaTeX Main font. The scaling factor is 1000.
+// That is, inside the font, that arrowhead is 522 units tall, which
+// corresponds to 0.522 em inside the document.
+
+
+var katexImagesData = {
+  //   path(s), minWidth, height, align
+  overrightarrow: [["rightarrow"], 0.888, 522, "xMaxYMin"],
+  overleftarrow: [["leftarrow"], 0.888, 522, "xMinYMin"],
+  underrightarrow: [["rightarrow"], 0.888, 522, "xMaxYMin"],
+  underleftarrow: [["leftarrow"], 0.888, 522, "xMinYMin"],
+  xrightarrow: [["rightarrow"], 1.469, 522, "xMaxYMin"],
+  xleftarrow: [["leftarrow"], 1.469, 522, "xMinYMin"],
+  Overrightarrow: [["doublerightarrow"], 0.888, 560, "xMaxYMin"],
+  xRightarrow: [["doublerightarrow"], 1.526, 560, "xMaxYMin"],
+  xLeftarrow: [["doubleleftarrow"], 1.526, 560, "xMinYMin"],
+  overleftharpoon: [["leftharpoon"], 0.888, 522, "xMinYMin"],
+  xleftharpoonup: [["leftharpoon"], 0.888, 522, "xMinYMin"],
+  xleftharpoondown: [["leftharpoondown"], 0.888, 522, "xMinYMin"],
+  overrightharpoon: [["rightharpoon"], 0.888, 522, "xMaxYMin"],
+  xrightharpoonup: [["rightharpoon"], 0.888, 522, "xMaxYMin"],
+  xrightharpoondown: [["rightharpoondown"], 0.888, 522, "xMaxYMin"],
+  xlongequal: [["longequal"], 0.888, 334, "xMinYMin"],
+  xtwoheadleftarrow: [["twoheadleftarrow"], 0.888, 334, "xMinYMin"],
+  xtwoheadrightarrow: [["twoheadrightarrow"], 0.888, 334, "xMaxYMin"],
+  overleftrightarrow: [["leftarrow", "rightarrow"], 0.888, 522],
+  overbrace: [["leftbrace", "midbrace", "rightbrace"], 1.6, 548],
+  underbrace: [["leftbraceunder", "midbraceunder", "rightbraceunder"], 1.6, 548],
+  underleftrightarrow: [["leftarrow", "rightarrow"], 0.888, 522],
+  xleftrightarrow: [["leftarrow", "rightarrow"], 1.75, 522],
+  xLeftrightarrow: [["doubleleftarrow", "doublerightarrow"], 1.75, 560],
+  xrightleftharpoons: [["leftharpoondownplus", "rightharpoonplus"], 1.75, 716],
+  xleftrightharpoons: [["leftharpoonplus", "rightharpoondownplus"], 1.75, 716],
+  xhookleftarrow: [["leftarrow", "righthook"], 1.08, 522],
+  xhookrightarrow: [["lefthook", "rightarrow"], 1.08, 522],
+  overlinesegment: [["leftlinesegment", "rightlinesegment"], 0.888, 522],
+  underlinesegment: [["leftlinesegment", "rightlinesegment"], 0.888, 522],
+  overgroup: [["leftgroup", "rightgroup"], 0.888, 342],
+  undergroup: [["leftgroupunder", "rightgroupunder"], 0.888, 342],
+  xmapsto: [["leftmapsto", "rightarrow"], 1.5, 522],
+  xtofrom: [["leftToFrom", "rightToFrom"], 1.75, 528],
+  // The next three arrows are from the mhchem package.
+  // In mhchem.sty, min-length is 2.0em. But these arrows might appear in the
+  // document as \xrightarrow or \xrightleftharpoons. Those have
+  // min-length = 1.75em, so we set min-length on these next three to match.
+  xrightleftarrows: [["baraboveleftarrow", "rightarrowabovebar"], 1.75, 901],
+  xrightequilibrium: [["baraboveshortleftharpoon", "rightharpoonaboveshortbar"], 1.75, 716],
+  xleftequilibrium: [["shortbaraboveleftharpoon", "shortrightharpoonabovebar"], 1.75, 716]
+};
+
+var groupLength = function groupLength(arg) {
+  if (arg.type === "ordgroup") {
+    return arg.body.length;
+  } else {
+    return 1;
+  }
+};
+
+var stretchy_svgSpan = function svgSpan(group, options) {
+  // Create a span with inline SVG for the element.
+  function buildSvgSpan_() {
+    var viewBoxWidth = 400000; // default
+
+    var label = group.label.substr(1);
+
+    if (utils.contains(["widehat", "widecheck", "widetilde", "utilde"], label)) {
+      // Each type in the `if` statement corresponds to one of the ParseNode
+      // types below. This narrowing is required to access `grp.base`.
+      var grp = group; // There are four SVG images available for each function.
+      // Choose a taller image when there are more characters.
+
+      var numChars = groupLength(grp.base);
+      var viewBoxHeight;
+      var pathName;
+
+      var _height;
+
+      if (numChars > 5) {
+        if (label === "widehat" || label === "widecheck") {
+          viewBoxHeight = 420;
+          viewBoxWidth = 2364;
+          _height = 0.42;
+          pathName = label + "4";
+        } else {
+          viewBoxHeight = 312;
+          viewBoxWidth = 2340;
+          _height = 0.34;
+          pathName = "tilde4";
+        }
+      } else {
+        var imgIndex = [1, 1, 2, 2, 3, 3][numChars];
+
+        if (label === "widehat" || label === "widecheck") {
+          viewBoxWidth = [0, 1062, 2364, 2364, 2364][imgIndex];
+          viewBoxHeight = [0, 239, 300, 360, 420][imgIndex];
+          _height = [0, 0.24, 0.3, 0.3, 0.36, 0.42][imgIndex];
+          pathName = label + imgIndex;
+        } else {
+          viewBoxWidth = [0, 600, 1033, 2339, 2340][imgIndex];
+          viewBoxHeight = [0, 260, 286, 306, 312][imgIndex];
+          _height = [0, 0.26, 0.286, 0.3, 0.306, 0.34][imgIndex];
+          pathName = "tilde" + imgIndex;
+        }
+      }
+
+      var path = new domTree_PathNode(pathName);
+      var svgNode = new SvgNode([path], {
+        "width": "100%",
+        "height": _height + "em",
+        "viewBox": "0 0 " + viewBoxWidth + " " + viewBoxHeight,
+        "preserveAspectRatio": "none"
+      });
+      return {
+        span: buildCommon.makeSvgSpan([], [svgNode], options),
+        minWidth: 0,
+        height: _height
+      };
+    } else {
+      var spans = [];
+      var data = katexImagesData[label];
+      var paths = data[0],
+          _minWidth = data[1],
+          _viewBoxHeight = data[2];
+
+      var _height2 = _viewBoxHeight / 1000;
+
+      var numSvgChildren = paths.length;
+      var widthClasses;
+      var aligns;
+
+      if (numSvgChildren === 1) {
+        // $FlowFixMe: All these cases must be of the 4-tuple type.
+        var align1 = data[3];
+        widthClasses = ["hide-tail"];
+        aligns = [align1];
+      } else if (numSvgChildren === 2) {
+        widthClasses = ["halfarrow-left", "halfarrow-right"];
+        aligns = ["xMinYMin", "xMaxYMin"];
+      } else if (numSvgChildren === 3) {
+        widthClasses = ["brace-left", "brace-center", "brace-right"];
+        aligns = ["xMinYMin", "xMidYMin", "xMaxYMin"];
+      } else {
+        throw new Error("Correct katexImagesData or update code here to support\n                    " + numSvgChildren + " children.");
+      }
+
+      for (var i = 0; i < numSvgChildren; i++) {
+        var _path = new domTree_PathNode(paths[i]);
+
+        var _svgNode = new SvgNode([_path], {
+          "width": "400em",
+          "height": _height2 + "em",
+          "viewBox": "0 0 " + viewBoxWidth + " " + _viewBoxHeight,
+          "preserveAspectRatio": aligns[i] + " slice"
+        });
+
+        var _span = buildCommon.makeSvgSpan([widthClasses[i]], [_svgNode], options);
+
+        if (numSvgChildren === 1) {
+          return {
+            span: _span,
+            minWidth: _minWidth,
+            height: _height2
+          };
+        } else {
+          _span.style.height = _height2 + "em";
+          spans.push(_span);
+        }
+      }
+
+      return {
+        span: buildCommon.makeSpan(["stretchy"], spans, options),
+        minWidth: _minWidth,
+        height: _height2
+      };
+    }
+  } // buildSvgSpan_()
+
+
+  var _buildSvgSpan_ = buildSvgSpan_(),
+      span = _buildSvgSpan_.span,
+      minWidth = _buildSvgSpan_.minWidth,
+      height = _buildSvgSpan_.height; // Note that we are returning span.depth = 0.
+  // Any adjustments relative to the baseline must be done in buildHTML.
+
+
+  span.height = height;
+  span.style.height = height + "em";
+
+  if (minWidth > 0) {
+    span.style.minWidth = minWidth + "em";
+  }
+
+  return span;
+};
+
+var stretchy_encloseSpan = function encloseSpan(inner, label, pad, options) {
+  // Return an image span for \cancel, \bcancel, \xcancel, or \fbox
+  var img;
+  var totalHeight = inner.height + inner.depth + 2 * pad;
+
+  if (/fbox|color/.test(label)) {
+    img = buildCommon.makeSpan(["stretchy", label], [], options);
+
+    if (label === "fbox") {
+      var color = options.color && options.getColor();
+
+      if (color) {
+        img.style.borderColor = color;
+      }
+    }
+  } else {
+    // \cancel, \bcancel, or \xcancel
+    // Since \cancel's SVG is inline and it omits the viewBox attribute,
+    // its stroke-width will not vary with span area.
+    var lines = [];
+
+    if (/^[bx]cancel$/.test(label)) {
+      lines.push(new LineNode({
+        "x1": "0",
+        "y1": "0",
+        "x2": "100%",
+        "y2": "100%",
+        "stroke-width": "0.046em"
+      }));
+    }
+
+    if (/^x?cancel$/.test(label)) {
+      lines.push(new LineNode({
+        "x1": "0",
+        "y1": "100%",
+        "x2": "100%",
+        "y2": "0",
+        "stroke-width": "0.046em"
+      }));
+    }
+
+    var svgNode = new SvgNode(lines, {
+      "width": "100%",
+      "height": totalHeight + "em"
+    });
+    img = buildCommon.makeSvgSpan([], [svgNode], options);
+  }
+
+  img.height = totalHeight;
+  img.style.height = totalHeight + "em";
+  return img;
+};
+
+/* harmony default export */ var stretchy = ({
+  encloseSpan: stretchy_encloseSpan,
+  mathMLnode: stretchy_mathMLnode,
+  svgSpan: stretchy_svgSpan
+});
+// CONCATENATED MODULE: ./src/parseNode.js
+
+
+/**
+ * Asserts that the node is of the given type and returns it with stricter
+ * typing. Throws if the node's type does not match.
+ */
+function assertNodeType(node, type) {
+  if (!node || node.type !== type) {
+    throw new Error("Expected node of type " + type + ", but got " + (node ? "node of type " + node.type : String(node)));
+  }
+
+  return node;
+}
+/**
+ * Returns the node more strictly typed iff it is of the given type. Otherwise,
+ * returns null.
+ */
+
+function assertSymbolNodeType(node) {
+  var typedNode = checkSymbolNodeType(node);
+
+  if (!typedNode) {
+    throw new Error("Expected node of symbol group type, but got " + (node ? "node of type " + node.type : String(node)));
+  }
+
+  return typedNode;
+}
+/**
+ * Returns the node more strictly typed iff it is of the given type. Otherwise,
+ * returns null.
+ */
+
+function checkSymbolNodeType(node) {
+  if (node && (node.type === "atom" || NON_ATOMS.hasOwnProperty(node.type))) {
+    // $FlowFixMe
+    return node;
+  }
+
+  return null;
+}
+// CONCATENATED MODULE: ./src/functions/accent.js
+
+
+
+
+
+
+
+
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only "accent", but
+var accent_htmlBuilder = function htmlBuilder(grp, options) {
+  // Accents are handled in the TeXbook pg. 443, rule 12.
+  var base;
+  var group;
+  var supSubGroup;
+
+  if (grp && grp.type === "supsub") {
+    // If our base is a character box, and we have superscripts and
+    // subscripts, the supsub will defer to us. In particular, we want
+    // to attach the superscripts and subscripts to the inner body (so
+    // that the position of the superscripts and subscripts won't be
+    // affected by the height of the accent). We accomplish this by
+    // sticking the base of the accent into the base of the supsub, and
+    // rendering that, while keeping track of where the accent is.
+    // The real accent group is the base of the supsub group
+    group = assertNodeType(grp.base, "accent"); // The character box is the base of the accent group
+
+    base = group.base; // Stick the character box into the base of the supsub group
+
+    grp.base = base; // Rerender the supsub group with its new base, and store that
+    // result.
+
+    supSubGroup = assertSpan(buildHTML_buildGroup(grp, options)); // reset original base
+
+    grp.base = group;
+  } else {
+    group = assertNodeType(grp, "accent");
+    base = group.base;
+  } // Build the base group
+
+
+  var body = buildHTML_buildGroup(base, options.havingCrampedStyle()); // Does the accent need to shift for the skew of a character?
+
+  var mustShift = group.isShifty && utils.isCharacterBox(base); // Calculate the skew of the accent. This is based on the line "If the
+  // nucleus is not a single character, let s = 0; otherwise set s to the
+  // kern amount for the nucleus followed by the \skewchar of its font."
+  // Note that our skew metrics are just the kern between each character
+  // and the skewchar.
+
+  var skew = 0;
+
+  if (mustShift) {
+    // If the base is a character box, then we want the skew of the
+    // innermost character. To do that, we find the innermost character:
+    var baseChar = utils.getBaseElem(base); // Then, we render its group to get the symbol inside it
+
+    var baseGroup = buildHTML_buildGroup(baseChar, options.havingCrampedStyle()); // Finally, we pull the skew off of the symbol.
+
+    skew = assertSymbolDomNode(baseGroup).skew; // Note that we now throw away baseGroup, because the layers we
+    // removed with getBaseElem might contain things like \color which
+    // we can't get rid of.
+    // TODO(emily): Find a better way to get the skew
+  } // calculate the amount of space between the body and the accent
+
+
+  var clearance = Math.min(body.height, options.fontMetrics().xHeight); // Build the accent
+
+  var accentBody;
+
+  if (!group.isStretchy) {
+    var accent;
+    var width;
+
+    if (group.label === "\\vec") {
+      // Before version 0.9, \vec used the combining font glyph U+20D7.
+      // But browsers, especially Safari, are not consistent in how they
+      // render combining characters when not preceded by a character.
+      // So now we use an SVG.
+      // If Safari reforms, we should consider reverting to the glyph.
+      accent = buildCommon.staticSvg("vec", options);
+      width = buildCommon.svgData.vec[1];
+    } else {
+      accent = buildCommon.makeOrd({
+        mode: group.mode,
+        text: group.label
+      }, options, "textord");
+      accent = assertSymbolDomNode(accent); // Remove the italic correction of the accent, because it only serves to
+      // shift the accent over to a place we don't want.
+
+      accent.italic = 0;
+      width = accent.width;
+    }
+
+    accentBody = buildCommon.makeSpan(["accent-body"], [accent]); // "Full" accents expand the width of the resulting symbol to be
+    // at least the width of the accent, and overlap directly onto the
+    // character without any vertical offset.
+
+    var accentFull = group.label === "\\textcircled";
+
+    if (accentFull) {
+      accentBody.classes.push('accent-full');
+      clearance = body.height;
+    } // Shift the accent over by the skew.
+
+
+    var left = skew; // CSS defines `.katex .accent .accent-body:not(.accent-full) { width: 0 }`
+    // so that the accent doesn't contribute to the bounding box.
+    // We need to shift the character by its width (effectively half
+    // its width) to compensate.
+
+    if (!accentFull) {
+      left -= width / 2;
+    }
+
+    accentBody.style.left = left + "em"; // \textcircled uses the \bigcirc glyph, so it needs some
+    // vertical adjustment to match LaTeX.
+
+    if (group.label === "\\textcircled") {
+      accentBody.style.top = ".2em";
+    }
+
+    accentBody = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "kern",
+        size: -clearance
+      }, {
+        type: "elem",
+        elem: accentBody
+      }]
+    }, options);
+  } else {
+    accentBody = stretchy.svgSpan(group, options);
+    accentBody = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "elem",
+        elem: accentBody,
+        wrapperClasses: ["svg-align"],
+        wrapperStyle: skew > 0 ? {
+          width: "calc(100% - " + 2 * skew + "em)",
+          marginLeft: 2 * skew + "em"
+        } : undefined
+      }]
+    }, options);
+  }
+
+  var accentWrap = buildCommon.makeSpan(["mord", "accent"], [accentBody], options);
+
+  if (supSubGroup) {
+    // Here, we replace the "base" child of the supsub with our newly
+    // generated accent.
+    supSubGroup.children[0] = accentWrap; // Since we don't rerun the height calculation after replacing the
+    // accent, we manually recalculate height.
+
+    supSubGroup.height = Math.max(accentWrap.height, supSubGroup.height); // Accents should always be ords, even when their innards are not.
+
+    supSubGroup.classes[0] = "mord";
+    return supSubGroup;
+  } else {
+    return accentWrap;
+  }
+};
+
+var accent_mathmlBuilder = function mathmlBuilder(group, options) {
+  var accentNode = group.isStretchy ? stretchy.mathMLnode(group.label) : new mathMLTree.MathNode("mo", [buildMathML_makeText(group.label, group.mode)]);
+  var node = new mathMLTree.MathNode("mover", [buildMathML_buildGroup(group.base, options), accentNode]);
+  node.setAttribute("accent", "true");
+  return node;
+};
+
+var NON_STRETCHY_ACCENT_REGEX = new RegExp(["\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve", "\\check", "\\hat", "\\vec", "\\dot", "\\mathring"].map(function (accent) {
+  return "\\" + accent;
+}).join("|")); // Accents
+
+defineFunction({
+  type: "accent",
+  names: ["\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve", "\\check", "\\hat", "\\vec", "\\dot", "\\mathring", "\\widecheck", "\\widehat", "\\widetilde", "\\overrightarrow", "\\overleftarrow", "\\Overrightarrow", "\\overleftrightarrow", "\\overgroup", "\\overlinesegment", "\\overleftharpoon", "\\overrightharpoon"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    var base = args[0];
+    var isStretchy = !NON_STRETCHY_ACCENT_REGEX.test(context.funcName);
+    var isShifty = !isStretchy || context.funcName === "\\widehat" || context.funcName === "\\widetilde" || context.funcName === "\\widecheck";
+    return {
+      type: "accent",
+      mode: context.parser.mode,
+      label: context.funcName,
+      isStretchy: isStretchy,
+      isShifty: isShifty,
+      base: base
+    };
+  },
+  htmlBuilder: accent_htmlBuilder,
+  mathmlBuilder: accent_mathmlBuilder
+}); // Text-mode accents
+
+defineFunction({
+  type: "accent",
+  names: ["\\'", "\\`", "\\^", "\\~", "\\=", "\\u", "\\.", '\\"', "\\r", "\\H", "\\v", "\\textcircled"],
+  props: {
+    numArgs: 1,
+    allowedInText: true,
+    allowedInMath: false
+  },
+  handler: function handler(context, args) {
+    var base = args[0];
+    return {
+      type: "accent",
+      mode: context.parser.mode,
+      label: context.funcName,
+      isStretchy: false,
+      isShifty: true,
+      base: base
+    };
+  },
+  htmlBuilder: accent_htmlBuilder,
+  mathmlBuilder: accent_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/accentunder.js
+// Horizontal overlap functions
+
+
+
+
+
+
+defineFunction({
+  type: "accentUnder",
+  names: ["\\underleftarrow", "\\underrightarrow", "\\underleftrightarrow", "\\undergroup", "\\underlinesegment", "\\utilde"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var base = args[0];
+    return {
+      type: "accentUnder",
+      mode: parser.mode,
+      label: funcName,
+      base: base
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Treat under accents much like underlines.
+    var innerGroup = buildHTML_buildGroup(group.base, options);
+    var accentBody = stretchy.svgSpan(group, options);
+    var kern = group.label === "\\utilde" ? 0.12 : 0; // Generate the vlist, with the appropriate kerns
+
+    var vlist = buildCommon.makeVList({
+      positionType: "top",
+      positionData: innerGroup.height,
+      children: [{
+        type: "elem",
+        elem: accentBody,
+        wrapperClasses: ["svg-align"]
+      }, {
+        type: "kern",
+        size: kern
+      }, {
+        type: "elem",
+        elem: innerGroup
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "accentunder"], [vlist], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var accentNode = stretchy.mathMLnode(group.label);
+    var node = new mathMLTree.MathNode("munder", [buildMathML_buildGroup(group.base, options), accentNode]);
+    node.setAttribute("accentunder", "true");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/arrow.js
+
+
+
+
+
+
+
+// Helper function
+var arrow_paddedNode = function paddedNode(group) {
+  var node = new mathMLTree.MathNode("mpadded", group ? [group] : []);
+  node.setAttribute("width", "+0.6em");
+  node.setAttribute("lspace", "0.3em");
+  return node;
+}; // Stretchy arrows with an optional argument
+
+
+defineFunction({
+  type: "xArrow",
+  names: ["\\xleftarrow", "\\xrightarrow", "\\xLeftarrow", "\\xRightarrow", "\\xleftrightarrow", "\\xLeftrightarrow", "\\xhookleftarrow", "\\xhookrightarrow", "\\xmapsto", "\\xrightharpoondown", "\\xrightharpoonup", "\\xleftharpoondown", "\\xleftharpoonup", "\\xrightleftharpoons", "\\xleftrightharpoons", "\\xlongequal", "\\xtwoheadrightarrow", "\\xtwoheadleftarrow", "\\xtofrom", // The next 3 functions are here to support the mhchem extension.
+  // Direct use of these functions is discouraged and may break someday.
+  "\\xrightleftarrows", "\\xrightequilibrium", "\\xleftequilibrium"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    return {
+      type: "xArrow",
+      mode: parser.mode,
+      label: funcName,
+      body: args[0],
+      below: optArgs[0]
+    };
+  },
+  // Flow is unable to correctly infer the type of `group`, even though it's
+  // unamibiguously determined from the passed-in `type` above.
+  htmlBuilder: function htmlBuilder(group, options) {
+    var style = options.style; // Build the argument groups in the appropriate style.
+    // Ref: amsmath.dtx:   \hbox{$\scriptstyle\mkern#3mu{#6}\mkern#4mu$}%
+    // Some groups can return document fragments.  Handle those by wrapping
+    // them in a span.
+
+    var newOptions = options.havingStyle(style.sup());
+    var upperGroup = buildCommon.wrapFragment(buildHTML_buildGroup(group.body, newOptions, options), options);
+    upperGroup.classes.push("x-arrow-pad");
+    var lowerGroup;
+
+    if (group.below) {
+      // Build the lower group
+      newOptions = options.havingStyle(style.sub());
+      lowerGroup = buildCommon.wrapFragment(buildHTML_buildGroup(group.below, newOptions, options), options);
+      lowerGroup.classes.push("x-arrow-pad");
+    }
+
+    var arrowBody = stretchy.svgSpan(group, options); // Re shift: Note that stretchy.svgSpan returned arrowBody.depth = 0.
+    // The point we want on the math axis is at 0.5 * arrowBody.height.
+
+    var arrowShift = -options.fontMetrics().axisHeight + 0.5 * arrowBody.height; // 2 mu kern. Ref: amsmath.dtx: #7\if0#2\else\mkern#2mu\fi
+
+    var upperShift = -options.fontMetrics().axisHeight - 0.5 * arrowBody.height - 0.111; // 0.111 em = 2 mu
+
+    if (upperGroup.depth > 0.25 || group.label === "\\xleftequilibrium") {
+      upperShift -= upperGroup.depth; // shift up if depth encroaches
+    } // Generate the vlist
+
+
+    var vlist;
+
+    if (lowerGroup) {
+      var lowerShift = -options.fontMetrics().axisHeight + lowerGroup.height + 0.5 * arrowBody.height + 0.111;
+      vlist = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: upperGroup,
+          shift: upperShift
+        }, {
+          type: "elem",
+          elem: arrowBody,
+          shift: arrowShift
+        }, {
+          type: "elem",
+          elem: lowerGroup,
+          shift: lowerShift
+        }]
+      }, options);
+    } else {
+      vlist = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: upperGroup,
+          shift: upperShift
+        }, {
+          type: "elem",
+          elem: arrowBody,
+          shift: arrowShift
+        }]
+      }, options);
+    } // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+
+    vlist.children[0].children[0].children[1].classes.push("svg-align");
+    return buildCommon.makeSpan(["mrel", "x-arrow"], [vlist], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var arrowNode = stretchy.mathMLnode(group.label);
+    var node;
+
+    if (group.body) {
+      var upperNode = arrow_paddedNode(buildMathML_buildGroup(group.body, options));
+
+      if (group.below) {
+        var lowerNode = arrow_paddedNode(buildMathML_buildGroup(group.below, options));
+        node = new mathMLTree.MathNode("munderover", [arrowNode, lowerNode, upperNode]);
+      } else {
+        node = new mathMLTree.MathNode("mover", [arrowNode, upperNode]);
+      }
+    } else if (group.below) {
+      var _lowerNode = arrow_paddedNode(buildMathML_buildGroup(group.below, options));
+
+      node = new mathMLTree.MathNode("munder", [arrowNode, _lowerNode]);
+    } else {
+      // This should never happen.
+      // Parser.js throws an error if there is no argument.
+      node = arrow_paddedNode();
+      node = new mathMLTree.MathNode("mover", [arrowNode, node]);
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/char.js
+
+
+ // \@char is an internal function that takes a grouped decimal argument like
+// {123} and converts into symbol with code 123.  It is used by the *macro*
+// \char defined in macros.js.
+
+defineFunction({
+  type: "textord",
+  names: ["\\@char"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var arg = assertNodeType(args[0], "ordgroup");
+    var group = arg.body;
+    var number = "";
+
+    for (var i = 0; i < group.length; i++) {
+      var node = assertNodeType(group[i], "textord");
+      number += node.text;
+    }
+
+    var code = parseInt(number);
+
+    if (isNaN(code)) {
+      throw new src_ParseError("\\@char has non-numeric argument " + number);
+    }
+
+    return {
+      type: "textord",
+      mode: parser.mode,
+      text: String.fromCharCode(code)
+    };
+  }
+});
+// CONCATENATED MODULE: ./src/functions/color.js
+
+
+
+
+
+
+
+var color_htmlBuilder = function htmlBuilder(group, options) {
+  var elements = buildHTML_buildExpression(group.body, options.withColor(group.color), false); // \color isn't supposed to affect the type of the elements it contains.
+  // To accomplish this, we wrap the results in a fragment, so the inner
+  // elements will be able to directly interact with their neighbors. For
+  // example, `\color{red}{2 +} 3` has the same spacing as `2 + 3`
+
+  return buildCommon.makeFragment(elements);
+};
+
+var color_mathmlBuilder = function mathmlBuilder(group, options) {
+  var inner = buildMathML_buildExpression(group.body, options.withColor(group.color));
+  var node = new mathMLTree.MathNode("mstyle", inner);
+  node.setAttribute("mathcolor", group.color);
+  return node;
+};
+
+defineFunction({
+  type: "color",
+  names: ["\\textcolor"],
+  props: {
+    numArgs: 2,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "original"]
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var color = assertNodeType(args[0], "color-token").color;
+    var body = args[1];
+    return {
+      type: "color",
+      mode: parser.mode,
+      color: color,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: color_htmlBuilder,
+  mathmlBuilder: color_mathmlBuilder
+});
+defineFunction({
+  type: "color",
+  names: ["\\color"],
+  props: {
+    numArgs: 1,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color"]
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser,
+        breakOnTokenText = _ref2.breakOnTokenText;
+    var color = assertNodeType(args[0], "color-token").color; // Set macro \current@color in current namespace to store the current
+    // color, mimicking the behavior of color.sty.
+    // This is currently used just to correctly color a \right
+    // that follows a \color command.
+
+    parser.gullet.macros.set("\\current@color", color); // Parse out the implicit body that should be colored.
+
+    var body = parser.parseExpression(true, breakOnTokenText);
+    return {
+      type: "color",
+      mode: parser.mode,
+      color: color,
+      body: body
+    };
+  },
+  htmlBuilder: color_htmlBuilder,
+  mathmlBuilder: color_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/cr.js
+// Row breaks within tabular environments, and line breaks at top level
+
+
+
+
+
+ // \\ is a macro mapping to either \cr or \newline.  Because they have the
+// same signature, we implement them as one megafunction, with newRow
+// indicating whether we're in the \cr case, and newLine indicating whether
+// to break the line in the \newline case.
+
+defineFunction({
+  type: "cr",
+  names: ["\\cr", "\\newline"],
+  props: {
+    numArgs: 0,
+    numOptionalArgs: 1,
+    argTypes: ["size"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var size = optArgs[0];
+    var newRow = funcName === "\\cr";
+    var newLine = false;
+
+    if (!newRow) {
+      if (parser.settings.displayMode && parser.settings.useStrictBehavior("newLineInDisplayMode", "In LaTeX, \\\\ or \\newline " + "does nothing in display mode")) {
+        newLine = false;
+      } else {
+        newLine = true;
+      }
+    }
+
+    return {
+      type: "cr",
+      mode: parser.mode,
+      newLine: newLine,
+      newRow: newRow,
+      size: size && assertNodeType(size, "size").value
+    };
+  },
+  // The following builders are called only at the top level,
+  // not within tabular/array environments.
+  htmlBuilder: function htmlBuilder(group, options) {
+    if (group.newRow) {
+      throw new src_ParseError("\\cr valid only within a tabular/array environment");
+    }
+
+    var span = buildCommon.makeSpan(["mspace"], [], options);
+
+    if (group.newLine) {
+      span.classes.push("newline");
+
+      if (group.size) {
+        span.style.marginTop = units_calculateSize(group.size, options) + "em";
+      }
+    }
+
+    return span;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mspace");
+
+    if (group.newLine) {
+      node.setAttribute("linebreak", "newline");
+
+      if (group.size) {
+        node.setAttribute("height", units_calculateSize(group.size, options) + "em");
+      }
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/def.js
+
+
+
+var globalMap = {
+  "\\global": "\\global",
+  "\\long": "\\\\globallong",
+  "\\\\globallong": "\\\\globallong",
+  "\\def": "\\gdef",
+  "\\gdef": "\\gdef",
+  "\\edef": "\\xdef",
+  "\\xdef": "\\xdef",
+  "\\let": "\\\\globallet",
+  "\\futurelet": "\\\\globalfuture"
+};
+
+var def_checkControlSequence = function checkControlSequence(tok) {
+  var name = tok.text;
+
+  if (/^(?:[\\{}$&#^_]|EOF)$/.test(name)) {
+    throw new src_ParseError("Expected a control sequence", tok);
+  }
+
+  return name;
+};
+
+var getRHS = function getRHS(parser) {
+  var tok = parser.gullet.popToken();
+
+  if (tok.text === "=") {
+    // consume optional equals
+    tok = parser.gullet.popToken();
+
+    if (tok.text === " ") {
+      // consume one optional space
+      tok = parser.gullet.popToken();
+    }
+  }
+
+  return tok;
+};
+
+var letCommand = function letCommand(parser, name, tok, global) {
+  var macro = parser.gullet.macros.get(tok.text);
+
+  if (macro == null) {
+    // don't expand it later even if a macro with the same name is defined
+    // e.g., \let\foo=\frac \def\frac{\relax} \frac12
+    tok.noexpand = true;
+    macro = {
+      tokens: [tok],
+      numArgs: 0,
+      // reproduce the same behavior in expansion
+      unexpandable: !parser.gullet.isExpandable(tok.text)
+    };
+  }
+
+  parser.gullet.macros.set(name, macro, global);
+}; // <assignment> -> <non-macro assignment>|<macro assignment>
+// <non-macro assignment> -> <simple assignment>|\global<non-macro assignment>
+// <macro assignment> -> <definition>|<prefix><macro assignment>
+// <prefix> -> \global|\long|\outer
+
+
+defineFunction({
+  type: "internal",
+  names: ["\\global", "\\long", "\\\\globallong"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    parser.consumeSpaces();
+    var token = parser.fetch();
+
+    if (globalMap[token.text]) {
+      // KaTeX doesn't have \par, so ignore \long
+      if (funcName === "\\global" || funcName === "\\\\globallong") {
+        token.text = globalMap[token.text];
+      }
+
+      return assertNodeType(parser.parseFunction(), "internal");
+    }
+
+    throw new src_ParseError("Invalid token after macro prefix", token);
+  }
+}); // Basic support for macro definitions: \def, \gdef, \edef, \xdef
+// <definition> -> <def><control sequence><definition text>
+// <def> -> \def|\gdef|\edef|\xdef
+// <definition text> -> <parameter text><left brace><balanced text><right brace>
+
+defineFunction({
+  type: "internal",
+  names: ["\\def", "\\gdef", "\\edef", "\\xdef"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref2) {
+    var parser = _ref2.parser,
+        funcName = _ref2.funcName;
+    var arg = parser.gullet.consumeArgs(1)[0];
+
+    if (arg.length !== 1) {
+      throw new src_ParseError("\\gdef's first argument must be a macro name");
+    }
+
+    var name = arg[0].text; // Count argument specifiers, and check they are in the order #1 #2 ...
+
+    var numArgs = 0;
+    arg = parser.gullet.consumeArgs(1)[0];
+
+    while (arg.length === 1 && arg[0].text === "#") {
+      arg = parser.gullet.consumeArgs(1)[0];
+
+      if (arg.length !== 1) {
+        throw new src_ParseError("Invalid argument number length \"" + arg.length + "\"");
+      }
+
+      if (!/^[1-9]$/.test(arg[0].text)) {
+        throw new src_ParseError("Invalid argument number \"" + arg[0].text + "\"");
+      }
+
+      numArgs++;
+
+      if (parseInt(arg[0].text) !== numArgs) {
+        throw new src_ParseError("Argument number \"" + arg[0].text + "\" out of order");
+      }
+
+      arg = parser.gullet.consumeArgs(1)[0];
+    }
+
+    if (funcName === "\\edef" || funcName === "\\xdef") {
+      arg = parser.gullet.expandTokens(arg);
+      arg.reverse(); // to fit in with stack order
+    } // Final arg is the expansion of the macro
+
+
+    parser.gullet.macros.set(name, {
+      tokens: arg,
+      numArgs: numArgs
+    }, funcName === globalMap[funcName]);
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+}); // <simple assignment> -> <let assignment>
+// <let assignment> -> \futurelet<control sequence><token><token>
+//     | \let<control sequence><equals><one optional space><token>
+// <equals> -> <optional spaces>|<optional spaces>=
+
+defineFunction({
+  type: "internal",
+  names: ["\\let", "\\\\globallet"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref3) {
+    var parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    var name = def_checkControlSequence(parser.gullet.popToken());
+    parser.gullet.consumeSpaces();
+    var tok = getRHS(parser);
+    letCommand(parser, name, tok, funcName === "\\\\globallet");
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+}); // ref: https://www.tug.org/TUGboat/tb09-3/tb22bechtolsheim.pdf
+
+defineFunction({
+  type: "internal",
+  names: ["\\futurelet", "\\\\globalfuture"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref4) {
+    var parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    var name = def_checkControlSequence(parser.gullet.popToken());
+    var middle = parser.gullet.popToken();
+    var tok = parser.gullet.popToken();
+    letCommand(parser, name, tok, funcName === "\\\\globalfuture");
+    parser.gullet.pushToken(tok);
+    parser.gullet.pushToken(middle);
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+});
+// CONCATENATED MODULE: ./src/delimiter.js
+/**
+ * This file deals with creating delimiters of various sizes. The TeXbook
+ * discusses these routines on page 441-442, in the "Another subroutine sets box
+ * x to a specified variable delimiter" paragraph.
+ *
+ * There are three main routines here. `makeSmallDelim` makes a delimiter in the
+ * normal font, but in either text, script, or scriptscript style.
+ * `makeLargeDelim` makes a delimiter in textstyle, but in one of the Size1,
+ * Size2, Size3, or Size4 fonts. `makeStackedDelim` makes a delimiter out of
+ * smaller pieces that are stacked on top of one another.
+ *
+ * The functions take a parameter `center`, which determines if the delimiter
+ * should be centered around the axis.
+ *
+ * Then, there are three exposed functions. `sizedDelim` makes a delimiter in
+ * one of the given sizes. This is used for things like `\bigl`.
+ * `customSizedDelim` makes a delimiter with a given total height+depth. It is
+ * called in places like `\sqrt`. `leftRightDelim` makes an appropriate
+ * delimiter which surrounds an expression of a given height an depth. It is
+ * used in `\left` and `\right`.
+ */
+
+
+
+
+
+
+
+
+
+/**
+ * Get the metrics for a given symbol and font, after transformation (i.e.
+ * after following replacement from symbols.js)
+ */
+var delimiter_getMetrics = function getMetrics(symbol, font, mode) {
+  var replace = src_symbols.math[symbol] && src_symbols.math[symbol].replace;
+  var metrics = getCharacterMetrics(replace || symbol, font, mode);
+
+  if (!metrics) {
+    throw new Error("Unsupported symbol " + symbol + " and font size " + font + ".");
+  }
+
+  return metrics;
+};
+/**
+ * Puts a delimiter span in a given style, and adds appropriate height, depth,
+ * and maxFontSizes.
+ */
+
+
+var delimiter_styleWrap = function styleWrap(delim, toStyle, options, classes) {
+  var newOptions = options.havingBaseStyle(toStyle);
+  var span = buildCommon.makeSpan(classes.concat(newOptions.sizingClasses(options)), [delim], options);
+  var delimSizeMultiplier = newOptions.sizeMultiplier / options.sizeMultiplier;
+  span.height *= delimSizeMultiplier;
+  span.depth *= delimSizeMultiplier;
+  span.maxFontSize = newOptions.sizeMultiplier;
+  return span;
+};
+
+var centerSpan = function centerSpan(span, options, style) {
+  var newOptions = options.havingBaseStyle(style);
+  var shift = (1 - options.sizeMultiplier / newOptions.sizeMultiplier) * options.fontMetrics().axisHeight;
+  span.classes.push("delimcenter");
+  span.style.top = shift + "em";
+  span.height -= shift;
+  span.depth += shift;
+};
+/**
+ * Makes a small delimiter. This is a delimiter that comes in the Main-Regular
+ * font, but is restyled to either be in textstyle, scriptstyle, or
+ * scriptscriptstyle.
+ */
+
+
+var delimiter_makeSmallDelim = function makeSmallDelim(delim, style, center, options, mode, classes) {
+  var text = buildCommon.makeSymbol(delim, "Main-Regular", mode, options);
+  var span = delimiter_styleWrap(text, style, options, classes);
+
+  if (center) {
+    centerSpan(span, options, style);
+  }
+
+  return span;
+};
+/**
+ * Builds a symbol in the given font size (note size is an integer)
+ */
+
+
+var delimiter_mathrmSize = function mathrmSize(value, size, mode, options) {
+  return buildCommon.makeSymbol(value, "Size" + size + "-Regular", mode, options);
+};
+/**
+ * Makes a large delimiter. This is a delimiter that comes in the Size1, Size2,
+ * Size3, or Size4 fonts. It is always rendered in textstyle.
+ */
+
+
+var delimiter_makeLargeDelim = function makeLargeDelim(delim, size, center, options, mode, classes) {
+  var inner = delimiter_mathrmSize(delim, size, mode, options);
+  var span = delimiter_styleWrap(buildCommon.makeSpan(["delimsizing", "size" + size], [inner], options), src_Style.TEXT, options, classes);
+
+  if (center) {
+    centerSpan(span, options, src_Style.TEXT);
+  }
+
+  return span;
+};
+/**
+ * Make an inner span with the given offset and in the given font. This is used
+ * in `makeStackedDelim` to make the stacking pieces for the delimiter.
+ */
+
+
+var delimiter_makeInner = function makeInner(symbol, font, mode) {
+  var sizeClass; // Apply the correct CSS class to choose the right font.
+
+  if (font === "Size1-Regular") {
+    sizeClass = "delim-size1";
+  } else
+    /* if (font === "Size4-Regular") */
+    {
+      sizeClass = "delim-size4";
+    }
+
+  var inner = buildCommon.makeSpan(["delimsizinginner", sizeClass], [buildCommon.makeSpan([], [buildCommon.makeSymbol(symbol, font, mode)])]); // Since this will be passed into `makeVList` in the end, wrap the element
+  // in the appropriate tag that VList uses.
+
+  return {
+    type: "elem",
+    elem: inner
+  };
+}; // Helper for makeStackedDelim
+
+
+var lap = {
+  type: "kern",
+  size: -0.005
+};
+/**
+ * Make a stacked delimiter out of a given delimiter, with the total height at
+ * least `heightTotal`. This routine is mentioned on page 442 of the TeXbook.
+ */
+
+var delimiter_makeStackedDelim = function makeStackedDelim(delim, heightTotal, center, options, mode, classes) {
+  // There are four parts, the top, an optional middle, a repeated part, and a
+  // bottom.
+  var top;
+  var middle;
+  var repeat;
+  var bottom;
+  top = repeat = bottom = delim;
+  middle = null; // Also keep track of what font the delimiters are in
+
+  var font = "Size1-Regular"; // We set the parts and font based on the symbol. Note that we use
+  // '\u23d0' instead of '|' and '\u2016' instead of '\\|' for the
+  // repeats of the arrows
+
+  if (delim === "\\uparrow") {
+    repeat = bottom = "\u23D0";
+  } else if (delim === "\\Uparrow") {
+    repeat = bottom = "\u2016";
+  } else if (delim === "\\downarrow") {
+    top = repeat = "\u23D0";
+  } else if (delim === "\\Downarrow") {
+    top = repeat = "\u2016";
+  } else if (delim === "\\updownarrow") {
+    top = "\\uparrow";
+    repeat = "\u23D0";
+    bottom = "\\downarrow";
+  } else if (delim === "\\Updownarrow") {
+    top = "\\Uparrow";
+    repeat = "\u2016";
+    bottom = "\\Downarrow";
+  } else if (delim === "[" || delim === "\\lbrack") {
+    top = "\u23A1";
+    repeat = "\u23A2";
+    bottom = "\u23A3";
+    font = "Size4-Regular";
+  } else if (delim === "]" || delim === "\\rbrack") {
+    top = "\u23A4";
+    repeat = "\u23A5";
+    bottom = "\u23A6";
+    font = "Size4-Regular";
+  } else if (delim === "\\lfloor" || delim === "\u230A") {
+    repeat = top = "\u23A2";
+    bottom = "\u23A3";
+    font = "Size4-Regular";
+  } else if (delim === "\\lceil" || delim === "\u2308") {
+    top = "\u23A1";
+    repeat = bottom = "\u23A2";
+    font = "Size4-Regular";
+  } else if (delim === "\\rfloor" || delim === "\u230B") {
+    repeat = top = "\u23A5";
+    bottom = "\u23A6";
+    font = "Size4-Regular";
+  } else if (delim === "\\rceil" || delim === "\u2309") {
+    top = "\u23A4";
+    repeat = bottom = "\u23A5";
+    font = "Size4-Regular";
+  } else if (delim === "(" || delim === "\\lparen") {
+    top = "\u239B";
+    repeat = "\u239C";
+    bottom = "\u239D";
+    font = "Size4-Regular";
+  } else if (delim === ")" || delim === "\\rparen") {
+    top = "\u239E";
+    repeat = "\u239F";
+    bottom = "\u23A0";
+    font = "Size4-Regular";
+  } else if (delim === "\\{" || delim === "\\lbrace") {
+    top = "\u23A7";
+    middle = "\u23A8";
+    bottom = "\u23A9";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\}" || delim === "\\rbrace") {
+    top = "\u23AB";
+    middle = "\u23AC";
+    bottom = "\u23AD";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\lgroup" || delim === "\u27EE") {
+    top = "\u23A7";
+    bottom = "\u23A9";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\rgroup" || delim === "\u27EF") {
+    top = "\u23AB";
+    bottom = "\u23AD";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\lmoustache" || delim === "\u23B0") {
+    top = "\u23A7";
+    bottom = "\u23AD";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\rmoustache" || delim === "\u23B1") {
+    top = "\u23AB";
+    bottom = "\u23A9";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } // Get the metrics of the four sections
+
+
+  var topMetrics = delimiter_getMetrics(top, font, mode);
+  var topHeightTotal = topMetrics.height + topMetrics.depth;
+  var repeatMetrics = delimiter_getMetrics(repeat, font, mode);
+  var repeatHeightTotal = repeatMetrics.height + repeatMetrics.depth;
+  var bottomMetrics = delimiter_getMetrics(bottom, font, mode);
+  var bottomHeightTotal = bottomMetrics.height + bottomMetrics.depth;
+  var middleHeightTotal = 0;
+  var middleFactor = 1;
+
+  if (middle !== null) {
+    var middleMetrics = delimiter_getMetrics(middle, font, mode);
+    middleHeightTotal = middleMetrics.height + middleMetrics.depth;
+    middleFactor = 2; // repeat symmetrically above and below middle
+  } // Calcuate the minimal height that the delimiter can have.
+  // It is at least the size of the top, bottom, and optional middle combined.
+
+
+  var minHeight = topHeightTotal + bottomHeightTotal + middleHeightTotal; // Compute the number of copies of the repeat symbol we will need
+
+  var repeatCount = Math.max(0, Math.ceil((heightTotal - minHeight) / (middleFactor * repeatHeightTotal))); // Compute the total height of the delimiter including all the symbols
+
+  var realHeightTotal = minHeight + repeatCount * middleFactor * repeatHeightTotal; // The center of the delimiter is placed at the center of the axis. Note
+  // that in this context, "center" means that the delimiter should be
+  // centered around the axis in the current style, while normally it is
+  // centered around the axis in textstyle.
+
+  var axisHeight = options.fontMetrics().axisHeight;
+
+  if (center) {
+    axisHeight *= options.sizeMultiplier;
+  } // Calculate the depth
+
+
+  var depth = realHeightTotal / 2 - axisHeight; // This function differs from the TeX procedure in one way.
+  // We shift each repeat element downwards by 0.005em, to prevent a gap
+  // due to browser floating point rounding error.
+  // Then, at the last element-to element joint, we add one extra repeat
+  // element to cover the gap created by the shifts.
+  // Find the shift needed to align the upper end of the extra element at a point
+  // 0.005em above the lower end of the top element.
+
+  var shiftOfExtraElement = (repeatCount + 1) * 0.005 - repeatHeightTotal; // Now, we start building the pieces that will go into the vlist
+  // Keep a list of the inner pieces
+
+  var inners = []; // Add the bottom symbol
+
+  inners.push(delimiter_makeInner(bottom, font, mode));
+
+  if (middle === null) {
+    // Add that many symbols
+    for (var i = 0; i < repeatCount; i++) {
+      inners.push(lap); // overlap
+
+      inners.push(delimiter_makeInner(repeat, font, mode));
+    }
+  } else {
+    // When there is a middle bit, we need the middle part and two repeated
+    // sections
+    for (var _i = 0; _i < repeatCount; _i++) {
+      inners.push(lap);
+      inners.push(delimiter_makeInner(repeat, font, mode));
+    } // Insert one extra repeat element.
+
+
+    inners.push({
+      type: "kern",
+      size: shiftOfExtraElement
+    });
+    inners.push(delimiter_makeInner(repeat, font, mode));
+    inners.push(lap); // Now insert the middle of the brace.
+
+    inners.push(delimiter_makeInner(middle, font, mode));
+
+    for (var _i2 = 0; _i2 < repeatCount; _i2++) {
+      inners.push(lap);
+      inners.push(delimiter_makeInner(repeat, font, mode));
+    }
+  } // To cover the gap create by the overlaps, insert one more repeat element,
+  // at a position that juts 0.005 above the bottom of the top element.
+
+
+  if ((repeat === "\u239C" || repeat === "\u239F") && repeatCount === 0) {
+    // Parentheses need a short repeat element in order to avoid an overrun.
+    // We'll make a 0.3em tall element from a SVG.
+    var overlap = buildCommon.svgData.leftParenInner[2] / 2;
+    inners.push({
+      type: "kern",
+      size: -overlap
+    });
+    var pathName = repeat === "\u239C" ? "leftParenInner" : "rightParenInner";
+    var innerSpan = buildCommon.staticSvg(pathName, options);
+    inners.push({
+      type: "elem",
+      elem: innerSpan
+    });
+    inners.push({
+      type: "kern",
+      size: -overlap
+    });
+  } else {
+    inners.push({
+      type: "kern",
+      size: shiftOfExtraElement
+    });
+    inners.push(delimiter_makeInner(repeat, font, mode));
+    inners.push(lap);
+  } // Add the top symbol
+
+
+  inners.push(delimiter_makeInner(top, font, mode)); // Finally, build the vlist
+
+  var newOptions = options.havingBaseStyle(src_Style.TEXT);
+  var inner = buildCommon.makeVList({
+    positionType: "bottom",
+    positionData: depth,
+    children: inners
+  }, newOptions);
+  return delimiter_styleWrap(buildCommon.makeSpan(["delimsizing", "mult"], [inner], newOptions), src_Style.TEXT, options, classes);
+}; // All surds have 0.08em padding above the viniculum inside the SVG.
+// That keeps browser span height rounding error from pinching the line.
+
+
+var vbPad = 80; // padding above the surd, measured inside the viewBox.
+
+var emPad = 0.08; // padding, in ems, measured in the document.
+
+var delimiter_sqrtSvg = function sqrtSvg(sqrtName, height, viewBoxHeight, extraViniculum, options) {
+  var path = sqrtPath(sqrtName, extraViniculum, viewBoxHeight);
+  var pathNode = new domTree_PathNode(sqrtName, path);
+  var svg = new SvgNode([pathNode], {
+    // Note: 1000:1 ratio of viewBox to document em width.
+    "width": "400em",
+    "height": height + "em",
+    "viewBox": "0 0 400000 " + viewBoxHeight,
+    "preserveAspectRatio": "xMinYMin slice"
+  });
+  return buildCommon.makeSvgSpan(["hide-tail"], [svg], options);
+};
+/**
+ * Make a sqrt image of the given height,
+ */
+
+
+var makeSqrtImage = function makeSqrtImage(height, options) {
+  // Define a newOptions that removes the effect of size changes such as \Huge.
+  // We don't pick different a height surd for \Huge. For it, we scale up.
+  var newOptions = options.havingBaseSizing(); // Pick the desired surd glyph from a sequence of surds.
+
+  var delim = traverseSequence("\\surd", height * newOptions.sizeMultiplier, stackLargeDelimiterSequence, newOptions);
+  var sizeMultiplier = newOptions.sizeMultiplier; // default
+  // The standard sqrt SVGs each have a 0.04em thick viniculum.
+  // If Settings.minRuleThickness is larger than that, we add extraViniculum.
+
+  var extraViniculum = Math.max(0, options.minRuleThickness - options.fontMetrics().sqrtRuleThickness); // Create a span containing an SVG image of a sqrt symbol.
+
+  var span;
+  var spanHeight = 0;
+  var texHeight = 0;
+  var viewBoxHeight = 0;
+  var advanceWidth; // We create viewBoxes with 80 units of "padding" above each surd.
+  // Then browser rounding error on the parent span height will not
+  // encroach on the ink of the viniculum. But that padding is not
+  // included in the TeX-like `height` used for calculation of
+  // vertical alignment. So texHeight = span.height < span.style.height.
+
+  if (delim.type === "small") {
+    // Get an SVG that is derived from glyph U+221A in font KaTeX-Main.
+    // 1000 unit normal glyph height.
+    viewBoxHeight = 1000 + 1000 * extraViniculum + vbPad;
+
+    if (height < 1.0) {
+      sizeMultiplier = 1.0; // mimic a \textfont radical
+    } else if (height < 1.4) {
+      sizeMultiplier = 0.7; // mimic a \scriptfont radical
+    }
+
+    spanHeight = (1.0 + extraViniculum + emPad) / sizeMultiplier;
+    texHeight = (1.00 + extraViniculum) / sizeMultiplier;
+    span = delimiter_sqrtSvg("sqrtMain", spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "0.853em";
+    advanceWidth = 0.833 / sizeMultiplier; // from the font.
+  } else if (delim.type === "large") {
+    // These SVGs come from fonts: KaTeX_Size1, _Size2, etc.
+    viewBoxHeight = (1000 + vbPad) * sizeToMaxHeight[delim.size];
+    texHeight = (sizeToMaxHeight[delim.size] + extraViniculum) / sizeMultiplier;
+    spanHeight = (sizeToMaxHeight[delim.size] + extraViniculum + emPad) / sizeMultiplier;
+    span = delimiter_sqrtSvg("sqrtSize" + delim.size, spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "1.02em";
+    advanceWidth = 1.0 / sizeMultiplier; // 1.0 from the font.
+  } else {
+    // Tall sqrt. In TeX, this would be stacked using multiple glyphs.
+    // We'll use a single SVG to accomplish the same thing.
+    spanHeight = height + extraViniculum + emPad;
+    texHeight = height + extraViniculum;
+    viewBoxHeight = Math.floor(1000 * height + extraViniculum) + vbPad;
+    span = delimiter_sqrtSvg("sqrtTall", spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "0.742em";
+    advanceWidth = 1.056;
+  }
+
+  span.height = texHeight;
+  span.style.height = spanHeight + "em";
+  return {
+    span: span,
+    advanceWidth: advanceWidth,
+    // Calculate the actual line width.
+    // This actually should depend on the chosen font -- e.g. \boldmath
+    // should use the thicker surd symbols from e.g. KaTeX_Main-Bold, and
+    // have thicker rules.
+    ruleWidth: (options.fontMetrics().sqrtRuleThickness + extraViniculum) * sizeMultiplier
+  };
+}; // There are three kinds of delimiters, delimiters that stack when they become
+// too large
+
+
+var stackLargeDelimiters = ["(", "\\lparen", ")", "\\rparen", "[", "\\lbrack", "]", "\\rbrack", "\\{", "\\lbrace", "\\}", "\\rbrace", "\\lfloor", "\\rfloor", "\u230A", "\u230B", "\\lceil", "\\rceil", "\u2308", "\u2309", "\\surd"]; // delimiters that always stack
+
+var stackAlwaysDelimiters = ["\\uparrow", "\\downarrow", "\\updownarrow", "\\Uparrow", "\\Downarrow", "\\Updownarrow", "|", "\\|", "\\vert", "\\Vert", "\\lvert", "\\rvert", "\\lVert", "\\rVert", "\\lgroup", "\\rgroup", "\u27EE", "\u27EF", "\\lmoustache", "\\rmoustache", "\u23B0", "\u23B1"]; // and delimiters that never stack
+
+var stackNeverDelimiters = ["<", ">", "\\langle", "\\rangle", "/", "\\backslash", "\\lt", "\\gt"]; // Metrics of the different sizes. Found by looking at TeX's output of
+// $\bigl| // \Bigl| \biggl| \Biggl| \showlists$
+// Used to create stacked delimiters of appropriate sizes in makeSizedDelim.
+
+var sizeToMaxHeight = [0, 1.2, 1.8, 2.4, 3.0];
+/**
+ * Used to create a delimiter of a specific size, where `size` is 1, 2, 3, or 4.
+ */
+
+var delimiter_makeSizedDelim = function makeSizedDelim(delim, size, options, mode, classes) {
+  // < and > turn into \langle and \rangle in delimiters
+  if (delim === "<" || delim === "\\lt" || delim === "\u27E8") {
+    delim = "\\langle";
+  } else if (delim === ">" || delim === "\\gt" || delim === "\u27E9") {
+    delim = "\\rangle";
+  } // Sized delimiters are never centered.
+
+
+  if (utils.contains(stackLargeDelimiters, delim) || utils.contains(stackNeverDelimiters, delim)) {
+    return delimiter_makeLargeDelim(delim, size, false, options, mode, classes);
+  } else if (utils.contains(stackAlwaysDelimiters, delim)) {
+    return delimiter_makeStackedDelim(delim, sizeToMaxHeight[size], false, options, mode, classes);
+  } else {
+    throw new src_ParseError("Illegal delimiter: '" + delim + "'");
+  }
+};
+/**
+ * There are three different sequences of delimiter sizes that the delimiters
+ * follow depending on the kind of delimiter. This is used when creating custom
+ * sized delimiters to decide whether to create a small, large, or stacked
+ * delimiter.
+ *
+ * In real TeX, these sequences aren't explicitly defined, but are instead
+ * defined inside the font metrics. Since there are only three sequences that
+ * are possible for the delimiters that TeX defines, it is easier to just encode
+ * them explicitly here.
+ */
+
+
+// Delimiters that never stack try small delimiters and large delimiters only
+var stackNeverDelimiterSequence = [{
+  type: "small",
+  style: src_Style.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: src_Style.SCRIPT
+}, {
+  type: "small",
+  style: src_Style.TEXT
+}, {
+  type: "large",
+  size: 1
+}, {
+  type: "large",
+  size: 2
+}, {
+  type: "large",
+  size: 3
+}, {
+  type: "large",
+  size: 4
+}]; // Delimiters that always stack try the small delimiters first, then stack
+
+var stackAlwaysDelimiterSequence = [{
+  type: "small",
+  style: src_Style.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: src_Style.SCRIPT
+}, {
+  type: "small",
+  style: src_Style.TEXT
+}, {
+  type: "stack"
+}]; // Delimiters that stack when large try the small and then large delimiters, and
+// stack afterwards
+
+var stackLargeDelimiterSequence = [{
+  type: "small",
+  style: src_Style.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: src_Style.SCRIPT
+}, {
+  type: "small",
+  style: src_Style.TEXT
+}, {
+  type: "large",
+  size: 1
+}, {
+  type: "large",
+  size: 2
+}, {
+  type: "large",
+  size: 3
+}, {
+  type: "large",
+  size: 4
+}, {
+  type: "stack"
+}];
+/**
+ * Get the font used in a delimiter based on what kind of delimiter it is.
+ * TODO(#963) Use more specific font family return type once that is introduced.
+ */
+
+var delimTypeToFont = function delimTypeToFont(type) {
+  if (type.type === "small") {
+    return "Main-Regular";
+  } else if (type.type === "large") {
+    return "Size" + type.size + "-Regular";
+  } else if (type.type === "stack") {
+    return "Size4-Regular";
+  } else {
+    throw new Error("Add support for delim type '" + type.type + "' here.");
+  }
+};
+/**
+ * Traverse a sequence of types of delimiters to decide what kind of delimiter
+ * should be used to create a delimiter of the given height+depth.
+ */
+
+
+var traverseSequence = function traverseSequence(delim, height, sequence, options) {
+  // Here, we choose the index we should start at in the sequences. In smaller
+  // sizes (which correspond to larger numbers in style.size) we start earlier
+  // in the sequence. Thus, scriptscript starts at index 3-3=0, script starts
+  // at index 3-2=1, text starts at 3-1=2, and display starts at min(2,3-0)=2
+  var start = Math.min(2, 3 - options.style.size);
+
+  for (var i = start; i < sequence.length; i++) {
+    if (sequence[i].type === "stack") {
+      // This is always the last delimiter, so we just break the loop now.
+      break;
+    }
+
+    var metrics = delimiter_getMetrics(delim, delimTypeToFont(sequence[i]), "math");
+    var heightDepth = metrics.height + metrics.depth; // Small delimiters are scaled down versions of the same font, so we
+    // account for the style change size.
+
+    if (sequence[i].type === "small") {
+      var newOptions = options.havingBaseStyle(sequence[i].style);
+      heightDepth *= newOptions.sizeMultiplier;
+    } // Check if the delimiter at this size works for the given height.
+
+
+    if (heightDepth > height) {
+      return sequence[i];
+    }
+  } // If we reached the end of the sequence, return the last sequence element.
+
+
+  return sequence[sequence.length - 1];
+};
+/**
+ * Make a delimiter of a given height+depth, with optional centering. Here, we
+ * traverse the sequences, and create a delimiter that the sequence tells us to.
+ */
+
+
+var delimiter_makeCustomSizedDelim = function makeCustomSizedDelim(delim, height, center, options, mode, classes) {
+  if (delim === "<" || delim === "\\lt" || delim === "\u27E8") {
+    delim = "\\langle";
+  } else if (delim === ">" || delim === "\\gt" || delim === "\u27E9") {
+    delim = "\\rangle";
+  } // Decide what sequence to use
+
+
+  var sequence;
+
+  if (utils.contains(stackNeverDelimiters, delim)) {
+    sequence = stackNeverDelimiterSequence;
+  } else if (utils.contains(stackLargeDelimiters, delim)) {
+    sequence = stackLargeDelimiterSequence;
+  } else {
+    sequence = stackAlwaysDelimiterSequence;
+  } // Look through the sequence
+
+
+  var delimType = traverseSequence(delim, height, sequence, options); // Get the delimiter from font glyphs.
+  // Depending on the sequence element we decided on, call the
+  // appropriate function.
+
+  if (delimType.type === "small") {
+    return delimiter_makeSmallDelim(delim, delimType.style, center, options, mode, classes);
+  } else if (delimType.type === "large") {
+    return delimiter_makeLargeDelim(delim, delimType.size, center, options, mode, classes);
+  } else
+    /* if (delimType.type === "stack") */
+    {
+      return delimiter_makeStackedDelim(delim, height, center, options, mode, classes);
+    }
+};
+/**
+ * Make a delimiter for use with `\left` and `\right`, given a height and depth
+ * of an expression that the delimiters surround.
+ */
+
+
+var makeLeftRightDelim = function makeLeftRightDelim(delim, height, depth, options, mode, classes) {
+  // We always center \left/\right delimiters, so the axis is always shifted
+  var axisHeight = options.fontMetrics().axisHeight * options.sizeMultiplier; // Taken from TeX source, tex.web, function make_left_right
+
+  var delimiterFactor = 901;
+  var delimiterExtend = 5.0 / options.fontMetrics().ptPerEm;
+  var maxDistFromAxis = Math.max(height - axisHeight, depth + axisHeight);
+  var totalHeight = Math.max( // In real TeX, calculations are done using integral values which are
+  // 65536 per pt, or 655360 per em. So, the division here truncates in
+  // TeX but doesn't here, producing different results. If we wanted to
+  // exactly match TeX's calculation, we could do
+  //   Math.floor(655360 * maxDistFromAxis / 500) *
+  //    delimiterFactor / 655360
+  // (To see the difference, compare
+  //    x^{x^{\left(\rule{0.1em}{0.68em}\right)}}
+  // in TeX and KaTeX)
+  maxDistFromAxis / 500 * delimiterFactor, 2 * maxDistFromAxis - delimiterExtend); // Finally, we defer to `makeCustomSizedDelim` with our calculated total
+  // height
+
+  return delimiter_makeCustomSizedDelim(delim, totalHeight, true, options, mode, classes);
+};
+
+/* harmony default export */ var delimiter = ({
+  sqrtImage: makeSqrtImage,
+  sizedDelim: delimiter_makeSizedDelim,
+  customSizedDelim: delimiter_makeCustomSizedDelim,
+  leftRightDelim: makeLeftRightDelim
+});
+// CONCATENATED MODULE: ./src/functions/delimsizing.js
+
+
+
+
+
+
+
+
+
+// Extra data needed for the delimiter handler down below
+var delimiterSizes = {
+  "\\bigl": {
+    mclass: "mopen",
+    size: 1
+  },
+  "\\Bigl": {
+    mclass: "mopen",
+    size: 2
+  },
+  "\\biggl": {
+    mclass: "mopen",
+    size: 3
+  },
+  "\\Biggl": {
+    mclass: "mopen",
+    size: 4
+  },
+  "\\bigr": {
+    mclass: "mclose",
+    size: 1
+  },
+  "\\Bigr": {
+    mclass: "mclose",
+    size: 2
+  },
+  "\\biggr": {
+    mclass: "mclose",
+    size: 3
+  },
+  "\\Biggr": {
+    mclass: "mclose",
+    size: 4
+  },
+  "\\bigm": {
+    mclass: "mrel",
+    size: 1
+  },
+  "\\Bigm": {
+    mclass: "mrel",
+    size: 2
+  },
+  "\\biggm": {
+    mclass: "mrel",
+    size: 3
+  },
+  "\\Biggm": {
+    mclass: "mrel",
+    size: 4
+  },
+  "\\big": {
+    mclass: "mord",
+    size: 1
+  },
+  "\\Big": {
+    mclass: "mord",
+    size: 2
+  },
+  "\\bigg": {
+    mclass: "mord",
+    size: 3
+  },
+  "\\Bigg": {
+    mclass: "mord",
+    size: 4
+  }
+};
+var delimiters = ["(", "\\lparen", ")", "\\rparen", "[", "\\lbrack", "]", "\\rbrack", "\\{", "\\lbrace", "\\}", "\\rbrace", "\\lfloor", "\\rfloor", "\u230A", "\u230B", "\\lceil", "\\rceil", "\u2308", "\u2309", "<", ">", "\\langle", "\u27E8", "\\rangle", "\u27E9", "\\lt", "\\gt", "\\lvert", "\\rvert", "\\lVert", "\\rVert", "\\lgroup", "\\rgroup", "\u27EE", "\u27EF", "\\lmoustache", "\\rmoustache", "\u23B0", "\u23B1", "/", "\\backslash", "|", "\\vert", "\\|", "\\Vert", "\\uparrow", "\\Uparrow", "\\downarrow", "\\Downarrow", "\\updownarrow", "\\Updownarrow", "."];
+
+// Delimiter functions
+function checkDelimiter(delim, context) {
+  var symDelim = checkSymbolNodeType(delim);
+
+  if (symDelim && utils.contains(delimiters, symDelim.text)) {
+    return symDelim;
+  } else if (symDelim) {
+    throw new src_ParseError("Invalid delimiter '" + symDelim.text + "' after '" + context.funcName + "'", delim);
+  } else {
+    throw new src_ParseError("Invalid delimiter type '" + delim.type + "'", delim);
+  }
+}
+
+defineFunction({
+  type: "delimsizing",
+  names: ["\\bigl", "\\Bigl", "\\biggl", "\\Biggl", "\\bigr", "\\Bigr", "\\biggr", "\\Biggr", "\\bigm", "\\Bigm", "\\biggm", "\\Biggm", "\\big", "\\Big", "\\bigg", "\\Bigg"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    var delim = checkDelimiter(args[0], context);
+    return {
+      type: "delimsizing",
+      mode: context.parser.mode,
+      size: delimiterSizes[context.funcName].size,
+      mclass: delimiterSizes[context.funcName].mclass,
+      delim: delim.text
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    if (group.delim === ".") {
+      // Empty delimiters still count as elements, even though they don't
+      // show anything.
+      return buildCommon.makeSpan([group.mclass]);
+    } // Use delimiter.sizedDelim to generate the delimiter.
+
+
+    return delimiter.sizedDelim(group.delim, group.size, options, group.mode, [group.mclass]);
+  },
+  mathmlBuilder: function mathmlBuilder(group) {
+    var children = [];
+
+    if (group.delim !== ".") {
+      children.push(buildMathML_makeText(group.delim, group.mode));
+    }
+
+    var node = new mathMLTree.MathNode("mo", children);
+
+    if (group.mclass === "mopen" || group.mclass === "mclose") {
+      // Only some of the delimsizing functions act as fences, and they
+      // return "mopen" or "mclose" mclass.
+      node.setAttribute("fence", "true");
+    } else {
+      // Explicitly disable fencing if it's not a fence, to override the
+      // defaults.
+      node.setAttribute("fence", "false");
+    }
+
+    return node;
+  }
+});
+
+function assertParsed(group) {
+  if (!group.body) {
+    throw new Error("Bug: The leftright ParseNode wasn't fully parsed.");
+  }
+}
+
+defineFunction({
+  type: "leftright-right",
+  names: ["\\right"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    // \left case below triggers parsing of \right in
+    //   `const right = parser.parseFunction();`
+    // uses this return value.
+    var color = context.parser.gullet.macros.get("\\current@color");
+
+    if (color && typeof color !== "string") {
+      throw new src_ParseError("\\current@color set to non-string in \\right");
+    }
+
+    return {
+      type: "leftright-right",
+      mode: context.parser.mode,
+      delim: checkDelimiter(args[0], context).text,
+      color: color // undefined if not set via \color
+
+    };
+  }
+});
+defineFunction({
+  type: "leftright",
+  names: ["\\left"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    var delim = checkDelimiter(args[0], context);
+    var parser = context.parser; // Parse out the implicit body
+
+    ++parser.leftrightDepth; // parseExpression stops before '\\right'
+
+    var body = parser.parseExpression(false);
+    --parser.leftrightDepth; // Check the next token
+
+    parser.expect("\\right", false);
+    var right = assertNodeType(parser.parseFunction(), "leftright-right");
+    return {
+      type: "leftright",
+      mode: parser.mode,
+      body: body,
+      left: delim.text,
+      right: right.delim,
+      rightColor: right.color
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    assertParsed(group); // Build the inner expression
+
+    var inner = buildHTML_buildExpression(group.body, options, true, ["mopen", "mclose"]);
+    var innerHeight = 0;
+    var innerDepth = 0;
+    var hadMiddle = false; // Calculate its height and depth
+
+    for (var i = 0; i < inner.length; i++) {
+      // Property `isMiddle` not defined on `span`. See comment in
+      // "middle"'s htmlBuilder.
+      // $FlowFixMe
+      if (inner[i].isMiddle) {
+        hadMiddle = true;
+      } else {
+        innerHeight = Math.max(inner[i].height, innerHeight);
+        innerDepth = Math.max(inner[i].depth, innerDepth);
+      }
+    } // The size of delimiters is the same, regardless of what style we are
+    // in. Thus, to correctly calculate the size of delimiter we need around
+    // a group, we scale down the inner size based on the size.
+
+
+    innerHeight *= options.sizeMultiplier;
+    innerDepth *= options.sizeMultiplier;
+    var leftDelim;
+
+    if (group.left === ".") {
+      // Empty delimiters in \left and \right make null delimiter spaces.
+      leftDelim = makeNullDelimiter(options, ["mopen"]);
+    } else {
+      // Otherwise, use leftRightDelim to generate the correct sized
+      // delimiter.
+      leftDelim = delimiter.leftRightDelim(group.left, innerHeight, innerDepth, options, group.mode, ["mopen"]);
+    } // Add it to the beginning of the expression
+
+
+    inner.unshift(leftDelim); // Handle middle delimiters
+
+    if (hadMiddle) {
+      for (var _i = 1; _i < inner.length; _i++) {
+        var middleDelim = inner[_i]; // Property `isMiddle` not defined on `span`. See comment in
+        // "middle"'s htmlBuilder.
+        // $FlowFixMe
+
+        var isMiddle = middleDelim.isMiddle;
+
+        if (isMiddle) {
+          // Apply the options that were active when \middle was called
+          inner[_i] = delimiter.leftRightDelim(isMiddle.delim, innerHeight, innerDepth, isMiddle.options, group.mode, []);
+        }
+      }
+    }
+
+    var rightDelim; // Same for the right delimiter, but using color specified by \color
+
+    if (group.right === ".") {
+      rightDelim = makeNullDelimiter(options, ["mclose"]);
+    } else {
+      var colorOptions = group.rightColor ? options.withColor(group.rightColor) : options;
+      rightDelim = delimiter.leftRightDelim(group.right, innerHeight, innerDepth, colorOptions, group.mode, ["mclose"]);
+    } // Add it to the end of the expression.
+
+
+    inner.push(rightDelim);
+    return buildCommon.makeSpan(["minner"], inner, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    assertParsed(group);
+    var inner = buildMathML_buildExpression(group.body, options);
+
+    if (group.left !== ".") {
+      var leftNode = new mathMLTree.MathNode("mo", [buildMathML_makeText(group.left, group.mode)]);
+      leftNode.setAttribute("fence", "true");
+      inner.unshift(leftNode);
+    }
+
+    if (group.right !== ".") {
+      var rightNode = new mathMLTree.MathNode("mo", [buildMathML_makeText(group.right, group.mode)]);
+      rightNode.setAttribute("fence", "true");
+
+      if (group.rightColor) {
+        rightNode.setAttribute("mathcolor", group.rightColor);
+      }
+
+      inner.push(rightNode);
+    }
+
+    return buildMathML_makeRow(inner);
+  }
+});
+defineFunction({
+  type: "middle",
+  names: ["\\middle"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    var delim = checkDelimiter(args[0], context);
+
+    if (!context.parser.leftrightDepth) {
+      throw new src_ParseError("\\middle without preceding \\left", delim);
+    }
+
+    return {
+      type: "middle",
+      mode: context.parser.mode,
+      delim: delim.text
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var middleDelim;
+
+    if (group.delim === ".") {
+      middleDelim = makeNullDelimiter(options, []);
+    } else {
+      middleDelim = delimiter.sizedDelim(group.delim, 1, options, group.mode, []);
+      var isMiddle = {
+        delim: group.delim,
+        options: options
+      }; // Property `isMiddle` not defined on `span`. It is only used in
+      // this file above.
+      // TODO: Fix this violation of the `span` type and possibly rename
+      // things since `isMiddle` sounds like a boolean, but is a struct.
+      // $FlowFixMe
+
+      middleDelim.isMiddle = isMiddle;
+    }
+
+    return middleDelim;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    // A Firefox \middle will strech a character vertically only if it
+    // is in the fence part of the operator dictionary at:
+    // https://www.w3.org/TR/MathML3/appendixc.html.
+    // So we need to avoid U+2223 and use plain "|" instead.
+    var textNode = group.delim === "\\vert" || group.delim === "|" ? buildMathML_makeText("|", "text") : buildMathML_makeText(group.delim, group.mode);
+    var middleNode = new mathMLTree.MathNode("mo", [textNode]);
+    middleNode.setAttribute("fence", "true"); // MathML gives 5/18em spacing to each <mo> element.
+    // \middle should get delimiter spacing instead.
+
+    middleNode.setAttribute("lspace", "0.05em");
+    middleNode.setAttribute("rspace", "0.05em");
+    return middleNode;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/enclose.js
+
+
+
+
+
+
+
+
+
+var enclose_htmlBuilder = function htmlBuilder(group, options) {
+  // \cancel, \bcancel, \xcancel, \sout, \fbox, \colorbox, \fcolorbox
+  // Some groups can return document fragments.  Handle those by wrapping
+  // them in a span.
+  var inner = buildCommon.wrapFragment(buildHTML_buildGroup(group.body, options), options);
+  var label = group.label.substr(1);
+  var scale = options.sizeMultiplier;
+  var img;
+  var imgShift = 0; // In the LaTeX cancel package, line geometry is slightly different
+  // depending on whether the subject is wider than it is tall, or vice versa.
+  // We don't know the width of a group, so as a proxy, we test if
+  // the subject is a single character. This captures most of the
+  // subjects that should get the "tall" treatment.
+
+  var isSingleChar = utils.isCharacterBox(group.body);
+
+  if (label === "sout") {
+    img = buildCommon.makeSpan(["stretchy", "sout"]);
+    img.height = options.fontMetrics().defaultRuleThickness / scale;
+    imgShift = -0.5 * options.fontMetrics().xHeight;
+  } else {
+    // Add horizontal padding
+    if (/cancel/.test(label)) {
+      if (!isSingleChar) {
+        inner.classes.push("cancel-pad");
+      }
+    } else {
+      inner.classes.push("boxpad");
+    } // Add vertical padding
+
+
+    var vertPad = 0;
+    var ruleThickness = 0; // ref: cancel package: \advance\totalheight2\p@ % "+2"
+
+    if (/box/.test(label)) {
+      ruleThickness = Math.max(options.fontMetrics().fboxrule, // default
+      options.minRuleThickness // User override.
+      );
+      vertPad = options.fontMetrics().fboxsep + (label === "colorbox" ? 0 : ruleThickness);
+    } else {
+      vertPad = isSingleChar ? 0.2 : 0;
+    }
+
+    img = stretchy.encloseSpan(inner, label, vertPad, options);
+
+    if (/fbox|boxed|fcolorbox/.test(label)) {
+      img.style.borderStyle = "solid";
+      img.style.borderWidth = ruleThickness + "em";
+    }
+
+    imgShift = inner.depth + vertPad;
+
+    if (group.backgroundColor) {
+      img.style.backgroundColor = group.backgroundColor;
+
+      if (group.borderColor) {
+        img.style.borderColor = group.borderColor;
+      }
+    }
+  }
+
+  var vlist;
+
+  if (group.backgroundColor) {
+    vlist = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [// Put the color background behind inner;
+      {
+        type: "elem",
+        elem: img,
+        shift: imgShift
+      }, {
+        type: "elem",
+        elem: inner,
+        shift: 0
+      }]
+    }, options);
+  } else {
+    vlist = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [// Write the \cancel stroke on top of inner.
+      {
+        type: "elem",
+        elem: inner,
+        shift: 0
+      }, {
+        type: "elem",
+        elem: img,
+        shift: imgShift,
+        wrapperClasses: /cancel/.test(label) ? ["svg-align"] : []
+      }]
+    }, options);
+  }
+
+  if (/cancel/.test(label)) {
+    // The cancel package documentation says that cancel lines add their height
+    // to the expression, but tests show that isn't how it actually works.
+    vlist.height = inner.height;
+    vlist.depth = inner.depth;
+  }
+
+  if (/cancel/.test(label) && !isSingleChar) {
+    // cancel does not create horiz space for its line extension.
+    return buildCommon.makeSpan(["mord", "cancel-lap"], [vlist], options);
+  } else {
+    return buildCommon.makeSpan(["mord"], [vlist], options);
+  }
+};
+
+var enclose_mathmlBuilder = function mathmlBuilder(group, options) {
+  var fboxsep = 0;
+  var node = new mathMLTree.MathNode(group.label.indexOf("colorbox") > -1 ? "mpadded" : "menclose", [buildMathML_buildGroup(group.body, options)]);
+
+  switch (group.label) {
+    case "\\cancel":
+      node.setAttribute("notation", "updiagonalstrike");
+      break;
+
+    case "\\bcancel":
+      node.setAttribute("notation", "downdiagonalstrike");
+      break;
+
+    case "\\sout":
+      node.setAttribute("notation", "horizontalstrike");
+      break;
+
+    case "\\fbox":
+      node.setAttribute("notation", "box");
+      break;
+
+    case "\\fcolorbox":
+    case "\\colorbox":
+      // <menclose> doesn't have a good notation option. So use <mpadded>
+      // instead. Set some attributes that come included with <menclose>.
+      fboxsep = options.fontMetrics().fboxsep * options.fontMetrics().ptPerEm;
+      node.setAttribute("width", "+" + 2 * fboxsep + "pt");
+      node.setAttribute("height", "+" + 2 * fboxsep + "pt");
+      node.setAttribute("lspace", fboxsep + "pt"); //
+
+      node.setAttribute("voffset", fboxsep + "pt");
+
+      if (group.label === "\\fcolorbox") {
+        var thk = Math.max(options.fontMetrics().fboxrule, // default
+        options.minRuleThickness // user override
+        );
+        node.setAttribute("style", "border: " + thk + "em solid " + String(group.borderColor));
+      }
+
+      break;
+
+    case "\\xcancel":
+      node.setAttribute("notation", "updiagonalstrike downdiagonalstrike");
+      break;
+  }
+
+  if (group.backgroundColor) {
+    node.setAttribute("mathbackground", group.backgroundColor);
+  }
+
+  return node;
+};
+
+defineFunction({
+  type: "enclose",
+  names: ["\\colorbox"],
+  props: {
+    numArgs: 2,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "text"]
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var color = assertNodeType(args[0], "color-token").color;
+    var body = args[1];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      backgroundColor: color,
+      body: body
+    };
+  },
+  htmlBuilder: enclose_htmlBuilder,
+  mathmlBuilder: enclose_mathmlBuilder
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\fcolorbox"],
+  props: {
+    numArgs: 3,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "color", "text"]
+  },
+  handler: function handler(_ref2, args, optArgs) {
+    var parser = _ref2.parser,
+        funcName = _ref2.funcName;
+    var borderColor = assertNodeType(args[0], "color-token").color;
+    var backgroundColor = assertNodeType(args[1], "color-token").color;
+    var body = args[2];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      backgroundColor: backgroundColor,
+      borderColor: borderColor,
+      body: body
+    };
+  },
+  htmlBuilder: enclose_htmlBuilder,
+  mathmlBuilder: enclose_mathmlBuilder
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\fbox"],
+  props: {
+    numArgs: 1,
+    argTypes: ["hbox"],
+    allowedInText: true
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser;
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: "\\fbox",
+      body: args[0]
+    };
+  }
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\cancel", "\\bcancel", "\\xcancel", "\\sout"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref4, args, optArgs) {
+    var parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    var body = args[0];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      body: body
+    };
+  },
+  htmlBuilder: enclose_htmlBuilder,
+  mathmlBuilder: enclose_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/defineEnvironment.js
+
+
+/**
+ * All registered environments.
+ * `environments.js` exports this same dictionary again and makes it public.
+ * `Parser.js` requires this dictionary via `environments.js`.
+ */
+var _environments = {};
+function defineEnvironment(_ref) {
+  var type = _ref.type,
+      names = _ref.names,
+      props = _ref.props,
+      handler = _ref.handler,
+      htmlBuilder = _ref.htmlBuilder,
+      mathmlBuilder = _ref.mathmlBuilder;
+  // Set default values of environments.
+  var data = {
+    type: type,
+    numArgs: props.numArgs || 0,
+    greediness: 1,
+    allowedInText: false,
+    numOptionalArgs: 0,
+    handler: handler
+  };
+
+  for (var i = 0; i < names.length; ++i) {
+    // TODO: The value type of _environments should be a type union of all
+    // possible `EnvSpec<>` possibilities instead of `EnvSpec<*>`, which is
+    // an existential type.
+    // $FlowFixMe
+    _environments[names[i]] = data;
+  }
+
+  if (htmlBuilder) {
+    _htmlGroupBuilders[type] = htmlBuilder;
+  }
+
+  if (mathmlBuilder) {
+    _mathmlGroupBuilders[type] = mathmlBuilder;
+  }
+}
+// CONCATENATED MODULE: ./src/environments/array.js
+
+
+
+
+
+
+
+
+
+
+
+
+
+function getHLines(parser) {
+  // Return an array. The array length = number of hlines.
+  // Each element in the array tells if the line is dashed.
+  var hlineInfo = [];
+  parser.consumeSpaces();
+  var nxt = parser.fetch().text;
+
+  while (nxt === "\\hline" || nxt === "\\hdashline") {
+    parser.consume();
+    hlineInfo.push(nxt === "\\hdashline");
+    parser.consumeSpaces();
+    nxt = parser.fetch().text;
+  }
+
+  return hlineInfo;
+}
+/**
+ * Parse the body of the environment, with rows delimited by \\ and
+ * columns delimited by &, and create a nested list in row-major order
+ * with one group per cell.  If given an optional argument style
+ * ("text", "display", etc.), then each cell is cast into that style.
+ */
+
+
+function parseArray(parser, _ref, style) {
+  var hskipBeforeAndAfter = _ref.hskipBeforeAndAfter,
+      addJot = _ref.addJot,
+      cols = _ref.cols,
+      arraystretch = _ref.arraystretch,
+      colSeparationType = _ref.colSeparationType;
+  // Parse body of array with \\ temporarily mapped to \cr
+  parser.gullet.beginGroup();
+  parser.gullet.macros.set("\\\\", "\\cr"); // Get current arraystretch if it's not set by the environment
+
+  if (!arraystretch) {
+    var stretch = parser.gullet.expandMacroAsText("\\arraystretch");
+
+    if (stretch == null) {
+      // Default \arraystretch from lttab.dtx
+      arraystretch = 1;
+    } else {
+      arraystretch = parseFloat(stretch);
+
+      if (!arraystretch || arraystretch < 0) {
+        throw new src_ParseError("Invalid \\arraystretch: " + stretch);
+      }
+    }
+  } // Start group for first cell
+
+
+  parser.gullet.beginGroup();
+  var row = [];
+  var body = [row];
+  var rowGaps = [];
+  var hLinesBeforeRow = []; // Test for \hline at the top of the array.
+
+  hLinesBeforeRow.push(getHLines(parser));
+
+  while (true) {
+    // eslint-disable-line no-constant-condition
+    // Parse each cell in its own group (namespace)
+    var cell = parser.parseExpression(false, "\\cr");
+    parser.gullet.endGroup();
+    parser.gullet.beginGroup();
+    cell = {
+      type: "ordgroup",
+      mode: parser.mode,
+      body: cell
+    };
+
+    if (style) {
+      cell = {
+        type: "styling",
+        mode: parser.mode,
+        style: style,
+        body: [cell]
+      };
+    }
+
+    row.push(cell);
+    var next = parser.fetch().text;
+
+    if (next === "&") {
+      parser.consume();
+    } else if (next === "\\end") {
+      // Arrays terminate newlines with `\crcr` which consumes a `\cr` if
+      // the last line is empty.
+      // NOTE: Currently, `cell` is the last item added into `row`.
+      if (row.length === 1 && cell.type === "styling" && cell.body[0].body.length === 0) {
+        body.pop();
+      }
+
+      if (hLinesBeforeRow.length < body.length + 1) {
+        hLinesBeforeRow.push([]);
+      }
+
+      break;
+    } else if (next === "\\cr") {
+      var cr = assertNodeType(parser.parseFunction(), "cr");
+      rowGaps.push(cr.size); // check for \hline(s) following the row separator
+
+      hLinesBeforeRow.push(getHLines(parser));
+      row = [];
+      body.push(row);
+    } else {
+      throw new src_ParseError("Expected & or \\\\ or \\cr or \\end", parser.nextToken);
+    }
+  } // End cell group
+
+
+  parser.gullet.endGroup(); // End array group defining \\
+
+  parser.gullet.endGroup();
+  return {
+    type: "array",
+    mode: parser.mode,
+    addJot: addJot,
+    arraystretch: arraystretch,
+    body: body,
+    cols: cols,
+    rowGaps: rowGaps,
+    hskipBeforeAndAfter: hskipBeforeAndAfter,
+    hLinesBeforeRow: hLinesBeforeRow,
+    colSeparationType: colSeparationType
+  };
+} // Decides on a style for cells in an array according to whether the given
+// environment name starts with the letter 'd'.
+
+
+function dCellStyle(envName) {
+  if (envName.substr(0, 1) === "d") {
+    return "display";
+  } else {
+    return "text";
+  }
+}
+
+var array_htmlBuilder = function htmlBuilder(group, options) {
+  var r;
+  var c;
+  var nr = group.body.length;
+  var hLinesBeforeRow = group.hLinesBeforeRow;
+  var nc = 0;
+  var body = new Array(nr);
+  var hlines = [];
+  var ruleThickness = Math.max( // From LaTeX \showthe\arrayrulewidth. Equals 0.04 em.
+  options.fontMetrics().arrayRuleWidth, options.minRuleThickness // User override.
+  ); // Horizontal spacing
+
+  var pt = 1 / options.fontMetrics().ptPerEm;
+  var arraycolsep = 5 * pt; // default value, i.e. \arraycolsep in article.cls
+
+  if (group.colSeparationType && group.colSeparationType === "small") {
+    // We're in a {smallmatrix}. Default column space is \thickspace,
+    // i.e. 5/18em = 0.2778em, per amsmath.dtx for {smallmatrix}.
+    // But that needs adjustment because LaTeX applies \scriptstyle to the
+    // entire array, including the colspace, but this function applies
+    // \scriptstyle only inside each element.
+    var localMultiplier = options.havingStyle(src_Style.SCRIPT).sizeMultiplier;
+    arraycolsep = 0.2778 * (localMultiplier / options.sizeMultiplier);
+  } // Vertical spacing
+
+
+  var baselineskip = 12 * pt; // see size10.clo
+  // Default \jot from ltmath.dtx
+  // TODO(edemaine): allow overriding \jot via \setlength (#687)
+
+  var jot = 3 * pt;
+  var arrayskip = group.arraystretch * baselineskip;
+  var arstrutHeight = 0.7 * arrayskip; // \strutbox in ltfsstrc.dtx and
+
+  var arstrutDepth = 0.3 * arrayskip; // \@arstrutbox in lttab.dtx
+
+  var totalHeight = 0; // Set a position for \hline(s) at the top of the array, if any.
+
+  function setHLinePos(hlinesInGap) {
+    for (var i = 0; i < hlinesInGap.length; ++i) {
+      if (i > 0) {
+        totalHeight += 0.25;
+      }
+
+      hlines.push({
+        pos: totalHeight,
+        isDashed: hlinesInGap[i]
+      });
+    }
+  }
+
+  setHLinePos(hLinesBeforeRow[0]);
+
+  for (r = 0; r < group.body.length; ++r) {
+    var inrow = group.body[r];
+    var height = arstrutHeight; // \@array adds an \@arstrut
+
+    var depth = arstrutDepth; // to each tow (via the template)
+
+    if (nc < inrow.length) {
+      nc = inrow.length;
+    }
+
+    var outrow = new Array(inrow.length);
+
+    for (c = 0; c < inrow.length; ++c) {
+      var elt = buildHTML_buildGroup(inrow[c], options);
+
+      if (depth < elt.depth) {
+        depth = elt.depth;
+      }
+
+      if (height < elt.height) {
+        height = elt.height;
+      }
+
+      outrow[c] = elt;
+    }
+
+    var rowGap = group.rowGaps[r];
+    var gap = 0;
+
+    if (rowGap) {
+      gap = units_calculateSize(rowGap, options);
+
+      if (gap > 0) {
+        // \@argarraycr
+        gap += arstrutDepth;
+
+        if (depth < gap) {
+          depth = gap; // \@xargarraycr
+        }
+
+        gap = 0;
+      }
+    } // In AMS multiline environments such as aligned and gathered, rows
+    // correspond to lines that have additional \jot added to the
+    // \baselineskip via \openup.
+
+
+    if (group.addJot) {
+      depth += jot;
+    }
+
+    outrow.height = height;
+    outrow.depth = depth;
+    totalHeight += height;
+    outrow.pos = totalHeight;
+    totalHeight += depth + gap; // \@yargarraycr
+
+    body[r] = outrow; // Set a position for \hline(s), if any.
+
+    setHLinePos(hLinesBeforeRow[r + 1]);
+  }
+
+  var offset = totalHeight / 2 + options.fontMetrics().axisHeight;
+  var colDescriptions = group.cols || [];
+  var cols = [];
+  var colSep;
+  var colDescrNum;
+
+  for (c = 0, colDescrNum = 0; // Continue while either there are more columns or more column
+  // descriptions, so trailing separators don't get lost.
+  c < nc || colDescrNum < colDescriptions.length; ++c, ++colDescrNum) {
+    var colDescr = colDescriptions[colDescrNum] || {};
+    var firstSeparator = true;
+
+    while (colDescr.type === "separator") {
+      // If there is more than one separator in a row, add a space
+      // between them.
+      if (!firstSeparator) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = options.fontMetrics().doubleRuleSep + "em";
+        cols.push(colSep);
+      }
+
+      if (colDescr.separator === "|" || colDescr.separator === ":") {
+        var lineType = colDescr.separator === "|" ? "solid" : "dashed";
+        var separator = buildCommon.makeSpan(["vertical-separator"], [], options);
+        separator.style.height = totalHeight + "em";
+        separator.style.borderRightWidth = ruleThickness + "em";
+        separator.style.borderRightStyle = lineType;
+        separator.style.margin = "0 -" + ruleThickness / 2 + "em";
+        separator.style.verticalAlign = -(totalHeight - offset) + "em";
+        cols.push(separator);
+      } else {
+        throw new src_ParseError("Invalid separator type: " + colDescr.separator);
+      }
+
+      colDescrNum++;
+      colDescr = colDescriptions[colDescrNum] || {};
+      firstSeparator = false;
+    }
+
+    if (c >= nc) {
+      continue;
+    }
+
+    var sepwidth = void 0;
+
+    if (c > 0 || group.hskipBeforeAndAfter) {
+      sepwidth = utils.deflt(colDescr.pregap, arraycolsep);
+
+      if (sepwidth !== 0) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = sepwidth + "em";
+        cols.push(colSep);
+      }
+    }
+
+    var col = [];
+
+    for (r = 0; r < nr; ++r) {
+      var row = body[r];
+      var elem = row[c];
+
+      if (!elem) {
+        continue;
+      }
+
+      var shift = row.pos - offset;
+      elem.depth = row.depth;
+      elem.height = row.height;
+      col.push({
+        type: "elem",
+        elem: elem,
+        shift: shift
+      });
+    }
+
+    col = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: col
+    }, options);
+    col = buildCommon.makeSpan(["col-align-" + (colDescr.align || "c")], [col]);
+    cols.push(col);
+
+    if (c < nc - 1 || group.hskipBeforeAndAfter) {
+      sepwidth = utils.deflt(colDescr.postgap, arraycolsep);
+
+      if (sepwidth !== 0) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = sepwidth + "em";
+        cols.push(colSep);
+      }
+    }
+  }
+
+  body = buildCommon.makeSpan(["mtable"], cols); // Add \hline(s), if any.
+
+  if (hlines.length > 0) {
+    var line = buildCommon.makeLineSpan("hline", options, ruleThickness);
+    var dashes = buildCommon.makeLineSpan("hdashline", options, ruleThickness);
+    var vListElems = [{
+      type: "elem",
+      elem: body,
+      shift: 0
+    }];
+
+    while (hlines.length > 0) {
+      var hline = hlines.pop();
+      var lineShift = hline.pos - offset;
+
+      if (hline.isDashed) {
+        vListElems.push({
+          type: "elem",
+          elem: dashes,
+          shift: lineShift
+        });
+      } else {
+        vListElems.push({
+          type: "elem",
+          elem: line,
+          shift: lineShift
+        });
+      }
+    }
+
+    body = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: vListElems
+    }, options);
+  }
+
+  return buildCommon.makeSpan(["mord"], [body], options);
+};
+
+var alignMap = {
+  c: "center ",
+  l: "left ",
+  r: "right "
+};
+
+var array_mathmlBuilder = function mathmlBuilder(group, options) {
+  var table = new mathMLTree.MathNode("mtable", group.body.map(function (row) {
+    return new mathMLTree.MathNode("mtr", row.map(function (cell) {
+      return new mathMLTree.MathNode("mtd", [buildMathML_buildGroup(cell, options)]);
+    }));
+  })); // Set column alignment, row spacing, column spacing, and
+  // array lines by setting attributes on the table element.
+  // Set the row spacing. In MathML, we specify a gap distance.
+  // We do not use rowGap[] because MathML automatically increases
+  // cell height with the height/depth of the element content.
+  // LaTeX \arraystretch multiplies the row baseline-to-baseline distance.
+  // We simulate this by adding (arraystretch - 1)em to the gap. This
+  // does a reasonable job of adjusting arrays containing 1 em tall content.
+  // The 0.16 and 0.09 values are found emprically. They produce an array
+  // similar to LaTeX and in which content does not interfere with \hines.
+
+  var gap = group.arraystretch === 0.5 ? 0.1 // {smallmatrix}, {subarray}
+  : 0.16 + group.arraystretch - 1 + (group.addJot ? 0.09 : 0);
+  table.setAttribute("rowspacing", gap + "em"); // MathML table lines go only between cells.
+  // To place a line on an edge we'll use <menclose>, if necessary.
+
+  var menclose = "";
+  var align = "";
+
+  if (group.cols && group.cols.length > 0) {
+    // Find column alignment, column spacing, and  vertical lines.
+    var cols = group.cols;
+    var columnLines = "";
+    var prevTypeWasAlign = false;
+    var iStart = 0;
+    var iEnd = cols.length;
+
+    if (cols[0].type === "separator") {
+      menclose += "top ";
+      iStart = 1;
+    }
+
+    if (cols[cols.length - 1].type === "separator") {
+      menclose += "bottom ";
+      iEnd -= 1;
+    }
+
+    for (var i = iStart; i < iEnd; i++) {
+      if (cols[i].type === "align") {
+        align += alignMap[cols[i].align];
+
+        if (prevTypeWasAlign) {
+          columnLines += "none ";
+        }
+
+        prevTypeWasAlign = true;
+      } else if (cols[i].type === "separator") {
+        // MathML accepts only single lines between cells.
+        // So we read only the first of consecutive separators.
+        if (prevTypeWasAlign) {
+          columnLines += cols[i].separator === "|" ? "solid " : "dashed ";
+          prevTypeWasAlign = false;
+        }
+      }
+    }
+
+    table.setAttribute("columnalign", align.trim());
+
+    if (/[sd]/.test(columnLines)) {
+      table.setAttribute("columnlines", columnLines.trim());
+    }
+  } // Set column spacing.
+
+
+  if (group.colSeparationType === "align") {
+    var _cols = group.cols || [];
+
+    var spacing = "";
+
+    for (var _i = 1; _i < _cols.length; _i++) {
+      spacing += _i % 2 ? "0em " : "1em ";
+    }
+
+    table.setAttribute("columnspacing", spacing.trim());
+  } else if (group.colSeparationType === "alignat") {
+    table.setAttribute("columnspacing", "0em");
+  } else if (group.colSeparationType === "small") {
+    table.setAttribute("columnspacing", "0.2778em");
+  } else {
+    table.setAttribute("columnspacing", "1em");
+  } // Address \hline and \hdashline
+
+
+  var rowLines = "";
+  var hlines = group.hLinesBeforeRow;
+  menclose += hlines[0].length > 0 ? "left " : "";
+  menclose += hlines[hlines.length - 1].length > 0 ? "right " : "";
+
+  for (var _i2 = 1; _i2 < hlines.length - 1; _i2++) {
+    rowLines += hlines[_i2].length === 0 ? "none " // MathML accepts only a single line between rows. Read one element.
+    : hlines[_i2][0] ? "dashed " : "solid ";
+  }
+
+  if (/[sd]/.test(rowLines)) {
+    table.setAttribute("rowlines", rowLines.trim());
+  }
+
+  if (menclose !== "") {
+    table = new mathMLTree.MathNode("menclose", [table]);
+    table.setAttribute("notation", menclose.trim());
+  }
+
+  if (group.arraystretch && group.arraystretch < 1) {
+    // A small array. Wrap in scriptstyle so row gap is not too large.
+    table = new mathMLTree.MathNode("mstyle", [table]);
+    table.setAttribute("scriptlevel", "1");
+  }
+
+  return table;
+}; // Convenience function for aligned and alignedat environments.
+
+
+var array_alignedHandler = function alignedHandler(context, args) {
+  var cols = [];
+  var res = parseArray(context.parser, {
+    cols: cols,
+    addJot: true
+  }, "display"); // Determining number of columns.
+  // 1. If the first argument is given, we use it as a number of columns,
+  //    and makes sure that each row doesn't exceed that number.
+  // 2. Otherwise, just count number of columns = maximum number
+  //    of cells in each row ("aligned" mode -- isAligned will be true).
+  //
+  // At the same time, prepend empty group {} at beginning of every second
+  // cell in each row (starting with second cell) so that operators become
+  // binary.  This behavior is implemented in amsmath's \start@aligned.
+
+  var numMaths;
+  var numCols = 0;
+  var emptyGroup = {
+    type: "ordgroup",
+    mode: context.mode,
+    body: []
+  };
+
+  if (args[0] && args[0].type === "ordgroup") {
+    var arg0 = "";
+
+    for (var i = 0; i < args[0].body.length; i++) {
+      var textord = assertNodeType(args[0].body[i], "textord");
+      arg0 += textord.text;
+    }
+
+    numMaths = Number(arg0);
+    numCols = numMaths * 2;
+  }
+
+  var isAligned = !numCols;
+  res.body.forEach(function (row) {
+    for (var _i3 = 1; _i3 < row.length; _i3 += 2) {
+      // Modify ordgroup node within styling node
+      var styling = assertNodeType(row[_i3], "styling");
+      var ordgroup = assertNodeType(styling.body[0], "ordgroup");
+      ordgroup.body.unshift(emptyGroup);
+    }
+
+    if (!isAligned) {
+      // Case 1
+      var curMaths = row.length / 2;
+
+      if (numMaths < curMaths) {
+        throw new src_ParseError("Too many math in a row: " + ("expected " + numMaths + ", but got " + curMaths), row[0]);
+      }
+    } else if (numCols < row.length) {
+      // Case 2
+      numCols = row.length;
+    }
+  }); // Adjusting alignment.
+  // In aligned mode, we add one \qquad between columns;
+  // otherwise we add nothing.
+
+  for (var _i4 = 0; _i4 < numCols; ++_i4) {
+    var align = "r";
+    var pregap = 0;
+
+    if (_i4 % 2 === 1) {
+      align = "l";
+    } else if (_i4 > 0 && isAligned) {
+      // "aligned" mode.
+      pregap = 1; // add one \quad
+    }
+
+    cols[_i4] = {
+      type: "align",
+      align: align,
+      pregap: pregap,
+      postgap: 0
+    };
+  }
+
+  res.colSeparationType = isAligned ? "align" : "alignat";
+  return res;
+}; // Arrays are part of LaTeX, defined in lttab.dtx so its documentation
+// is part of the source2e.pdf file of LaTeX2e source documentation.
+// {darray} is an {array} environment where cells are set in \displaystyle,
+// as defined in nccmath.sty.
+
+
+defineEnvironment({
+  type: "array",
+  names: ["array", "darray"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    // Since no types are specified above, the two possibilities are
+    // - The argument is wrapped in {} or [], in which case Parser's
+    //   parseGroup() returns an "ordgroup" wrapping some symbol node.
+    // - The argument is a bare symbol node.
+    var symNode = checkSymbolNodeType(args[0]);
+    var colalign = symNode ? [args[0]] : assertNodeType(args[0], "ordgroup").body;
+    var cols = colalign.map(function (nde) {
+      var node = assertSymbolNodeType(nde);
+      var ca = node.text;
+
+      if ("lcr".indexOf(ca) !== -1) {
+        return {
+          type: "align",
+          align: ca
+        };
+      } else if (ca === "|") {
+        return {
+          type: "separator",
+          separator: "|"
+        };
+      } else if (ca === ":") {
+        return {
+          type: "separator",
+          separator: ":"
+        };
+      }
+
+      throw new src_ParseError("Unknown column alignment: " + ca, nde);
+    });
+    var res = {
+      cols: cols,
+      hskipBeforeAndAfter: true // \@preamble in lttab.dtx
+
+    };
+    return parseArray(context.parser, res, dCellStyle(context.envName));
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // The matrix environments of amsmath builds on the array environment
+// of LaTeX, which is discussed above.
+
+defineEnvironment({
+  type: "array",
+  names: ["matrix", "pmatrix", "bmatrix", "Bmatrix", "vmatrix", "Vmatrix"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(context) {
+    var delimiters = {
+      "matrix": null,
+      "pmatrix": ["(", ")"],
+      "bmatrix": ["[", "]"],
+      "Bmatrix": ["\\{", "\\}"],
+      "vmatrix": ["|", "|"],
+      "Vmatrix": ["\\Vert", "\\Vert"]
+    }[context.envName]; // \hskip -\arraycolsep in amsmath
+
+    var payload = {
+      hskipBeforeAndAfter: false
+    };
+    var res = parseArray(context.parser, payload, dCellStyle(context.envName));
+    return delimiters ? {
+      type: "leftright",
+      mode: context.mode,
+      body: [res],
+      left: delimiters[0],
+      right: delimiters[1],
+      rightColor: undefined // \right uninfluenced by \color in array
+
+    } : res;
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+});
+defineEnvironment({
+  type: "array",
+  names: ["smallmatrix"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(context) {
+    var payload = {
+      arraystretch: 0.5
+    };
+    var res = parseArray(context.parser, payload, "script");
+    res.colSeparationType = "small";
+    return res;
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+});
+defineEnvironment({
+  type: "array",
+  names: ["subarray"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    // Parsing of {subarray} is similar to {array}
+    var symNode = checkSymbolNodeType(args[0]);
+    var colalign = symNode ? [args[0]] : assertNodeType(args[0], "ordgroup").body;
+    var cols = colalign.map(function (nde) {
+      var node = assertSymbolNodeType(nde);
+      var ca = node.text; // {subarray} only recognizes "l" & "c"
+
+      if ("lc".indexOf(ca) !== -1) {
+        return {
+          type: "align",
+          align: ca
+        };
+      }
+
+      throw new src_ParseError("Unknown column alignment: " + ca, nde);
+    });
+
+    if (cols.length > 1) {
+      throw new src_ParseError("{subarray} can contain only one column");
+    }
+
+    var res = {
+      cols: cols,
+      hskipBeforeAndAfter: false,
+      arraystretch: 0.5
+    };
+    res = parseArray(context.parser, res, "script");
+
+    if (res.body.length > 0 && res.body[0].length > 1) {
+      throw new src_ParseError("{subarray} can contain only one column");
+    }
+
+    return res;
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // A cases environment (in amsmath.sty) is almost equivalent to
+// \def\arraystretch{1.2}%
+// \left\{\begin{array}{@{}l@{\quad}l@{}} … \end{array}\right.
+// {dcases} is a {cases} environment where cells are set in \displaystyle,
+// as defined in mathtools.sty.
+// {rcases} is another mathtools environment. It's brace is on the right side.
+
+defineEnvironment({
+  type: "array",
+  names: ["cases", "dcases", "rcases", "drcases"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(context) {
+    var payload = {
+      arraystretch: 1.2,
+      cols: [{
+        type: "align",
+        align: "l",
+        pregap: 0,
+        // TODO(kevinb) get the current style.
+        // For now we use the metrics for TEXT style which is what we were
+        // doing before.  Before attempting to get the current style we
+        // should look at TeX's behavior especially for \over and matrices.
+        postgap: 1.0
+        /* 1em quad */
+
+      }, {
+        type: "align",
+        align: "l",
+        pregap: 0,
+        postgap: 0
+      }]
+    };
+    var res = parseArray(context.parser, payload, dCellStyle(context.envName));
+    return {
+      type: "leftright",
+      mode: context.mode,
+      body: [res],
+      left: context.envName.indexOf("r") > -1 ? "." : "\\{",
+      right: context.envName.indexOf("r") > -1 ? "\\}" : ".",
+      rightColor: undefined
+    };
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // An aligned environment is like the align* environment
+// except it operates within math mode.
+// Note that we assume \nomallineskiplimit to be zero,
+// so that \strut@ is the same as \strut.
+
+defineEnvironment({
+  type: "array",
+  names: ["aligned"],
+  props: {
+    numArgs: 0
+  },
+  handler: array_alignedHandler,
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // A gathered environment is like an array environment with one centered
+// column, but where rows are considered lines so get \jot line spacing
+// and contents are set in \displaystyle.
+
+defineEnvironment({
+  type: "array",
+  names: ["gathered"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(context) {
+    var res = {
+      cols: [{
+        type: "align",
+        align: "c"
+      }],
+      addJot: true
+    };
+    return parseArray(context.parser, res, "display");
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // alignat environment is like an align environment, but one must explicitly
+// specify maximum number of columns in each row, and can adjust spacing between
+// each columns.
+
+defineEnvironment({
+  type: "array",
+  names: ["alignedat"],
+  // One for numbered and for unnumbered;
+  // but, KaTeX doesn't supports math numbering yet,
+  // they make no difference for now.
+  props: {
+    numArgs: 1
+  },
+  handler: array_alignedHandler,
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // Catch \hline outside array environment
+
+defineFunction({
+  type: "text",
+  // Doesn't matter what this is.
+  names: ["\\hline", "\\hdashline"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: true
+  },
+  handler: function handler(context, args) {
+    throw new src_ParseError(context.funcName + " valid only within array environment");
+  }
+});
+// CONCATENATED MODULE: ./src/environments.js
+
+var environments = _environments;
+/* harmony default export */ var src_environments = (environments); // All environment definitions should be imported below
+
+
+// CONCATENATED MODULE: ./src/functions/environment.js
+
+
+
+ // Environment delimiters. HTML/MathML rendering is defined in the corresponding
+// defineEnvironment definitions.
+// $FlowFixMe, "environment" handler returns an environment ParseNode
+
+defineFunction({
+  type: "environment",
+  names: ["\\begin", "\\end"],
+  props: {
+    numArgs: 1,
+    argTypes: ["text"]
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var nameGroup = args[0];
+
+    if (nameGroup.type !== "ordgroup") {
+      throw new src_ParseError("Invalid environment name", nameGroup);
+    }
+
+    var envName = "";
+
+    for (var i = 0; i < nameGroup.body.length; ++i) {
+      envName += assertNodeType(nameGroup.body[i], "textord").text;
+    }
+
+    if (funcName === "\\begin") {
+      // begin...end is similar to left...right
+      if (!src_environments.hasOwnProperty(envName)) {
+        throw new src_ParseError("No such environment: " + envName, nameGroup);
+      } // Build the environment object. Arguments and other information will
+      // be made available to the begin and end methods using properties.
+
+
+      var env = src_environments[envName];
+
+      var _parser$parseArgument = parser.parseArguments("\\begin{" + envName + "}", env),
+          _args = _parser$parseArgument.args,
+          optArgs = _parser$parseArgument.optArgs;
+
+      var context = {
+        mode: parser.mode,
+        envName: envName,
+        parser: parser
+      };
+      var result = env.handler(context, _args, optArgs);
+      parser.expect("\\end", false);
+      var endNameToken = parser.nextToken;
+      var end = assertNodeType(parser.parseFunction(), "environment");
+
+      if (end.name !== envName) {
+        throw new src_ParseError("Mismatch: \\begin{" + envName + "} matched by \\end{" + end.name + "}", endNameToken);
+      }
+
+      return result;
+    }
+
+    return {
+      type: "environment",
+      mode: parser.mode,
+      name: envName,
+      nameGroup: nameGroup
+    };
+  }
+});
+// CONCATENATED MODULE: ./src/functions/mclass.js
+
+
+
+
+
+
+var mclass_makeSpan = buildCommon.makeSpan;
+
+function mclass_htmlBuilder(group, options) {
+  var elements = buildHTML_buildExpression(group.body, options, true);
+  return mclass_makeSpan([group.mclass], elements, options);
+}
+
+function mclass_mathmlBuilder(group, options) {
+  var node;
+  var inner = buildMathML_buildExpression(group.body, options);
+
+  if (group.mclass === "minner") {
+    return mathMLTree.newDocumentFragment(inner);
+  } else if (group.mclass === "mord") {
+    if (group.isCharacterBox) {
+      node = inner[0];
+      node.type = "mi";
+    } else {
+      node = new mathMLTree.MathNode("mi", inner);
+    }
+  } else {
+    if (group.isCharacterBox) {
+      node = inner[0];
+      node.type = "mo";
+    } else {
+      node = new mathMLTree.MathNode("mo", inner);
+    } // Set spacing based on what is the most likely adjacent atom type.
+    // See TeXbook p170.
+
+
+    if (group.mclass === "mbin") {
+      node.attributes.lspace = "0.22em"; // medium space
+
+      node.attributes.rspace = "0.22em";
+    } else if (group.mclass === "mpunct") {
+      node.attributes.lspace = "0em";
+      node.attributes.rspace = "0.17em"; // thinspace
+    } else if (group.mclass === "mopen" || group.mclass === "mclose") {
+      node.attributes.lspace = "0em";
+      node.attributes.rspace = "0em";
+    } // MathML <mo> default space is 5/18 em, so <mrel> needs no action.
+    // Ref: https://developer.mozilla.org/en-US/docs/Web/MathML/Element/mo
+
+  }
+
+  return node;
+} // Math class commands except \mathop
+
+
+defineFunction({
+  type: "mclass",
+  names: ["\\mathord", "\\mathbin", "\\mathrel", "\\mathopen", "\\mathclose", "\\mathpunct", "\\mathinner"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: "m" + funcName.substr(5),
+      // TODO(kevinb): don't prefix with 'm'
+      body: ordargument(body),
+      isCharacterBox: utils.isCharacterBox(body)
+    };
+  },
+  htmlBuilder: mclass_htmlBuilder,
+  mathmlBuilder: mclass_mathmlBuilder
+});
+var binrelClass = function binrelClass(arg) {
+  // \binrel@ spacing varies with (bin|rel|ord) of the atom in the argument.
+  // (by rendering separately and with {}s before and after, and measuring
+  // the change in spacing).  We'll do roughly the same by detecting the
+  // atom type directly.
+  var atom = arg.type === "ordgroup" && arg.body.length ? arg.body[0] : arg;
+
+  if (atom.type === "atom" && (atom.family === "bin" || atom.family === "rel")) {
+    return "m" + atom.family;
+  } else {
+    return "mord";
+  }
+}; // \@binrel{x}{y} renders like y but as mbin/mrel/mord if x is mbin/mrel/mord.
+// This is equivalent to \binrel@{x}\binrel@@{y} in AMSTeX.
+
+defineFunction({
+  type: "mclass",
+  names: ["\\@binrel"],
+  props: {
+    numArgs: 2
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: binrelClass(args[0]),
+      body: [args[1]],
+      isCharacterBox: utils.isCharacterBox(args[1])
+    };
+  }
+}); // Build a relation or stacked op by placing one symbol on top of another
+
+defineFunction({
+  type: "mclass",
+  names: ["\\stackrel", "\\overset", "\\underset"],
+  props: {
+    numArgs: 2
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    var baseArg = args[1];
+    var shiftedArg = args[0];
+    var mclass;
+
+    if (funcName !== "\\stackrel") {
+      // LaTeX applies \binrel spacing to \overset and \underset.
+      mclass = binrelClass(baseArg);
+    } else {
+      mclass = "mrel"; // for \stackrel
+    }
+
+    var baseOp = {
+      type: "op",
+      mode: baseArg.mode,
+      limits: true,
+      alwaysHandleSupSub: true,
+      parentIsSupSub: false,
+      symbol: false,
+      suppressBaseShift: funcName !== "\\stackrel",
+      body: ordargument(baseArg)
+    };
+    var supsub = {
+      type: "supsub",
+      mode: shiftedArg.mode,
+      base: baseOp,
+      sup: funcName === "\\underset" ? null : shiftedArg,
+      sub: funcName === "\\underset" ? shiftedArg : null
+    };
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: mclass,
+      body: [supsub],
+      isCharacterBox: utils.isCharacterBox(supsub)
+    };
+  },
+  htmlBuilder: mclass_htmlBuilder,
+  mathmlBuilder: mclass_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/font.js
+// TODO(kevinb): implement \\sl and \\sc
+
+
+
+
+
+
+var font_htmlBuilder = function htmlBuilder(group, options) {
+  var font = group.font;
+  var newOptions = options.withFont(font);
+  return buildHTML_buildGroup(group.body, newOptions);
+};
+
+var font_mathmlBuilder = function mathmlBuilder(group, options) {
+  var font = group.font;
+  var newOptions = options.withFont(font);
+  return buildMathML_buildGroup(group.body, newOptions);
+};
+
+var fontAliases = {
+  "\\Bbb": "\\mathbb",
+  "\\bold": "\\mathbf",
+  "\\frak": "\\mathfrak",
+  "\\bm": "\\boldsymbol"
+};
+defineFunction({
+  type: "font",
+  names: [// styles, except \boldsymbol defined below
+  "\\mathrm", "\\mathit", "\\mathbf", "\\mathnormal", // families
+  "\\mathbb", "\\mathcal", "\\mathfrak", "\\mathscr", "\\mathsf", "\\mathtt", // aliases, except \bm defined below
+  "\\Bbb", "\\bold", "\\frak"],
+  props: {
+    numArgs: 1,
+    greediness: 2
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    var func = funcName;
+
+    if (func in fontAliases) {
+      func = fontAliases[func];
+    }
+
+    return {
+      type: "font",
+      mode: parser.mode,
+      font: func.slice(1),
+      body: body
+    };
+  },
+  htmlBuilder: font_htmlBuilder,
+  mathmlBuilder: font_mathmlBuilder
+});
+defineFunction({
+  type: "mclass",
+  names: ["\\boldsymbol", "\\bm"],
+  props: {
+    numArgs: 1,
+    greediness: 2
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    var body = args[0];
+    var isCharacterBox = utils.isCharacterBox(body); // amsbsy.sty's \boldsymbol uses \binrel spacing to inherit the
+    // argument's bin|rel|ord status
+
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: binrelClass(body),
+      body: [{
+        type: "font",
+        mode: parser.mode,
+        font: "boldsymbol",
+        body: body
+      }],
+      isCharacterBox: isCharacterBox
+    };
+  }
+}); // Old font changing functions
+
+defineFunction({
+  type: "font",
+  names: ["\\rm", "\\sf", "\\tt", "\\bf", "\\it", "\\cal"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser,
+        funcName = _ref3.funcName,
+        breakOnTokenText = _ref3.breakOnTokenText;
+    var mode = parser.mode;
+    var body = parser.parseExpression(true, breakOnTokenText);
+    var style = "math" + funcName.slice(1);
+    return {
+      type: "font",
+      mode: mode,
+      font: style,
+      body: {
+        type: "ordgroup",
+        mode: parser.mode,
+        body: body
+      }
+    };
+  },
+  htmlBuilder: font_htmlBuilder,
+  mathmlBuilder: font_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/genfrac.js
+
+
+
+
+
+
+
+
+
+
+
+var genfrac_adjustStyle = function adjustStyle(size, originalStyle) {
+  // Figure out what style this fraction should be in based on the
+  // function used
+  var style = originalStyle;
+
+  if (size === "display") {
+    // Get display style as a default.
+    // If incoming style is sub/sup, use style.text() to get correct size.
+    style = style.id >= src_Style.SCRIPT.id ? style.text() : src_Style.DISPLAY;
+  } else if (size === "text" && style.size === src_Style.DISPLAY.size) {
+    // We're in a \tfrac but incoming style is displaystyle, so:
+    style = src_Style.TEXT;
+  } else if (size === "script") {
+    style = src_Style.SCRIPT;
+  } else if (size === "scriptscript") {
+    style = src_Style.SCRIPTSCRIPT;
+  }
+
+  return style;
+};
+
+var genfrac_htmlBuilder = function htmlBuilder(group, options) {
+  // Fractions are handled in the TeXbook on pages 444-445, rules 15(a-e).
+  var style = genfrac_adjustStyle(group.size, options.style);
+  var nstyle = style.fracNum();
+  var dstyle = style.fracDen();
+  var newOptions;
+  newOptions = options.havingStyle(nstyle);
+  var numerm = buildHTML_buildGroup(group.numer, newOptions, options);
+
+  if (group.continued) {
+    // \cfrac inserts a \strut into the numerator.
+    // Get \strut dimensions from TeXbook page 353.
+    var hStrut = 8.5 / options.fontMetrics().ptPerEm;
+    var dStrut = 3.5 / options.fontMetrics().ptPerEm;
+    numerm.height = numerm.height < hStrut ? hStrut : numerm.height;
+    numerm.depth = numerm.depth < dStrut ? dStrut : numerm.depth;
+  }
+
+  newOptions = options.havingStyle(dstyle);
+  var denomm = buildHTML_buildGroup(group.denom, newOptions, options);
+  var rule;
+  var ruleWidth;
+  var ruleSpacing;
+
+  if (group.hasBarLine) {
+    if (group.barSize) {
+      ruleWidth = units_calculateSize(group.barSize, options);
+      rule = buildCommon.makeLineSpan("frac-line", options, ruleWidth);
+    } else {
+      rule = buildCommon.makeLineSpan("frac-line", options);
+    }
+
+    ruleWidth = rule.height;
+    ruleSpacing = rule.height;
+  } else {
+    rule = null;
+    ruleWidth = 0;
+    ruleSpacing = options.fontMetrics().defaultRuleThickness;
+  } // Rule 15b
+
+
+  var numShift;
+  var clearance;
+  var denomShift;
+
+  if (style.size === src_Style.DISPLAY.size || group.size === "display") {
+    numShift = options.fontMetrics().num1;
+
+    if (ruleWidth > 0) {
+      clearance = 3 * ruleSpacing;
+    } else {
+      clearance = 7 * ruleSpacing;
+    }
+
+    denomShift = options.fontMetrics().denom1;
+  } else {
+    if (ruleWidth > 0) {
+      numShift = options.fontMetrics().num2;
+      clearance = ruleSpacing;
+    } else {
+      numShift = options.fontMetrics().num3;
+      clearance = 3 * ruleSpacing;
+    }
+
+    denomShift = options.fontMetrics().denom2;
+  }
+
+  var frac;
+
+  if (!rule) {
+    // Rule 15c
+    var candidateClearance = numShift - numerm.depth - (denomm.height - denomShift);
+
+    if (candidateClearance < clearance) {
+      numShift += 0.5 * (clearance - candidateClearance);
+      denomShift += 0.5 * (clearance - candidateClearance);
+    }
+
+    frac = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [{
+        type: "elem",
+        elem: denomm,
+        shift: denomShift
+      }, {
+        type: "elem",
+        elem: numerm,
+        shift: -numShift
+      }]
+    }, options);
+  } else {
+    // Rule 15d
+    var axisHeight = options.fontMetrics().axisHeight;
+
+    if (numShift - numerm.depth - (axisHeight + 0.5 * ruleWidth) < clearance) {
+      numShift += clearance - (numShift - numerm.depth - (axisHeight + 0.5 * ruleWidth));
+    }
+
+    if (axisHeight - 0.5 * ruleWidth - (denomm.height - denomShift) < clearance) {
+      denomShift += clearance - (axisHeight - 0.5 * ruleWidth - (denomm.height - denomShift));
+    }
+
+    var midShift = -(axisHeight - 0.5 * ruleWidth);
+    frac = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [{
+        type: "elem",
+        elem: denomm,
+        shift: denomShift
+      }, {
+        type: "elem",
+        elem: rule,
+        shift: midShift
+      }, {
+        type: "elem",
+        elem: numerm,
+        shift: -numShift
+      }]
+    }, options);
+  } // Since we manually change the style sometimes (with \dfrac or \tfrac),
+  // account for the possible size change here.
+
+
+  newOptions = options.havingStyle(style);
+  frac.height *= newOptions.sizeMultiplier / options.sizeMultiplier;
+  frac.depth *= newOptions.sizeMultiplier / options.sizeMultiplier; // Rule 15e
+
+  var delimSize;
+
+  if (style.size === src_Style.DISPLAY.size) {
+    delimSize = options.fontMetrics().delim1;
+  } else {
+    delimSize = options.fontMetrics().delim2;
+  }
+
+  var leftDelim;
+  var rightDelim;
+
+  if (group.leftDelim == null) {
+    leftDelim = makeNullDelimiter(options, ["mopen"]);
+  } else {
+    leftDelim = delimiter.customSizedDelim(group.leftDelim, delimSize, true, options.havingStyle(style), group.mode, ["mopen"]);
+  }
+
+  if (group.continued) {
+    rightDelim = buildCommon.makeSpan([]); // zero width for \cfrac
+  } else if (group.rightDelim == null) {
+    rightDelim = makeNullDelimiter(options, ["mclose"]);
+  } else {
+    rightDelim = delimiter.customSizedDelim(group.rightDelim, delimSize, true, options.havingStyle(style), group.mode, ["mclose"]);
+  }
+
+  return buildCommon.makeSpan(["mord"].concat(newOptions.sizingClasses(options)), [leftDelim, buildCommon.makeSpan(["mfrac"], [frac]), rightDelim], options);
+};
+
+var genfrac_mathmlBuilder = function mathmlBuilder(group, options) {
+  var node = new mathMLTree.MathNode("mfrac", [buildMathML_buildGroup(group.numer, options), buildMathML_buildGroup(group.denom, options)]);
+
+  if (!group.hasBarLine) {
+    node.setAttribute("linethickness", "0px");
+  } else if (group.barSize) {
+    var ruleWidth = units_calculateSize(group.barSize, options);
+    node.setAttribute("linethickness", ruleWidth + "em");
+  }
+
+  var style = genfrac_adjustStyle(group.size, options.style);
+
+  if (style.size !== options.style.size) {
+    node = new mathMLTree.MathNode("mstyle", [node]);
+    var isDisplay = style.size === src_Style.DISPLAY.size ? "true" : "false";
+    node.setAttribute("displaystyle", isDisplay);
+    node.setAttribute("scriptlevel", "0");
+  }
+
+  if (group.leftDelim != null || group.rightDelim != null) {
+    var withDelims = [];
+
+    if (group.leftDelim != null) {
+      var leftOp = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(group.leftDelim.replace("\\", ""))]);
+      leftOp.setAttribute("fence", "true");
+      withDelims.push(leftOp);
+    }
+
+    withDelims.push(node);
+
+    if (group.rightDelim != null) {
+      var rightOp = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(group.rightDelim.replace("\\", ""))]);
+      rightOp.setAttribute("fence", "true");
+      withDelims.push(rightOp);
+    }
+
+    return buildMathML_makeRow(withDelims);
+  }
+
+  return node;
+};
+
+defineFunction({
+  type: "genfrac",
+  names: ["\\cfrac", "\\dfrac", "\\frac", "\\tfrac", "\\dbinom", "\\binom", "\\tbinom", "\\\\atopfrac", // can’t be entered directly
+  "\\\\bracefrac", "\\\\brackfrac"],
+  props: {
+    numArgs: 2,
+    greediness: 2
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var numer = args[0];
+    var denom = args[1];
+    var hasBarLine;
+    var leftDelim = null;
+    var rightDelim = null;
+    var size = "auto";
+
+    switch (funcName) {
+      case "\\cfrac":
+      case "\\dfrac":
+      case "\\frac":
+      case "\\tfrac":
+        hasBarLine = true;
+        break;
+
+      case "\\\\atopfrac":
+        hasBarLine = false;
+        break;
+
+      case "\\dbinom":
+      case "\\binom":
+      case "\\tbinom":
+        hasBarLine = false;
+        leftDelim = "(";
+        rightDelim = ")";
+        break;
+
+      case "\\\\bracefrac":
+        hasBarLine = false;
+        leftDelim = "\\{";
+        rightDelim = "\\}";
+        break;
+
+      case "\\\\brackfrac":
+        hasBarLine = false;
+        leftDelim = "[";
+        rightDelim = "]";
+        break;
+
+      default:
+        throw new Error("Unrecognized genfrac command");
+    }
+
+    switch (funcName) {
+      case "\\cfrac":
+      case "\\dfrac":
+      case "\\dbinom":
+        size = "display";
+        break;
+
+      case "\\tfrac":
+      case "\\tbinom":
+        size = "text";
+        break;
+    }
+
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      continued: funcName === "\\cfrac",
+      numer: numer,
+      denom: denom,
+      hasBarLine: hasBarLine,
+      leftDelim: leftDelim,
+      rightDelim: rightDelim,
+      size: size,
+      barSize: null
+    };
+  },
+  htmlBuilder: genfrac_htmlBuilder,
+  mathmlBuilder: genfrac_mathmlBuilder
+}); // Infix generalized fractions -- these are not rendered directly, but replaced
+// immediately by one of the variants above.
+
+defineFunction({
+  type: "infix",
+  names: ["\\over", "\\choose", "\\atop", "\\brace", "\\brack"],
+  props: {
+    numArgs: 0,
+    infix: true
+  },
+  handler: function handler(_ref2) {
+    var parser = _ref2.parser,
+        funcName = _ref2.funcName,
+        token = _ref2.token;
+    var replaceWith;
+
+    switch (funcName) {
+      case "\\over":
+        replaceWith = "\\frac";
+        break;
+
+      case "\\choose":
+        replaceWith = "\\binom";
+        break;
+
+      case "\\atop":
+        replaceWith = "\\\\atopfrac";
+        break;
+
+      case "\\brace":
+        replaceWith = "\\\\bracefrac";
+        break;
+
+      case "\\brack":
+        replaceWith = "\\\\brackfrac";
+        break;
+
+      default:
+        throw new Error("Unrecognized infix genfrac command");
+    }
+
+    return {
+      type: "infix",
+      mode: parser.mode,
+      replaceWith: replaceWith,
+      token: token
+    };
+  }
+});
+var stylArray = ["display", "text", "script", "scriptscript"];
+
+var delimFromValue = function delimFromValue(delimString) {
+  var delim = null;
+
+  if (delimString.length > 0) {
+    delim = delimString;
+    delim = delim === "." ? null : delim;
+  }
+
+  return delim;
+};
+
+defineFunction({
+  type: "genfrac",
+  names: ["\\genfrac"],
+  props: {
+    numArgs: 6,
+    greediness: 6,
+    argTypes: ["math", "math", "size", "text", "math", "math"]
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser;
+    var numer = args[4];
+    var denom = args[5]; // Look into the parse nodes to get the desired delimiters.
+
+    var leftDelim = args[0].type === "atom" && args[0].family === "open" ? delimFromValue(args[0].text) : null;
+    var rightDelim = args[1].type === "atom" && args[1].family === "close" ? delimFromValue(args[1].text) : null;
+    var barNode = assertNodeType(args[2], "size");
+    var hasBarLine;
+    var barSize = null;
+
+    if (barNode.isBlank) {
+      // \genfrac acts differently than \above.
+      // \genfrac treats an empty size group as a signal to use a
+      // standard bar size. \above would see size = 0 and omit the bar.
+      hasBarLine = true;
+    } else {
+      barSize = barNode.value;
+      hasBarLine = barSize.number > 0;
+    } // Find out if we want displaystyle, textstyle, etc.
+
+
+    var size = "auto";
+    var styl = args[3];
+
+    if (styl.type === "ordgroup") {
+      if (styl.body.length > 0) {
+        var textOrd = assertNodeType(styl.body[0], "textord");
+        size = stylArray[Number(textOrd.text)];
+      }
+    } else {
+      styl = assertNodeType(styl, "textord");
+      size = stylArray[Number(styl.text)];
+    }
+
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      numer: numer,
+      denom: denom,
+      continued: false,
+      hasBarLine: hasBarLine,
+      barSize: barSize,
+      leftDelim: leftDelim,
+      rightDelim: rightDelim,
+      size: size
+    };
+  },
+  htmlBuilder: genfrac_htmlBuilder,
+  mathmlBuilder: genfrac_mathmlBuilder
+}); // \above is an infix fraction that also defines a fraction bar size.
+
+defineFunction({
+  type: "infix",
+  names: ["\\above"],
+  props: {
+    numArgs: 1,
+    argTypes: ["size"],
+    infix: true
+  },
+  handler: function handler(_ref4, args) {
+    var parser = _ref4.parser,
+        funcName = _ref4.funcName,
+        token = _ref4.token;
+    return {
+      type: "infix",
+      mode: parser.mode,
+      replaceWith: "\\\\abovefrac",
+      size: assertNodeType(args[0], "size").value,
+      token: token
+    };
+  }
+});
+defineFunction({
+  type: "genfrac",
+  names: ["\\\\abovefrac"],
+  props: {
+    numArgs: 3,
+    argTypes: ["math", "size", "math"]
+  },
+  handler: function handler(_ref5, args) {
+    var parser = _ref5.parser,
+        funcName = _ref5.funcName;
+    var numer = args[0];
+    var barSize = assert(assertNodeType(args[1], "infix").size);
+    var denom = args[2];
+    var hasBarLine = barSize.number > 0;
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      numer: numer,
+      denom: denom,
+      continued: false,
+      hasBarLine: hasBarLine,
+      barSize: barSize,
+      leftDelim: null,
+      rightDelim: null,
+      size: "auto"
+    };
+  },
+  htmlBuilder: genfrac_htmlBuilder,
+  mathmlBuilder: genfrac_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/horizBrace.js
+
+
+
+
+
+
+
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only "horizBrace", but
+var horizBrace_htmlBuilder = function htmlBuilder(grp, options) {
+  var style = options.style; // Pull out the `ParseNode<"horizBrace">` if `grp` is a "supsub" node.
+
+  var supSubGroup;
+  var group;
+
+  if (grp.type === "supsub") {
+    // Ref: LaTeX source2e: }}}}\limits}
+    // i.e. LaTeX treats the brace similar to an op and passes it
+    // with \limits, so we need to assign supsub style.
+    supSubGroup = grp.sup ? buildHTML_buildGroup(grp.sup, options.havingStyle(style.sup()), options) : buildHTML_buildGroup(grp.sub, options.havingStyle(style.sub()), options);
+    group = assertNodeType(grp.base, "horizBrace");
+  } else {
+    group = assertNodeType(grp, "horizBrace");
+  } // Build the base group
+
+
+  var body = buildHTML_buildGroup(group.base, options.havingBaseStyle(src_Style.DISPLAY)); // Create the stretchy element
+
+  var braceBody = stretchy.svgSpan(group, options); // Generate the vlist, with the appropriate kerns        ┏━━━━━━━━┓
+  // This first vlist contains the content and the brace:   equation
+
+  var vlist;
+
+  if (group.isOver) {
+    vlist = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "kern",
+        size: 0.1
+      }, {
+        type: "elem",
+        elem: braceBody
+      }]
+    }, options); // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+    vlist.children[0].children[0].children[1].classes.push("svg-align");
+  } else {
+    vlist = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: body.depth + 0.1 + braceBody.height,
+      children: [{
+        type: "elem",
+        elem: braceBody
+      }, {
+        type: "kern",
+        size: 0.1
+      }, {
+        type: "elem",
+        elem: body
+      }]
+    }, options); // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+    vlist.children[0].children[0].children[0].classes.push("svg-align");
+  }
+
+  if (supSubGroup) {
+    // To write the supsub, wrap the first vlist in another vlist:
+    // They can't all go in the same vlist, because the note might be
+    // wider than the equation. We want the equation to control the
+    // brace width.
+    //      note          long note           long note
+    //   ┏━━━━━━━━┓   or    ┏━━━┓     not    ┏━━━━━━━━━┓
+    //    equation           eqn                 eqn
+    var vSpan = buildCommon.makeSpan(["mord", group.isOver ? "mover" : "munder"], [vlist], options);
+
+    if (group.isOver) {
+      vlist = buildCommon.makeVList({
+        positionType: "firstBaseline",
+        children: [{
+          type: "elem",
+          elem: vSpan
+        }, {
+          type: "kern",
+          size: 0.2
+        }, {
+          type: "elem",
+          elem: supSubGroup
+        }]
+      }, options);
+    } else {
+      vlist = buildCommon.makeVList({
+        positionType: "bottom",
+        positionData: vSpan.depth + 0.2 + supSubGroup.height + supSubGroup.depth,
+        children: [{
+          type: "elem",
+          elem: supSubGroup
+        }, {
+          type: "kern",
+          size: 0.2
+        }, {
+          type: "elem",
+          elem: vSpan
+        }]
+      }, options);
+    }
+  }
+
+  return buildCommon.makeSpan(["mord", group.isOver ? "mover" : "munder"], [vlist], options);
+};
+
+var horizBrace_mathmlBuilder = function mathmlBuilder(group, options) {
+  var accentNode = stretchy.mathMLnode(group.label);
+  return new mathMLTree.MathNode(group.isOver ? "mover" : "munder", [buildMathML_buildGroup(group.base, options), accentNode]);
+}; // Horizontal stretchy braces
+
+
+defineFunction({
+  type: "horizBrace",
+  names: ["\\overbrace", "\\underbrace"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    return {
+      type: "horizBrace",
+      mode: parser.mode,
+      label: funcName,
+      isOver: /^\\over/.test(funcName),
+      base: args[0]
+    };
+  },
+  htmlBuilder: horizBrace_htmlBuilder,
+  mathmlBuilder: horizBrace_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/href.js
+
+
+
+
+
+
+defineFunction({
+  type: "href",
+  names: ["\\href"],
+  props: {
+    numArgs: 2,
+    argTypes: ["url", "original"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var body = args[1];
+    var href = assertNodeType(args[0], "url").url;
+
+    if (!parser.settings.isTrusted({
+      command: "\\href",
+      url: href
+    })) {
+      return parser.formatUnsupportedCmd("\\href");
+    }
+
+    return {
+      type: "href",
+      mode: parser.mode,
+      href: href,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var elements = buildHTML_buildExpression(group.body, options, false);
+    return buildCommon.makeAnchor(group.href, [], elements, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var math = buildExpressionRow(group.body, options);
+
+    if (!(math instanceof mathMLTree_MathNode)) {
+      math = new mathMLTree_MathNode("mrow", [math]);
+    }
+
+    math.setAttribute("href", group.href);
+    return math;
+  }
+});
+defineFunction({
+  type: "href",
+  names: ["\\url"],
+  props: {
+    numArgs: 1,
+    argTypes: ["url"],
+    allowedInText: true
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    var href = assertNodeType(args[0], "url").url;
+
+    if (!parser.settings.isTrusted({
+      command: "\\url",
+      url: href
+    })) {
+      return parser.formatUnsupportedCmd("\\url");
+    }
+
+    var chars = [];
+
+    for (var i = 0; i < href.length; i++) {
+      var c = href[i];
+
+      if (c === "~") {
+        c = "\\textasciitilde";
+      }
+
+      chars.push({
+        type: "textord",
+        mode: "text",
+        text: c
+      });
+    }
+
+    var body = {
+      type: "text",
+      mode: parser.mode,
+      font: "\\texttt",
+      body: chars
+    };
+    return {
+      type: "href",
+      mode: parser.mode,
+      href: href,
+      body: ordargument(body)
+    };
+  }
+});
+// CONCATENATED MODULE: ./src/functions/html.js
+
+
+
+
+
+
+defineFunction({
+  type: "html",
+  names: ["\\htmlClass", "\\htmlId", "\\htmlStyle", "\\htmlData"],
+  props: {
+    numArgs: 2,
+    argTypes: ["raw", "original"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName,
+        token = _ref.token;
+    var value = assertNodeType(args[0], "raw").string;
+    var body = args[1];
+
+    if (parser.settings.strict) {
+      parser.settings.reportNonstrict("htmlExtension", "HTML extension is disabled on strict mode");
+    }
+
+    var trustContext;
+    var attributes = {};
+
+    switch (funcName) {
+      case "\\htmlClass":
+        attributes.class = value;
+        trustContext = {
+          command: "\\htmlClass",
+          class: value
+        };
+        break;
+
+      case "\\htmlId":
+        attributes.id = value;
+        trustContext = {
+          command: "\\htmlId",
+          id: value
+        };
+        break;
+
+      case "\\htmlStyle":
+        attributes.style = value;
+        trustContext = {
+          command: "\\htmlStyle",
+          style: value
+        };
+        break;
+
+      case "\\htmlData":
+        {
+          var data = value.split(",");
+
+          for (var i = 0; i < data.length; i++) {
+            var keyVal = data[i].split("=");
+
+            if (keyVal.length !== 2) {
+              throw new src_ParseError("Error parsing key-value for \\htmlData");
+            }
+
+            attributes["data-" + keyVal[0].trim()] = keyVal[1].trim();
+          }
+
+          trustContext = {
+            command: "\\htmlData",
+            attributes: attributes
+          };
+          break;
+        }
+
+      default:
+        throw new Error("Unrecognized html command");
+    }
+
+    if (!parser.settings.isTrusted(trustContext)) {
+      return parser.formatUnsupportedCmd(funcName);
+    }
+
+    return {
+      type: "html",
+      mode: parser.mode,
+      attributes: attributes,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var elements = buildHTML_buildExpression(group.body, options, false);
+    var classes = ["enclosing"];
+
+    if (group.attributes.class) {
+      classes.push.apply(classes, group.attributes.class.trim().split(/\s+/));
+    }
+
+    var span = buildCommon.makeSpan(classes, elements, options);
+
+    for (var attr in group.attributes) {
+      if (attr !== "class" && group.attributes.hasOwnProperty(attr)) {
+        span.setAttribute(attr, group.attributes[attr]);
+      }
+    }
+
+    return span;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    return buildExpressionRow(group.body, options);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/htmlmathml.js
+
+
+
+
+defineFunction({
+  type: "htmlmathml",
+  names: ["\\html@mathml"],
+  props: {
+    numArgs: 2,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    return {
+      type: "htmlmathml",
+      mode: parser.mode,
+      html: ordargument(args[0]),
+      mathml: ordargument(args[1])
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var elements = buildHTML_buildExpression(group.html, options, false);
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    return buildExpressionRow(group.mathml, options);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/includegraphics.js
+
+
+
+
+
+
+
+var includegraphics_sizeData = function sizeData(str) {
+  if (/^[-+]? *(\d+(\.\d*)?|\.\d+)$/.test(str)) {
+    // str is a number with no unit specified.
+    // default unit is bp, per graphix package.
+    return {
+      number: +str,
+      unit: "bp"
+    };
+  } else {
+    var match = /([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(str);
+
+    if (!match) {
+      throw new src_ParseError("Invalid size: '" + str + "' in \\includegraphics");
+    }
+
+    var data = {
+      number: +(match[1] + match[2]),
+      // sign + magnitude, cast to number
+      unit: match[3]
+    };
+
+    if (!validUnit(data)) {
+      throw new src_ParseError("Invalid unit: '" + data.unit + "' in \\includegraphics.");
+    }
+
+    return data;
+  }
+};
+
+defineFunction({
+  type: "includegraphics",
+  names: ["\\includegraphics"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1,
+    argTypes: ["raw", "url"],
+    allowedInText: false
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser;
+    var width = {
+      number: 0,
+      unit: "em"
+    };
+    var height = {
+      number: 0.9,
+      unit: "em"
+    }; // sorta character sized.
+
+    var totalheight = {
+      number: 0,
+      unit: "em"
+    };
+    var alt = "";
+
+    if (optArgs[0]) {
+      var attributeStr = assertNodeType(optArgs[0], "raw").string; // Parser.js does not parse key/value pairs. We get a string.
+
+      var attributes = attributeStr.split(",");
+
+      for (var i = 0; i < attributes.length; i++) {
+        var keyVal = attributes[i].split("=");
+
+        if (keyVal.length === 2) {
+          var str = keyVal[1].trim();
+
+          switch (keyVal[0].trim()) {
+            case "alt":
+              alt = str;
+              break;
+
+            case "width":
+              width = includegraphics_sizeData(str);
+              break;
+
+            case "height":
+              height = includegraphics_sizeData(str);
+              break;
+
+            case "totalheight":
+              totalheight = includegraphics_sizeData(str);
+              break;
+
+            default:
+              throw new src_ParseError("Invalid key: '" + keyVal[0] + "' in \\includegraphics.");
+          }
+        }
+      }
+    }
+
+    var src = assertNodeType(args[0], "url").url;
+
+    if (alt === "") {
+      // No alt given. Use the file name. Strip away the path.
+      alt = src;
+      alt = alt.replace(/^.*[\\/]/, '');
+      alt = alt.substring(0, alt.lastIndexOf('.'));
+    }
+
+    if (!parser.settings.isTrusted({
+      command: "\\includegraphics",
+      url: src
+    })) {
+      return parser.formatUnsupportedCmd("\\includegraphics");
+    }
+
+    return {
+      type: "includegraphics",
+      mode: parser.mode,
+      alt: alt,
+      width: width,
+      height: height,
+      totalheight: totalheight,
+      src: src
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var height = units_calculateSize(group.height, options);
+    var depth = 0;
+
+    if (group.totalheight.number > 0) {
+      depth = units_calculateSize(group.totalheight, options) - height;
+      depth = Number(depth.toFixed(2));
+    }
+
+    var width = 0;
+
+    if (group.width.number > 0) {
+      width = units_calculateSize(group.width, options);
+    }
+
+    var style = {
+      height: height + depth + "em"
+    };
+
+    if (width > 0) {
+      style.width = width + "em";
+    }
+
+    if (depth > 0) {
+      style.verticalAlign = -depth + "em";
+    }
+
+    var node = new domTree_Img(group.src, group.alt, style);
+    node.height = height;
+    node.depth = depth;
+    return node;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mglyph", []);
+    node.setAttribute("alt", group.alt);
+    var height = units_calculateSize(group.height, options);
+    var depth = 0;
+
+    if (group.totalheight.number > 0) {
+      depth = units_calculateSize(group.totalheight, options) - height;
+      depth = depth.toFixed(2);
+      node.setAttribute("valign", "-" + depth + "em");
+    }
+
+    node.setAttribute("height", height + depth + "em");
+
+    if (group.width.number > 0) {
+      var width = units_calculateSize(group.width, options);
+      node.setAttribute("width", width + "em");
+    }
+
+    node.setAttribute("src", group.src);
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/kern.js
+// Horizontal spacing commands
+
+
+
+
+ // TODO: \hskip and \mskip should support plus and minus in lengths
+
+defineFunction({
+  type: "kern",
+  names: ["\\kern", "\\mkern", "\\hskip", "\\mskip"],
+  props: {
+    numArgs: 1,
+    argTypes: ["size"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var size = assertNodeType(args[0], "size");
+
+    if (parser.settings.strict) {
+      var mathFunction = funcName[1] === 'm'; // \mkern, \mskip
+
+      var muUnit = size.value.unit === 'mu';
+
+      if (mathFunction) {
+        if (!muUnit) {
+          parser.settings.reportNonstrict("mathVsTextUnits", "LaTeX's " + funcName + " supports only mu units, " + ("not " + size.value.unit + " units"));
+        }
+
+        if (parser.mode !== "math") {
+          parser.settings.reportNonstrict("mathVsTextUnits", "LaTeX's " + funcName + " works only in math mode");
+        }
+      } else {
+        // !mathFunction
+        if (muUnit) {
+          parser.settings.reportNonstrict("mathVsTextUnits", "LaTeX's " + funcName + " doesn't support mu units");
+        }
+      }
+    }
+
+    return {
+      type: "kern",
+      mode: parser.mode,
+      dimension: size.value
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    return buildCommon.makeGlue(group.dimension, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var dimension = units_calculateSize(group.dimension, options);
+    return new mathMLTree.SpaceNode(dimension);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/lap.js
+// Horizontal overlap functions
+
+
+
+
+
+defineFunction({
+  type: "lap",
+  names: ["\\mathllap", "\\mathrlap", "\\mathclap"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    return {
+      type: "lap",
+      mode: parser.mode,
+      alignment: funcName.slice(5),
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // mathllap, mathrlap, mathclap
+    var inner;
+
+    if (group.alignment === "clap") {
+      // ref: https://www.math.lsu.edu/~aperlis/publications/mathclap/
+      inner = buildCommon.makeSpan([], [buildHTML_buildGroup(group.body, options)]); // wrap, since CSS will center a .clap > .inner > span
+
+      inner = buildCommon.makeSpan(["inner"], [inner], options);
+    } else {
+      inner = buildCommon.makeSpan(["inner"], [buildHTML_buildGroup(group.body, options)]);
+    }
+
+    var fix = buildCommon.makeSpan(["fix"], []);
+    var node = buildCommon.makeSpan([group.alignment], [inner, fix], options); // At this point, we have correctly set horizontal alignment of the
+    // two items involved in the lap.
+    // Next, use a strut to set the height of the HTML bounding box.
+    // Otherwise, a tall argument may be misplaced.
+    // This code resolved issue #1153
+
+    var strut = buildCommon.makeSpan(["strut"]);
+    strut.style.height = node.height + node.depth + "em";
+    strut.style.verticalAlign = -node.depth + "em";
+    node.children.unshift(strut); // Next, prevent vertical misplacement when next to something tall.
+    // This code resolves issue #1234
+
+    node = buildCommon.makeSpan(["thinbox"], [node], options);
+    return buildCommon.makeSpan(["mord", "vbox"], [node], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    // mathllap, mathrlap, mathclap
+    var node = new mathMLTree.MathNode("mpadded", [buildMathML_buildGroup(group.body, options)]);
+
+    if (group.alignment !== "rlap") {
+      var offset = group.alignment === "llap" ? "-1" : "-0.5";
+      node.setAttribute("lspace", offset + "width");
+    }
+
+    node.setAttribute("width", "0px");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/math.js
+
+ // Switching from text mode back to math mode
+
+defineFunction({
+  type: "styling",
+  names: ["\\(", "$"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: false
+  },
+  handler: function handler(_ref, args) {
+    var funcName = _ref.funcName,
+        parser = _ref.parser;
+    var outerMode = parser.mode;
+    parser.switchMode("math");
+    var close = funcName === "\\(" ? "\\)" : "$";
+    var body = parser.parseExpression(false, close);
+    parser.expect(close);
+    parser.switchMode(outerMode);
+    return {
+      type: "styling",
+      mode: parser.mode,
+      style: "text",
+      body: body
+    };
+  }
+}); // Check for extra closing math delimiters
+
+defineFunction({
+  type: "text",
+  // Doesn't matter what this is.
+  names: ["\\)", "\\]"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: false
+  },
+  handler: function handler(context, args) {
+    throw new src_ParseError("Mismatched " + context.funcName);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/mathchoice.js
+
+
+
+
+
+
+var mathchoice_chooseMathStyle = function chooseMathStyle(group, options) {
+  switch (options.style.size) {
+    case src_Style.DISPLAY.size:
+      return group.display;
+
+    case src_Style.TEXT.size:
+      return group.text;
+
+    case src_Style.SCRIPT.size:
+      return group.script;
+
+    case src_Style.SCRIPTSCRIPT.size:
+      return group.scriptscript;
+
+    default:
+      return group.text;
+  }
+};
+
+defineFunction({
+  type: "mathchoice",
+  names: ["\\mathchoice"],
+  props: {
+    numArgs: 4
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    return {
+      type: "mathchoice",
+      mode: parser.mode,
+      display: ordargument(args[0]),
+      text: ordargument(args[1]),
+      script: ordargument(args[2]),
+      scriptscript: ordargument(args[3])
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var body = mathchoice_chooseMathStyle(group, options);
+    var elements = buildHTML_buildExpression(body, options, false);
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var body = mathchoice_chooseMathStyle(group, options);
+    return buildExpressionRow(body, options);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/utils/assembleSupSub.js
+
+
+// For an operator with limits, assemble the base, sup, and sub into a span.
+var assembleSupSub_assembleSupSub = function assembleSupSub(base, supGroup, subGroup, options, style, slant, baseShift) {
+  base = buildCommon.makeSpan([], [base]);
+  var sub;
+  var sup; // We manually have to handle the superscripts and subscripts. This,
+  // aside from the kern calculations, is copied from supsub.
+
+  if (supGroup) {
+    var elem = buildHTML_buildGroup(supGroup, options.havingStyle(style.sup()), options);
+    sup = {
+      elem: elem,
+      kern: Math.max(options.fontMetrics().bigOpSpacing1, options.fontMetrics().bigOpSpacing3 - elem.depth)
+    };
+  }
+
+  if (subGroup) {
+    var _elem = buildHTML_buildGroup(subGroup, options.havingStyle(style.sub()), options);
+
+    sub = {
+      elem: _elem,
+      kern: Math.max(options.fontMetrics().bigOpSpacing2, options.fontMetrics().bigOpSpacing4 - _elem.height)
+    };
+  } // Build the final group as a vlist of the possible subscript, base,
+  // and possible superscript.
+
+
+  var finalGroup;
+
+  if (sup && sub) {
+    var bottom = options.fontMetrics().bigOpSpacing5 + sub.elem.height + sub.elem.depth + sub.kern + base.depth + baseShift;
+    finalGroup = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: bottom,
+      children: [{
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }, {
+        type: "elem",
+        elem: sub.elem,
+        marginLeft: -slant + "em"
+      }, {
+        type: "kern",
+        size: sub.kern
+      }, {
+        type: "elem",
+        elem: base
+      }, {
+        type: "kern",
+        size: sup.kern
+      }, {
+        type: "elem",
+        elem: sup.elem,
+        marginLeft: slant + "em"
+      }, {
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }]
+    }, options);
+  } else if (sub) {
+    var top = base.height - baseShift; // Shift the limits by the slant of the symbol. Note
+    // that we are supposed to shift the limits by 1/2 of the slant,
+    // but since we are centering the limits adding a full slant of
+    // margin will shift by 1/2 that.
+
+    finalGroup = buildCommon.makeVList({
+      positionType: "top",
+      positionData: top,
+      children: [{
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }, {
+        type: "elem",
+        elem: sub.elem,
+        marginLeft: -slant + "em"
+      }, {
+        type: "kern",
+        size: sub.kern
+      }, {
+        type: "elem",
+        elem: base
+      }]
+    }, options);
+  } else if (sup) {
+    var _bottom = base.depth + baseShift;
+
+    finalGroup = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: _bottom,
+      children: [{
+        type: "elem",
+        elem: base
+      }, {
+        type: "kern",
+        size: sup.kern
+      }, {
+        type: "elem",
+        elem: sup.elem,
+        marginLeft: slant + "em"
+      }, {
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }]
+    }, options);
+  } else {
+    // This case probably shouldn't occur (this would mean the
+    // supsub was sending us a group with no superscript or
+    // subscript) but be safe.
+    return base;
+  }
+
+  return buildCommon.makeSpan(["mop", "op-limits"], [finalGroup], options);
+};
+// CONCATENATED MODULE: ./src/functions/op.js
+// Limits, symbols
+
+
+
+
+
+
+
+
+
+
+// Most operators have a large successor symbol, but these don't.
+var noSuccessor = ["\\smallint"]; // NOTE: Unlike most `htmlBuilder`s, this one handles not only "op", but also
+// "supsub" since some of them (like \int) can affect super/subscripting.
+
+var op_htmlBuilder = function htmlBuilder(grp, options) {
+  // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+  var supGroup;
+  var subGroup;
+  var hasLimits = false;
+  var group;
+
+  if (grp.type === "supsub") {
+    // If we have limits, supsub will pass us its group to handle. Pull
+    // out the superscript and subscript and set the group to the op in
+    // its base.
+    supGroup = grp.sup;
+    subGroup = grp.sub;
+    group = assertNodeType(grp.base, "op");
+    hasLimits = true;
+  } else {
+    group = assertNodeType(grp, "op");
+  }
+
+  var style = options.style;
+  var large = false;
+
+  if (style.size === src_Style.DISPLAY.size && group.symbol && !utils.contains(noSuccessor, group.name)) {
+    // Most symbol operators get larger in displaystyle (rule 13)
+    large = true;
+  }
+
+  var base;
+
+  if (group.symbol) {
+    // If this is a symbol, create the symbol.
+    var fontName = large ? "Size2-Regular" : "Size1-Regular";
+    var stash = "";
+
+    if (group.name === "\\oiint" || group.name === "\\oiiint") {
+      // No font glyphs yet, so use a glyph w/o the oval.
+      // TODO: When font glyphs are available, delete this code.
+      stash = group.name.substr(1); // $FlowFixMe
+
+      group.name = stash === "oiint" ? "\\iint" : "\\iiint";
+    }
+
+    base = buildCommon.makeSymbol(group.name, fontName, "math", options, ["mop", "op-symbol", large ? "large-op" : "small-op"]);
+
+    if (stash.length > 0) {
+      // We're in \oiint or \oiiint. Overlay the oval.
+      // TODO: When font glyphs are available, delete this code.
+      var italic = base.italic;
+      var oval = buildCommon.staticSvg(stash + "Size" + (large ? "2" : "1"), options);
+      base = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: base,
+          shift: 0
+        }, {
+          type: "elem",
+          elem: oval,
+          shift: large ? 0.08 : 0
+        }]
+      }, options); // $FlowFixMe
+
+      group.name = "\\" + stash;
+      base.classes.unshift("mop"); // $FlowFixMe
+
+      base.italic = italic;
+    }
+  } else if (group.body) {
+    // If this is a list, compose that list.
+    var inner = buildHTML_buildExpression(group.body, options, true);
+
+    if (inner.length === 1 && inner[0] instanceof domTree_SymbolNode) {
+      base = inner[0];
+      base.classes[0] = "mop"; // replace old mclass
+    } else {
+      base = buildCommon.makeSpan(["mop"], buildCommon.tryCombineChars(inner), options);
+    }
+  } else {
+    // Otherwise, this is a text operator. Build the text from the
+    // operator's name.
+    // TODO(emily): Add a space in the middle of some of these
+    // operators, like \limsup
+    var output = [];
+
+    for (var i = 1; i < group.name.length; i++) {
+      output.push(buildCommon.mathsym(group.name[i], group.mode, options));
+    }
+
+    base = buildCommon.makeSpan(["mop"], output, options);
+  } // If content of op is a single symbol, shift it vertically.
+
+
+  var baseShift = 0;
+  var slant = 0;
+
+  if ((base instanceof domTree_SymbolNode || group.name === "\\oiint" || group.name === "\\oiiint") && !group.suppressBaseShift) {
+    // We suppress the shift of the base of \overset and \underset. Otherwise,
+    // shift the symbol so its center lies on the axis (rule 13). It
+    // appears that our fonts have the centers of the symbols already
+    // almost on the axis, so these numbers are very small. Note we
+    // don't actually apply this here, but instead it is used either in
+    // the vlist creation or separately when there are no limits.
+    baseShift = (base.height - base.depth) / 2 - options.fontMetrics().axisHeight; // The slant of the symbol is just its italic correction.
+    // $FlowFixMe
+
+    slant = base.italic;
+  }
+
+  if (hasLimits) {
+    return assembleSupSub_assembleSupSub(base, supGroup, subGroup, options, style, slant, baseShift);
+  } else {
+    if (baseShift) {
+      base.style.position = "relative";
+      base.style.top = baseShift + "em";
+    }
+
+    return base;
+  }
+};
+
+var op_mathmlBuilder = function mathmlBuilder(group, options) {
+  var node;
+
+  if (group.symbol) {
+    // This is a symbol. Just add the symbol.
+    node = new mathMLTree_MathNode("mo", [buildMathML_makeText(group.name, group.mode)]);
+
+    if (utils.contains(noSuccessor, group.name)) {
+      node.setAttribute("largeop", "false");
+    }
+  } else if (group.body) {
+    // This is an operator with children. Add them.
+    node = new mathMLTree_MathNode("mo", buildMathML_buildExpression(group.body, options));
+  } else {
+    // This is a text operator. Add all of the characters from the
+    // operator's name.
+    node = new mathMLTree_MathNode("mi", [new mathMLTree_TextNode(group.name.slice(1))]); // Append an <mo>&ApplyFunction;</mo>.
+    // ref: https://www.w3.org/TR/REC-MathML/chap3_2.html#sec3.2.4
+
+    var operator = new mathMLTree_MathNode("mo", [buildMathML_makeText("\u2061", "text")]);
+
+    if (group.parentIsSupSub) {
+      node = new mathMLTree_MathNode("mo", [node, operator]);
+    } else {
+      node = newDocumentFragment([node, operator]);
+    }
+  }
+
+  return node;
+};
+
+var singleCharBigOps = {
+  "\u220F": "\\prod",
+  "\u2210": "\\coprod",
+  "\u2211": "\\sum",
+  "\u22C0": "\\bigwedge",
+  "\u22C1": "\\bigvee",
+  "\u22C2": "\\bigcap",
+  "\u22C3": "\\bigcup",
+  "\u2A00": "\\bigodot",
+  "\u2A01": "\\bigoplus",
+  "\u2A02": "\\bigotimes",
+  "\u2A04": "\\biguplus",
+  "\u2A06": "\\bigsqcup"
+};
+defineFunction({
+  type: "op",
+  names: ["\\coprod", "\\bigvee", "\\bigwedge", "\\biguplus", "\\bigcap", "\\bigcup", "\\intop", "\\prod", "\\sum", "\\bigotimes", "\\bigoplus", "\\bigodot", "\\bigsqcup", "\\smallint", "\u220F", "\u2210", "\u2211", "\u22C0", "\u22C1", "\u22C2", "\u22C3", "\u2A00", "\u2A01", "\u2A02", "\u2A04", "\u2A06"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var fName = funcName;
+
+    if (fName.length === 1) {
+      fName = singleCharBigOps[fName];
+    }
+
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: true,
+      parentIsSupSub: false,
+      symbol: true,
+      name: fName
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+}); // Note: calling defineFunction with a type that's already been defined only
+// works because the same htmlBuilder and mathmlBuilder are being used.
+
+defineFunction({
+  type: "op",
+  names: ["\\mathop"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    var body = args[0];
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: false,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+}); // There are 2 flags for operators; whether they produce limits in
+// displaystyle, and whether they are symbols and should grow in
+// displaystyle. These four groups cover the four possible choices.
+
+var singleCharIntegrals = {
+  "\u222B": "\\int",
+  "\u222C": "\\iint",
+  "\u222D": "\\iiint",
+  "\u222E": "\\oint",
+  "\u222F": "\\oiint",
+  "\u2230": "\\oiiint"
+}; // No limits, not symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\arcsin", "\\arccos", "\\arctan", "\\arctg", "\\arcctg", "\\arg", "\\ch", "\\cos", "\\cosec", "\\cosh", "\\cot", "\\cotg", "\\coth", "\\csc", "\\ctg", "\\cth", "\\deg", "\\dim", "\\exp", "\\hom", "\\ker", "\\lg", "\\ln", "\\log", "\\sec", "\\sin", "\\sinh", "\\sh", "\\tan", "\\tanh", "\\tg", "\\th"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(_ref3) {
+    var parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: false,
+      name: funcName
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+}); // Limits, not symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\det", "\\gcd", "\\inf", "\\lim", "\\max", "\\min", "\\Pr", "\\sup"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(_ref4) {
+    var parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: true,
+      parentIsSupSub: false,
+      symbol: false,
+      name: funcName
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+}); // No limits, symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\int", "\\iint", "\\iiint", "\\oint", "\\oiint", "\\oiiint", "\u222B", "\u222C", "\u222D", "\u222E", "\u222F", "\u2230"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(_ref5) {
+    var parser = _ref5.parser,
+        funcName = _ref5.funcName;
+    var fName = funcName;
+
+    if (fName.length === 1) {
+      fName = singleCharIntegrals[fName];
+    }
+
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: true,
+      name: fName
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/operatorname.js
+
+
+
+
+
+
+
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only
+// "operatorname", but also  "supsub" since \operatorname* can
+var operatorname_htmlBuilder = function htmlBuilder(grp, options) {
+  // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+  var supGroup;
+  var subGroup;
+  var hasLimits = false;
+  var group;
+
+  if (grp.type === "supsub") {
+    // If we have limits, supsub will pass us its group to handle. Pull
+    // out the superscript and subscript and set the group to the op in
+    // its base.
+    supGroup = grp.sup;
+    subGroup = grp.sub;
+    group = assertNodeType(grp.base, "operatorname");
+    hasLimits = true;
+  } else {
+    group = assertNodeType(grp, "operatorname");
+  }
+
+  var base;
+
+  if (group.body.length > 0) {
+    var body = group.body.map(function (child) {
+      // $FlowFixMe: Check if the node has a string `text` property.
+      var childText = child.text;
+
+      if (typeof childText === "string") {
+        return {
+          type: "textord",
+          mode: child.mode,
+          text: childText
+        };
+      } else {
+        return child;
+      }
+    }); // Consolidate function names into symbol characters.
+
+    var expression = buildHTML_buildExpression(body, options.withFont("mathrm"), true);
+
+    for (var i = 0; i < expression.length; i++) {
+      var child = expression[i];
+
+      if (child instanceof domTree_SymbolNode) {
+        // Per amsopn package,
+        // change minus to hyphen and \ast to asterisk
+        child.text = child.text.replace(/\u2212/, "-").replace(/\u2217/, "*");
+      }
+    }
+
+    base = buildCommon.makeSpan(["mop"], expression, options);
+  } else {
+    base = buildCommon.makeSpan(["mop"], [], options);
+  }
+
+  if (hasLimits) {
+    return assembleSupSub_assembleSupSub(base, supGroup, subGroup, options, options.style, 0, 0);
+  } else {
+    return base;
+  }
+};
+
+var operatorname_mathmlBuilder = function mathmlBuilder(group, options) {
+  // The steps taken here are similar to the html version.
+  var expression = buildMathML_buildExpression(group.body, options.withFont("mathrm")); // Is expression a string or has it something like a fraction?
+
+  var isAllString = true; // default
+
+  for (var i = 0; i < expression.length; i++) {
+    var node = expression[i];
+
+    if (node instanceof mathMLTree.SpaceNode) {// Do nothing
+    } else if (node instanceof mathMLTree.MathNode) {
+      switch (node.type) {
+        case "mi":
+        case "mn":
+        case "ms":
+        case "mspace":
+        case "mtext":
+          break;
+        // Do nothing yet.
+
+        case "mo":
+          {
+            var child = node.children[0];
+
+            if (node.children.length === 1 && child instanceof mathMLTree.TextNode) {
+              child.text = child.text.replace(/\u2212/, "-").replace(/\u2217/, "*");
+            } else {
+              isAllString = false;
+            }
+
+            break;
+          }
+
+        default:
+          isAllString = false;
+      }
+    } else {
+      isAllString = false;
+    }
+  }
+
+  if (isAllString) {
+    // Write a single TextNode instead of multiple nested tags.
+    var word = expression.map(function (node) {
+      return node.toText();
+    }).join("");
+    expression = [new mathMLTree.TextNode(word)];
+  }
+
+  var identifier = new mathMLTree.MathNode("mi", expression);
+  identifier.setAttribute("mathvariant", "normal"); // \u2061 is the same as &ApplyFunction;
+  // ref: https://www.w3schools.com/charsets/ref_html_entities_a.asp
+
+  var operator = new mathMLTree.MathNode("mo", [buildMathML_makeText("\u2061", "text")]);
+
+  if (group.parentIsSupSub) {
+    return new mathMLTree.MathNode("mo", [identifier, operator]);
+  } else {
+    return mathMLTree.newDocumentFragment([identifier, operator]);
+  }
+}; // \operatorname
+// amsopn.dtx: \mathop{#1\kern\z@\operator@font#3}\newmcodes@
+
+
+defineFunction({
+  type: "operatorname",
+  names: ["\\operatorname", "\\operatorname*"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    return {
+      type: "operatorname",
+      mode: parser.mode,
+      body: ordargument(body),
+      alwaysHandleSupSub: funcName === "\\operatorname*",
+      limits: false,
+      parentIsSupSub: false
+    };
+  },
+  htmlBuilder: operatorname_htmlBuilder,
+  mathmlBuilder: operatorname_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/ordgroup.js
+
+
+
+
+defineFunctionBuilders({
+  type: "ordgroup",
+  htmlBuilder: function htmlBuilder(group, options) {
+    if (group.semisimple) {
+      return buildCommon.makeFragment(buildHTML_buildExpression(group.body, options, false));
+    }
+
+    return buildCommon.makeSpan(["mord"], buildHTML_buildExpression(group.body, options, true), options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    return buildExpressionRow(group.body, options, true);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/overline.js
+
+
+
+
+
+defineFunction({
+  type: "overline",
+  names: ["\\overline"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var body = args[0];
+    return {
+      type: "overline",
+      mode: parser.mode,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Overlines are handled in the TeXbook pg 443, Rule 9.
+    // Build the inner group in the cramped style.
+    var innerGroup = buildHTML_buildGroup(group.body, options.havingCrampedStyle()); // Create the line above the body
+
+    var line = buildCommon.makeLineSpan("overline-line", options); // Generate the vlist, with the appropriate kerns
+
+    var defaultRuleThickness = options.fontMetrics().defaultRuleThickness;
+    var vlist = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: innerGroup
+      }, {
+        type: "kern",
+        size: 3 * defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: line
+      }, {
+        type: "kern",
+        size: defaultRuleThickness
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "overline"], [vlist], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var operator = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode("\u203E")]);
+    operator.setAttribute("stretchy", "true");
+    var node = new mathMLTree.MathNode("mover", [buildMathML_buildGroup(group.body, options), operator]);
+    node.setAttribute("accent", "true");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/phantom.js
+
+
+
+
+
+defineFunction({
+  type: "phantom",
+  names: ["\\phantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var body = args[0];
+    return {
+      type: "phantom",
+      mode: parser.mode,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var elements = buildHTML_buildExpression(group.body, options.withPhantom(), false); // \phantom isn't supposed to affect the elements it contains.
+    // See "color" for more details.
+
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var inner = buildMathML_buildExpression(group.body, options);
+    return new mathMLTree.MathNode("mphantom", inner);
+  }
+});
+defineFunction({
+  type: "hphantom",
+  names: ["\\hphantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    var body = args[0];
+    return {
+      type: "hphantom",
+      mode: parser.mode,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var node = buildCommon.makeSpan([], [buildHTML_buildGroup(group.body, options.withPhantom())]);
+    node.height = 0;
+    node.depth = 0;
+
+    if (node.children) {
+      for (var i = 0; i < node.children.length; i++) {
+        node.children[i].height = 0;
+        node.children[i].depth = 0;
+      }
+    } // See smash for comment re: use of makeVList
+
+
+    node = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: node
+      }]
+    }, options); // For spacing, TeX treats \smash as a math group (same spacing as ord).
+
+    return buildCommon.makeSpan(["mord"], [node], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var inner = buildMathML_buildExpression(ordargument(group.body), options);
+    var phantom = new mathMLTree.MathNode("mphantom", inner);
+    var node = new mathMLTree.MathNode("mpadded", [phantom]);
+    node.setAttribute("height", "0px");
+    node.setAttribute("depth", "0px");
+    return node;
+  }
+});
+defineFunction({
+  type: "vphantom",
+  names: ["\\vphantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser;
+    var body = args[0];
+    return {
+      type: "vphantom",
+      mode: parser.mode,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var inner = buildCommon.makeSpan(["inner"], [buildHTML_buildGroup(group.body, options.withPhantom())]);
+    var fix = buildCommon.makeSpan(["fix"], []);
+    return buildCommon.makeSpan(["mord", "rlap"], [inner, fix], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var inner = buildMathML_buildExpression(ordargument(group.body), options);
+    var phantom = new mathMLTree.MathNode("mphantom", inner);
+    var node = new mathMLTree.MathNode("mpadded", [phantom]);
+    node.setAttribute("width", "0px");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/raisebox.js
+
+
+
+
+
+
+ // Box manipulation
+
+defineFunction({
+  type: "raisebox",
+  names: ["\\raisebox"],
+  props: {
+    numArgs: 2,
+    argTypes: ["size", "hbox"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var amount = assertNodeType(args[0], "size").value;
+    var body = args[1];
+    return {
+      type: "raisebox",
+      mode: parser.mode,
+      dy: amount,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var body = buildHTML_buildGroup(group.body, options);
+    var dy = units_calculateSize(group.dy, options);
+    return buildCommon.makeVList({
+      positionType: "shift",
+      positionData: -dy,
+      children: [{
+        type: "elem",
+        elem: body
+      }]
+    }, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mpadded", [buildMathML_buildGroup(group.body, options)]);
+    var dy = group.dy.number + group.dy.unit;
+    node.setAttribute("voffset", dy);
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/rule.js
+
+
+
+
+
+defineFunction({
+  type: "rule",
+  names: ["\\rule"],
+  props: {
+    numArgs: 2,
+    numOptionalArgs: 1,
+    argTypes: ["size", "size", "size"]
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser;
+    var shift = optArgs[0];
+    var width = assertNodeType(args[0], "size");
+    var height = assertNodeType(args[1], "size");
+    return {
+      type: "rule",
+      mode: parser.mode,
+      shift: shift && assertNodeType(shift, "size").value,
+      width: width.value,
+      height: height.value
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Make an empty span for the rule
+    var rule = buildCommon.makeSpan(["mord", "rule"], [], options); // Calculate the shift, width, and height of the rule, and account for units
+
+    var width = units_calculateSize(group.width, options);
+    var height = units_calculateSize(group.height, options);
+    var shift = group.shift ? units_calculateSize(group.shift, options) : 0; // Style the rule to the right size
+
+    rule.style.borderRightWidth = width + "em";
+    rule.style.borderTopWidth = height + "em";
+    rule.style.bottom = shift + "em"; // Record the height and width
+
+    rule.width = width;
+    rule.height = height + shift;
+    rule.depth = -shift; // Font size is the number large enough that the browser will
+    // reserve at least `absHeight` space above the baseline.
+    // The 1.125 factor was empirically determined
+
+    rule.maxFontSize = height * 1.125 * options.sizeMultiplier;
+    return rule;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var width = units_calculateSize(group.width, options);
+    var height = units_calculateSize(group.height, options);
+    var shift = group.shift ? units_calculateSize(group.shift, options) : 0;
+    var color = options.color && options.getColor() || "black";
+    var rule = new mathMLTree.MathNode("mspace");
+    rule.setAttribute("mathbackground", color);
+    rule.setAttribute("width", width + "em");
+    rule.setAttribute("height", height + "em");
+    var wrapper = new mathMLTree.MathNode("mpadded", [rule]);
+
+    if (shift >= 0) {
+      wrapper.setAttribute("height", "+" + shift + "em");
+    } else {
+      wrapper.setAttribute("height", shift + "em");
+      wrapper.setAttribute("depth", "+" + -shift + "em");
+    }
+
+    wrapper.setAttribute("voffset", shift + "em");
+    return wrapper;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/sizing.js
+
+
+
+
+
+function sizingGroup(value, options, baseOptions) {
+  var inner = buildHTML_buildExpression(value, options, false);
+  var multiplier = options.sizeMultiplier / baseOptions.sizeMultiplier; // Add size-resetting classes to the inner list and set maxFontSize
+  // manually. Handle nested size changes.
+
+  for (var i = 0; i < inner.length; i++) {
+    var pos = inner[i].classes.indexOf("sizing");
+
+    if (pos < 0) {
+      Array.prototype.push.apply(inner[i].classes, options.sizingClasses(baseOptions));
+    } else if (inner[i].classes[pos + 1] === "reset-size" + options.size) {
+      // This is a nested size change: e.g., inner[i] is the "b" in
+      // `\Huge a \small b`. Override the old size (the `reset-` class)
+      // but not the new size.
+      inner[i].classes[pos + 1] = "reset-size" + baseOptions.size;
+    }
+
+    inner[i].height *= multiplier;
+    inner[i].depth *= multiplier;
+  }
+
+  return buildCommon.makeFragment(inner);
+}
+var sizeFuncs = ["\\tiny", "\\sixptsize", "\\scriptsize", "\\footnotesize", "\\small", "\\normalsize", "\\large", "\\Large", "\\LARGE", "\\huge", "\\Huge"];
+var sizing_htmlBuilder = function htmlBuilder(group, options) {
+  // Handle sizing operators like \Huge. Real TeX doesn't actually allow
+  // these functions inside of math expressions, so we do some special
+  // handling.
+  var newOptions = options.havingSize(group.size);
+  return sizingGroup(group.body, newOptions, options);
+};
+defineFunction({
+  type: "sizing",
+  names: sizeFuncs,
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var breakOnTokenText = _ref.breakOnTokenText,
+        funcName = _ref.funcName,
+        parser = _ref.parser;
+    var body = parser.parseExpression(false, breakOnTokenText);
+    return {
+      type: "sizing",
+      mode: parser.mode,
+      // Figure out what size to use based on the list of functions above
+      size: sizeFuncs.indexOf(funcName) + 1,
+      body: body
+    };
+  },
+  htmlBuilder: sizing_htmlBuilder,
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var newOptions = options.havingSize(group.size);
+    var inner = buildMathML_buildExpression(group.body, newOptions);
+    var node = new mathMLTree.MathNode("mstyle", inner); // TODO(emily): This doesn't produce the correct size for nested size
+    // changes, because we don't keep state of what style we're currently
+    // in, so we can't reset the size to normal before changing it.  Now
+    // that we're passing an options parameter we should be able to fix
+    // this.
+
+    node.setAttribute("mathsize", newOptions.sizeMultiplier + "em");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/smash.js
+// smash, with optional [tb], as in AMS
+
+
+
+
+
+
+defineFunction({
+  type: "smash",
+  names: ["\\smash"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser;
+    var smashHeight = false;
+    var smashDepth = false;
+    var tbArg = optArgs[0] && assertNodeType(optArgs[0], "ordgroup");
+
+    if (tbArg) {
+      // Optional [tb] argument is engaged.
+      // ref: amsmath: \renewcommand{\smash}[1][tb]{%
+      //               def\mb@t{\ht}\def\mb@b{\dp}\def\mb@tb{\ht\z@\z@\dp}%
+      var letter = "";
+
+      for (var i = 0; i < tbArg.body.length; ++i) {
+        var node = tbArg.body[i]; // $FlowFixMe: Not every node type has a `text` property.
+
+        letter = node.text;
+
+        if (letter === "t") {
+          smashHeight = true;
+        } else if (letter === "b") {
+          smashDepth = true;
+        } else {
+          smashHeight = false;
+          smashDepth = false;
+          break;
+        }
+      }
+    } else {
+      smashHeight = true;
+      smashDepth = true;
+    }
+
+    var body = args[0];
+    return {
+      type: "smash",
+      mode: parser.mode,
+      body: body,
+      smashHeight: smashHeight,
+      smashDepth: smashDepth
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var node = buildCommon.makeSpan([], [buildHTML_buildGroup(group.body, options)]);
+
+    if (!group.smashHeight && !group.smashDepth) {
+      return node;
+    }
+
+    if (group.smashHeight) {
+      node.height = 0; // In order to influence makeVList, we have to reset the children.
+
+      if (node.children) {
+        for (var i = 0; i < node.children.length; i++) {
+          node.children[i].height = 0;
+        }
+      }
+    }
+
+    if (group.smashDepth) {
+      node.depth = 0;
+
+      if (node.children) {
+        for (var _i = 0; _i < node.children.length; _i++) {
+          node.children[_i].depth = 0;
+        }
+      }
+    } // At this point, we've reset the TeX-like height and depth values.
+    // But the span still has an HTML line height.
+    // makeVList applies "display: table-cell", which prevents the browser
+    // from acting on that line height. So we'll call makeVList now.
+
+
+    var smashedNode = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: node
+      }]
+    }, options); // For spacing, TeX treats \hphantom as a math group (same spacing as ord).
+
+    return buildCommon.makeSpan(["mord"], [smashedNode], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mpadded", [buildMathML_buildGroup(group.body, options)]);
+
+    if (group.smashHeight) {
+      node.setAttribute("height", "0px");
+    }
+
+    if (group.smashDepth) {
+      node.setAttribute("depth", "0px");
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/sqrt.js
+
+
+
+
+
+
+
+defineFunction({
+  type: "sqrt",
+  names: ["\\sqrt"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser;
+    var index = optArgs[0];
+    var body = args[0];
+    return {
+      type: "sqrt",
+      mode: parser.mode,
+      body: body,
+      index: index
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Square roots are handled in the TeXbook pg. 443, Rule 11.
+    // First, we do the same steps as in overline to build the inner group
+    // and line
+    var inner = buildHTML_buildGroup(group.body, options.havingCrampedStyle());
+
+    if (inner.height === 0) {
+      // Render a small surd.
+      inner.height = options.fontMetrics().xHeight;
+    } // Some groups can return document fragments.  Handle those by wrapping
+    // them in a span.
+
+
+    inner = buildCommon.wrapFragment(inner, options); // Calculate the minimum size for the \surd delimiter
+
+    var metrics = options.fontMetrics();
+    var theta = metrics.defaultRuleThickness;
+    var phi = theta;
+
+    if (options.style.id < src_Style.TEXT.id) {
+      phi = options.fontMetrics().xHeight;
+    } // Calculate the clearance between the body and line
+
+
+    var lineClearance = theta + phi / 4;
+    var minDelimiterHeight = inner.height + inner.depth + lineClearance + theta; // Create a sqrt SVG of the required minimum size
+
+    var _delimiter$sqrtImage = delimiter.sqrtImage(minDelimiterHeight, options),
+        img = _delimiter$sqrtImage.span,
+        ruleWidth = _delimiter$sqrtImage.ruleWidth,
+        advanceWidth = _delimiter$sqrtImage.advanceWidth;
+
+    var delimDepth = img.height - ruleWidth; // Adjust the clearance based on the delimiter size
+
+    if (delimDepth > inner.height + inner.depth + lineClearance) {
+      lineClearance = (lineClearance + delimDepth - inner.height - inner.depth) / 2;
+    } // Shift the sqrt image
+
+
+    var imgShift = img.height - inner.height - lineClearance - ruleWidth;
+    inner.style.paddingLeft = advanceWidth + "em"; // Overlay the image and the argument.
+
+    var body = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: inner,
+        wrapperClasses: ["svg-align"]
+      }, {
+        type: "kern",
+        size: -(inner.height + imgShift)
+      }, {
+        type: "elem",
+        elem: img
+      }, {
+        type: "kern",
+        size: ruleWidth
+      }]
+    }, options);
+
+    if (!group.index) {
+      return buildCommon.makeSpan(["mord", "sqrt"], [body], options);
+    } else {
+      // Handle the optional root index
+      // The index is always in scriptscript style
+      var newOptions = options.havingStyle(src_Style.SCRIPTSCRIPT);
+      var rootm = buildHTML_buildGroup(group.index, newOptions, options); // The amount the index is shifted by. This is taken from the TeX
+      // source, in the definition of `\r@@t`.
+
+      var toShift = 0.6 * (body.height - body.depth); // Build a VList with the superscript shifted up correctly
+
+      var rootVList = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: -toShift,
+        children: [{
+          type: "elem",
+          elem: rootm
+        }]
+      }, options); // Add a class surrounding it so we can add on the appropriate
+      // kerning
+
+      var rootVListWrap = buildCommon.makeSpan(["root"], [rootVList]);
+      return buildCommon.makeSpan(["mord", "sqrt"], [rootVListWrap, body], options);
+    }
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var body = group.body,
+        index = group.index;
+    return index ? new mathMLTree.MathNode("mroot", [buildMathML_buildGroup(body, options), buildMathML_buildGroup(index, options)]) : new mathMLTree.MathNode("msqrt", [buildMathML_buildGroup(body, options)]);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/styling.js
+
+
+
+
+
+var styling_styleMap = {
+  "display": src_Style.DISPLAY,
+  "text": src_Style.TEXT,
+  "script": src_Style.SCRIPT,
+  "scriptscript": src_Style.SCRIPTSCRIPT
+};
+defineFunction({
+  type: "styling",
+  names: ["\\displaystyle", "\\textstyle", "\\scriptstyle", "\\scriptscriptstyle"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var breakOnTokenText = _ref.breakOnTokenText,
+        funcName = _ref.funcName,
+        parser = _ref.parser;
+    // parse out the implicit body
+    var body = parser.parseExpression(true, breakOnTokenText); // TODO: Refactor to avoid duplicating styleMap in multiple places (e.g.
+    // here and in buildHTML and de-dupe the enumeration of all the styles).
+    // $FlowFixMe: The names above exactly match the styles.
+
+    var style = funcName.slice(1, funcName.length - 5);
+    return {
+      type: "styling",
+      mode: parser.mode,
+      // Figure out what style to use by pulling out the style from
+      // the function name
+      style: style,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Style changes are handled in the TeXbook on pg. 442, Rule 3.
+    var newStyle = styling_styleMap[group.style];
+    var newOptions = options.havingStyle(newStyle).withFont('');
+    return sizingGroup(group.body, newOptions, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    // Figure out what style we're changing to.
+    var newStyle = styling_styleMap[group.style];
+    var newOptions = options.havingStyle(newStyle);
+    var inner = buildMathML_buildExpression(group.body, newOptions);
+    var node = new mathMLTree.MathNode("mstyle", inner);
+    var styleAttributes = {
+      "display": ["0", "true"],
+      "text": ["0", "false"],
+      "script": ["1", "false"],
+      "scriptscript": ["2", "false"]
+    };
+    var attr = styleAttributes[group.style];
+    node.setAttribute("scriptlevel", attr[0]);
+    node.setAttribute("displaystyle", attr[1]);
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/supsub.js
+
+
+
+
+
+
+
+
+
+
+
+
+
+/**
+ * Sometimes, groups perform special rules when they have superscripts or
+ * subscripts attached to them. This function lets the `supsub` group know that
+ * Sometimes, groups perform special rules when they have superscripts or
+ * its inner element should handle the superscripts and subscripts instead of
+ * handling them itself.
+ */
+var supsub_htmlBuilderDelegate = function htmlBuilderDelegate(group, options) {
+  var base = group.base;
+
+  if (!base) {
+    return null;
+  } else if (base.type === "op") {
+    // Operators handle supsubs differently when they have limits
+    // (e.g. `\displaystyle\sum_2^3`)
+    var delegate = base.limits && (options.style.size === src_Style.DISPLAY.size || base.alwaysHandleSupSub);
+    return delegate ? op_htmlBuilder : null;
+  } else if (base.type === "operatorname") {
+    var _delegate = base.alwaysHandleSupSub && (options.style.size === src_Style.DISPLAY.size || base.limits);
+
+    return _delegate ? operatorname_htmlBuilder : null;
+  } else if (base.type === "accent") {
+    return utils.isCharacterBox(base.base) ? accent_htmlBuilder : null;
+  } else if (base.type === "horizBrace") {
+    var isSup = !group.sub;
+    return isSup === base.isOver ? horizBrace_htmlBuilder : null;
+  } else {
+    return null;
+  }
+}; // Super scripts and subscripts, whose precise placement can depend on other
+// functions that precede them.
+
+
+defineFunctionBuilders({
+  type: "supsub",
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Superscript and subscripts are handled in the TeXbook on page
+    // 445-446, rules 18(a-f).
+    // Here is where we defer to the inner group if it should handle
+    // superscripts and subscripts itself.
+    var builderDelegate = supsub_htmlBuilderDelegate(group, options);
+
+    if (builderDelegate) {
+      return builderDelegate(group, options);
+    }
+
+    var valueBase = group.base,
+        valueSup = group.sup,
+        valueSub = group.sub;
+    var base = buildHTML_buildGroup(valueBase, options);
+    var supm;
+    var subm;
+    var metrics = options.fontMetrics(); // Rule 18a
+
+    var supShift = 0;
+    var subShift = 0;
+    var isCharacterBox = valueBase && utils.isCharacterBox(valueBase);
+
+    if (valueSup) {
+      var newOptions = options.havingStyle(options.style.sup());
+      supm = buildHTML_buildGroup(valueSup, newOptions, options);
+
+      if (!isCharacterBox) {
+        supShift = base.height - newOptions.fontMetrics().supDrop * newOptions.sizeMultiplier / options.sizeMultiplier;
+      }
+    }
+
+    if (valueSub) {
+      var _newOptions = options.havingStyle(options.style.sub());
+
+      subm = buildHTML_buildGroup(valueSub, _newOptions, options);
+
+      if (!isCharacterBox) {
+        subShift = base.depth + _newOptions.fontMetrics().subDrop * _newOptions.sizeMultiplier / options.sizeMultiplier;
+      }
+    } // Rule 18c
+
+
+    var minSupShift;
+
+    if (options.style === src_Style.DISPLAY) {
+      minSupShift = metrics.sup1;
+    } else if (options.style.cramped) {
+      minSupShift = metrics.sup3;
+    } else {
+      minSupShift = metrics.sup2;
+    } // scriptspace is a font-size-independent size, so scale it
+    // appropriately for use as the marginRight.
+
+
+    var multiplier = options.sizeMultiplier;
+    var marginRight = 0.5 / metrics.ptPerEm / multiplier + "em";
+    var marginLeft = null;
+
+    if (subm) {
+      // Subscripts shouldn't be shifted by the base's italic correction.
+      // Account for that by shifting the subscript back the appropriate
+      // amount. Note we only do this when the base is a single symbol.
+      var isOiint = group.base && group.base.type === "op" && group.base.name && (group.base.name === "\\oiint" || group.base.name === "\\oiiint");
+
+      if (base instanceof domTree_SymbolNode || isOiint) {
+        // $FlowFixMe
+        marginLeft = -base.italic + "em";
+      }
+    }
+
+    var supsub;
+
+    if (supm && subm) {
+      supShift = Math.max(supShift, minSupShift, supm.depth + 0.25 * metrics.xHeight);
+      subShift = Math.max(subShift, metrics.sub2);
+      var ruleWidth = metrics.defaultRuleThickness; // Rule 18e
+
+      var maxWidth = 4 * ruleWidth;
+
+      if (supShift - supm.depth - (subm.height - subShift) < maxWidth) {
+        subShift = maxWidth - (supShift - supm.depth) + subm.height;
+        var psi = 0.8 * metrics.xHeight - (supShift - supm.depth);
+
+        if (psi > 0) {
+          supShift += psi;
+          subShift -= psi;
+        }
+      }
+
+      var vlistElem = [{
+        type: "elem",
+        elem: subm,
+        shift: subShift,
+        marginRight: marginRight,
+        marginLeft: marginLeft
+      }, {
+        type: "elem",
+        elem: supm,
+        shift: -supShift,
+        marginRight: marginRight
+      }];
+      supsub = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: vlistElem
+      }, options);
+    } else if (subm) {
+      // Rule 18b
+      subShift = Math.max(subShift, metrics.sub1, subm.height - 0.8 * metrics.xHeight);
+      var _vlistElem = [{
+        type: "elem",
+        elem: subm,
+        marginLeft: marginLeft,
+        marginRight: marginRight
+      }];
+      supsub = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: subShift,
+        children: _vlistElem
+      }, options);
+    } else if (supm) {
+      // Rule 18c, d
+      supShift = Math.max(supShift, minSupShift, supm.depth + 0.25 * metrics.xHeight);
+      supsub = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: -supShift,
+        children: [{
+          type: "elem",
+          elem: supm,
+          marginRight: marginRight
+        }]
+      }, options);
+    } else {
+      throw new Error("supsub must have either sup or sub.");
+    } // Wrap the supsub vlist in a span.msupsub to reset text-align.
+
+
+    var mclass = getTypeOfDomTree(base, "right") || "mord";
+    return buildCommon.makeSpan([mclass], [base, buildCommon.makeSpan(["msupsub"], [supsub])], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    // Is the inner group a relevant horizonal brace?
+    var isBrace = false;
+    var isOver;
+    var isSup;
+
+    if (group.base && group.base.type === "horizBrace") {
+      isSup = !!group.sup;
+
+      if (isSup === group.base.isOver) {
+        isBrace = true;
+        isOver = group.base.isOver;
+      }
+    }
+
+    if (group.base && (group.base.type === "op" || group.base.type === "operatorname")) {
+      group.base.parentIsSupSub = true;
+    }
+
+    var children = [buildMathML_buildGroup(group.base, options)];
+
+    if (group.sub) {
+      children.push(buildMathML_buildGroup(group.sub, options));
+    }
+
+    if (group.sup) {
+      children.push(buildMathML_buildGroup(group.sup, options));
+    }
+
+    var nodeType;
+
+    if (isBrace) {
+      nodeType = isOver ? "mover" : "munder";
+    } else if (!group.sub) {
+      var base = group.base;
+
+      if (base && base.type === "op" && base.limits && (options.style === src_Style.DISPLAY || base.alwaysHandleSupSub)) {
+        nodeType = "mover";
+      } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub && (base.limits || options.style === src_Style.DISPLAY)) {
+        nodeType = "mover";
+      } else {
+        nodeType = "msup";
+      }
+    } else if (!group.sup) {
+      var _base = group.base;
+
+      if (_base && _base.type === "op" && _base.limits && (options.style === src_Style.DISPLAY || _base.alwaysHandleSupSub)) {
+        nodeType = "munder";
+      } else if (_base && _base.type === "operatorname" && _base.alwaysHandleSupSub && (_base.limits || options.style === src_Style.DISPLAY)) {
+        nodeType = "munder";
+      } else {
+        nodeType = "msub";
+      }
+    } else {
+      var _base2 = group.base;
+
+      if (_base2 && _base2.type === "op" && _base2.limits && options.style === src_Style.DISPLAY) {
+        nodeType = "munderover";
+      } else if (_base2 && _base2.type === "operatorname" && _base2.alwaysHandleSupSub && (options.style === src_Style.DISPLAY || _base2.limits)) {
+        nodeType = "munderover";
+      } else {
+        nodeType = "msubsup";
+      }
+    }
+
+    var node = new mathMLTree.MathNode(nodeType, children);
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/symbolsOp.js
+
+
+
+ // Operator ParseNodes created in Parser.js from symbol Groups in src/symbols.js.
+
+defineFunctionBuilders({
+  type: "atom",
+  htmlBuilder: function htmlBuilder(group, options) {
+    return buildCommon.mathsym(group.text, group.mode, options, ["m" + group.family]);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mo", [buildMathML_makeText(group.text, group.mode)]);
+
+    if (group.family === "bin") {
+      var variant = buildMathML_getVariant(group, options);
+
+      if (variant === "bold-italic") {
+        node.setAttribute("mathvariant", variant);
+      }
+    } else if (group.family === "punct") {
+      node.setAttribute("separator", "true");
+    } else if (group.family === "open" || group.family === "close") {
+      // Delims built here should not stretch vertically.
+      // See delimsizing.js for stretchy delims.
+      node.setAttribute("stretchy", "false");
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/symbolsOrd.js
+
+
+
+
+// "mathord" and "textord" ParseNodes created in Parser.js from symbol Groups in
+var defaultVariant = {
+  "mi": "italic",
+  "mn": "normal",
+  "mtext": "normal"
+};
+defineFunctionBuilders({
+  type: "mathord",
+  htmlBuilder: function htmlBuilder(group, options) {
+    return buildCommon.makeOrd(group, options, "mathord");
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mi", [buildMathML_makeText(group.text, group.mode, options)]);
+    var variant = buildMathML_getVariant(group, options) || "italic";
+
+    if (variant !== defaultVariant[node.type]) {
+      node.setAttribute("mathvariant", variant);
+    }
+
+    return node;
+  }
+});
+defineFunctionBuilders({
+  type: "textord",
+  htmlBuilder: function htmlBuilder(group, options) {
+    return buildCommon.makeOrd(group, options, "textord");
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var text = buildMathML_makeText(group.text, group.mode, options);
+    var variant = buildMathML_getVariant(group, options) || "normal";
+    var node;
+
+    if (group.mode === 'text') {
+      node = new mathMLTree.MathNode("mtext", [text]);
+    } else if (/[0-9]/.test(group.text)) {
+      // TODO(kevinb) merge adjacent <mn> nodes
+      // do it as a post processing step
+      node = new mathMLTree.MathNode("mn", [text]);
+    } else if (group.text === "\\prime") {
+      node = new mathMLTree.MathNode("mo", [text]);
+    } else {
+      node = new mathMLTree.MathNode("mi", [text]);
+    }
+
+    if (variant !== defaultVariant[node.type]) {
+      node.setAttribute("mathvariant", variant);
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/symbolsSpacing.js
+
+
+
+ // A map of CSS-based spacing functions to their CSS class.
+
+var cssSpace = {
+  "\\nobreak": "nobreak",
+  "\\allowbreak": "allowbreak"
+}; // A lookup table to determine whether a spacing function/symbol should be
+// treated like a regular space character.  If a symbol or command is a key
+// in this table, then it should be a regular space character.  Furthermore,
+// the associated value may have a `className` specifying an extra CSS class
+// to add to the created `span`.
+
+var regularSpace = {
+  " ": {},
+  "\\ ": {},
+  "~": {
+    className: "nobreak"
+  },
+  "\\space": {},
+  "\\nobreakspace": {
+    className: "nobreak"
+  }
+}; // ParseNode<"spacing"> created in Parser.js from the "spacing" symbol Groups in
+// src/symbols.js.
+
+defineFunctionBuilders({
+  type: "spacing",
+  htmlBuilder: function htmlBuilder(group, options) {
+    if (regularSpace.hasOwnProperty(group.text)) {
+      var className = regularSpace[group.text].className || ""; // Spaces are generated by adding an actual space. Each of these
+      // things has an entry in the symbols table, so these will be turned
+      // into appropriate outputs.
+
+      if (group.mode === "text") {
+        var ord = buildCommon.makeOrd(group, options, "textord");
+        ord.classes.push(className);
+        return ord;
+      } else {
+        return buildCommon.makeSpan(["mspace", className], [buildCommon.mathsym(group.text, group.mode, options)], options);
+      }
+    } else if (cssSpace.hasOwnProperty(group.text)) {
+      // Spaces based on just a CSS class.
+      return buildCommon.makeSpan(["mspace", cssSpace[group.text]], [], options);
+    } else {
+      throw new src_ParseError("Unknown type of space \"" + group.text + "\"");
+    }
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node;
+
+    if (regularSpace.hasOwnProperty(group.text)) {
+      node = new mathMLTree.MathNode("mtext", [new mathMLTree.TextNode("\xA0")]);
+    } else if (cssSpace.hasOwnProperty(group.text)) {
+      // CSS-based MathML spaces (\nobreak, \allowbreak) are ignored
+      return new mathMLTree.MathNode("mspace");
+    } else {
+      throw new src_ParseError("Unknown type of space \"" + group.text + "\"");
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/tag.js
+
+
+
+
+var tag_pad = function pad() {
+  var padNode = new mathMLTree.MathNode("mtd", []);
+  padNode.setAttribute("width", "50%");
+  return padNode;
+};
+
+defineFunctionBuilders({
+  type: "tag",
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var table = new mathMLTree.MathNode("mtable", [new mathMLTree.MathNode("mtr", [tag_pad(), new mathMLTree.MathNode("mtd", [buildExpressionRow(group.body, options)]), tag_pad(), new mathMLTree.MathNode("mtd", [buildExpressionRow(group.tag, options)])])]);
+    table.setAttribute("width", "100%");
+    return table; // TODO: Left-aligned tags.
+    // Currently, the group and options passed here do not contain
+    // enough info to set tag alignment. `leqno` is in Settings but it is
+    // not passed to Options. On the HTML side, leqno is
+    // set by a CSS class applied in buildTree.js. That would have worked
+    // in MathML if browsers supported <mlabeledtr>. Since they don't, we
+    // need to rewrite the way this function is called.
+  }
+});
+// CONCATENATED MODULE: ./src/functions/text.js
+
+
+
+ // Non-mathy text, possibly in a font
+
+var textFontFamilies = {
+  "\\text": undefined,
+  "\\textrm": "textrm",
+  "\\textsf": "textsf",
+  "\\texttt": "texttt",
+  "\\textnormal": "textrm"
+};
+var textFontWeights = {
+  "\\textbf": "textbf",
+  "\\textmd": "textmd"
+};
+var textFontShapes = {
+  "\\textit": "textit",
+  "\\textup": "textup"
+};
+
+var optionsWithFont = function optionsWithFont(group, options) {
+  var font = group.font; // Checks if the argument is a font family or a font style.
+
+  if (!font) {
+    return options;
+  } else if (textFontFamilies[font]) {
+    return options.withTextFontFamily(textFontFamilies[font]);
+  } else if (textFontWeights[font]) {
+    return options.withTextFontWeight(textFontWeights[font]);
+  } else {
+    return options.withTextFontShape(textFontShapes[font]);
+  }
+};
+
+defineFunction({
+  type: "text",
+  names: [// Font families
+  "\\text", "\\textrm", "\\textsf", "\\texttt", "\\textnormal", // Font weights
+  "\\textbf", "\\textmd", // Font Shapes
+  "\\textit", "\\textup"],
+  props: {
+    numArgs: 1,
+    argTypes: ["text"],
+    greediness: 2,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    return {
+      type: "text",
+      mode: parser.mode,
+      body: ordargument(body),
+      font: funcName
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var newOptions = optionsWithFont(group, options);
+    var inner = buildHTML_buildExpression(group.body, newOptions, true);
+    return buildCommon.makeSpan(["mord", "text"], buildCommon.tryCombineChars(inner), newOptions);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var newOptions = optionsWithFont(group, options);
+    return buildExpressionRow(group.body, newOptions);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/underline.js
+
+
+
+
+
+defineFunction({
+  type: "underline",
+  names: ["\\underline"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    return {
+      type: "underline",
+      mode: parser.mode,
+      body: args[0]
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Underlines are handled in the TeXbook pg 443, Rule 10.
+    // Build the inner group.
+    var innerGroup = buildHTML_buildGroup(group.body, options); // Create the line to go below the body
+
+    var line = buildCommon.makeLineSpan("underline-line", options); // Generate the vlist, with the appropriate kerns
+
+    var defaultRuleThickness = options.fontMetrics().defaultRuleThickness;
+    var vlist = buildCommon.makeVList({
+      positionType: "top",
+      positionData: innerGroup.height,
+      children: [{
+        type: "kern",
+        size: defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: line
+      }, {
+        type: "kern",
+        size: 3 * defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: innerGroup
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "underline"], [vlist], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var operator = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode("\u203E")]);
+    operator.setAttribute("stretchy", "true");
+    var node = new mathMLTree.MathNode("munder", [buildMathML_buildGroup(group.body, options), operator]);
+    node.setAttribute("accentunder", "true");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/verb.js
+
+
+
+
+defineFunction({
+  type: "verb",
+  names: ["\\verb"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(context, args, optArgs) {
+    // \verb and \verb* are dealt with directly in Parser.js.
+    // If we end up here, it's because of a failure to match the two delimiters
+    // in the regex in Lexer.js.  LaTeX raises the following error when \verb is
+    // terminated by end of line (or file).
+    throw new src_ParseError("\\verb ended by end of line instead of matching delimiter");
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var text = makeVerb(group);
+    var body = []; // \verb enters text mode and therefore is sized like \textstyle
+
+    var newOptions = options.havingStyle(options.style.text());
+
+    for (var i = 0; i < text.length; i++) {
+      var c = text[i];
+
+      if (c === '~') {
+        c = '\\textasciitilde';
+      }
+
+      body.push(buildCommon.makeSymbol(c, "Typewriter-Regular", group.mode, newOptions, ["mord", "texttt"]));
+    }
+
+    return buildCommon.makeSpan(["mord", "text"].concat(newOptions.sizingClasses(options)), buildCommon.tryCombineChars(body), newOptions);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var text = new mathMLTree.TextNode(makeVerb(group));
+    var node = new mathMLTree.MathNode("mtext", [text]);
+    node.setAttribute("mathvariant", "monospace");
+    return node;
+  }
+});
+/**
+ * Converts verb group into body string.
+ *
+ * \verb* replaces each space with an open box \u2423
+ * \verb replaces each space with a no-break space \xA0
+ */
+
+var makeVerb = function makeVerb(group) {
+  return group.body.replace(/ /g, group.star ? "\u2423" : '\xA0');
+};
+// CONCATENATED MODULE: ./src/functions.js
+/** Include this to ensure that all functions are defined. */
+
+var functions = _functions;
+/* harmony default export */ var src_functions = (functions); // TODO(kevinb): have functions return an object and call defineFunction with
+// that object in this file instead of relying on side-effects.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+// CONCATENATED MODULE: ./src/Lexer.js
+/**
+ * The Lexer class handles tokenizing the input in various ways. Since our
+ * parser expects us to be able to backtrack, the lexer allows lexing from any
+ * given starting point.
+ *
+ * Its main exposed function is the `lex` function, which takes a position to
+ * lex from and a type of token to lex. It defers to the appropriate `_innerLex`
+ * function.
+ *
+ * The various `_innerLex` functions perform the actual lexing of different
+ * kinds.
+ */
+
+
+
+
+/* The following tokenRegex
+ * - matches typical whitespace (but not NBSP etc.) using its first group
+ * - does not match any control character \x00-\x1f except whitespace
+ * - does not match a bare backslash
+ * - matches any ASCII character except those just mentioned
+ * - does not match the BMP private use area \uE000-\uF8FF
+ * - does not match bare surrogate code units
+ * - matches any BMP character except for those just described
+ * - matches any valid Unicode surrogate pair
+ * - matches a backslash followed by one or more letters
+ * - matches a backslash followed by any BMP character, including newline
+ * Just because the Lexer matches something doesn't mean it's valid input:
+ * If there is no matching function or symbol definition, the Parser will
+ * still reject the input.
+ */
+var spaceRegexString = "[ \r\n\t]";
+var controlWordRegexString = "\\\\[a-zA-Z@]+";
+var controlSymbolRegexString = "\\\\[^\uD800-\uDFFF]";
+var controlWordWhitespaceRegexString = "" + controlWordRegexString + spaceRegexString + "*";
+var controlWordWhitespaceRegex = new RegExp("^(" + controlWordRegexString + ")" + spaceRegexString + "*$");
+var combiningDiacriticalMarkString = "[\u0300-\u036F]";
+var combiningDiacriticalMarksEndRegex = new RegExp(combiningDiacriticalMarkString + "+$");
+var tokenRegexString = "(" + spaceRegexString + "+)|" + // whitespace
+"([!-\\[\\]-\u2027\u202A-\uD7FF\uF900-\uFFFF]" + ( // single codepoint
+combiningDiacriticalMarkString + "*") + // ...plus accents
+"|[\uD800-\uDBFF][\uDC00-\uDFFF]" + ( // surrogate pair
+combiningDiacriticalMarkString + "*") + // ...plus accents
+"|\\\\verb\\*([^]).*?\\3" + // \verb*
+"|\\\\verb([^*a-zA-Z]).*?\\4" + // \verb unstarred
+"|\\\\operatorname\\*" + ( // \operatorname*
+"|" + controlWordWhitespaceRegexString) + ( // \macroName + spaces
+"|" + controlSymbolRegexString + ")"); // \\, \', etc.
+
+/** Main Lexer class */
+
+var Lexer_Lexer =
+/*#__PURE__*/
+function () {
+  // category codes, only supports comment characters (14) for now
+  function Lexer(input, settings) {
+    this.input = void 0;
+    this.settings = void 0;
+    this.tokenRegex = void 0;
+    this.catcodes = void 0;
+    // Separate accents from characters
+    this.input = input;
+    this.settings = settings;
+    this.tokenRegex = new RegExp(tokenRegexString, 'g');
+    this.catcodes = {
+      "%": 14 // comment character
+
+    };
+  }
+
+  var _proto = Lexer.prototype;
+
+  _proto.setCatcode = function setCatcode(char, code) {
+    this.catcodes[char] = code;
+  }
+  /**
+   * This function lexes a single token.
+   */
+  ;
+
+  _proto.lex = function lex() {
+    var input = this.input;
+    var pos = this.tokenRegex.lastIndex;
+
+    if (pos === input.length) {
+      return new Token_Token("EOF", new SourceLocation(this, pos, pos));
+    }
+
+    var match = this.tokenRegex.exec(input);
+
+    if (match === null || match.index !== pos) {
+      throw new src_ParseError("Unexpected character: '" + input[pos] + "'", new Token_Token(input[pos], new SourceLocation(this, pos, pos + 1)));
+    }
+
+    var text = match[2] || " ";
+
+    if (this.catcodes[text] === 14) {
+      // comment character
+      var nlIndex = input.indexOf('\n', this.tokenRegex.lastIndex);
+
+      if (nlIndex === -1) {
+        this.tokenRegex.lastIndex = input.length; // EOF
+
+        this.settings.reportNonstrict("commentAtEnd", "% comment has no terminating newline; LaTeX would " + "fail because of commenting the end of math mode (e.g. $)");
+      } else {
+        this.tokenRegex.lastIndex = nlIndex + 1;
+      }
+
+      return this.lex();
+    } // Trim any trailing whitespace from control word match
+
+
+    var controlMatch = text.match(controlWordWhitespaceRegex);
+
+    if (controlMatch) {
+      text = controlMatch[1];
+    }
+
+    return new Token_Token(text, new SourceLocation(this, pos, this.tokenRegex.lastIndex));
+  };
+
+  return Lexer;
+}();
+
+
+// CONCATENATED MODULE: ./src/Namespace.js
+/**
+ * A `Namespace` refers to a space of nameable things like macros or lengths,
+ * which can be `set` either globally or local to a nested group, using an
+ * undo stack similar to how TeX implements this functionality.
+ * Performance-wise, `get` and local `set` take constant time, while global
+ * `set` takes time proportional to the depth of group nesting.
+ */
+
+
+var Namespace_Namespace =
+/*#__PURE__*/
+function () {
+  /**
+   * Both arguments are optional.  The first argument is an object of
+   * built-in mappings which never change.  The second argument is an object
+   * of initial (global-level) mappings, which will constantly change
+   * according to any global/top-level `set`s done.
+   */
+  function Namespace(builtins, globalMacros) {
+    if (builtins === void 0) {
+      builtins = {};
+    }
+
+    if (globalMacros === void 0) {
+      globalMacros = {};
+    }
+
+    this.current = void 0;
+    this.builtins = void 0;
+    this.undefStack = void 0;
+    this.current = globalMacros;
+    this.builtins = builtins;
+    this.undefStack = [];
+  }
+  /**
+   * Start a new nested group, affecting future local `set`s.
+   */
+
+
+  var _proto = Namespace.prototype;
+
+  _proto.beginGroup = function beginGroup() {
+    this.undefStack.push({});
+  }
+  /**
+   * End current nested group, restoring values before the group began.
+   */
+  ;
+
+  _proto.endGroup = function endGroup() {
+    if (this.undefStack.length === 0) {
+      throw new src_ParseError("Unbalanced namespace destruction: attempt " + "to pop global namespace; please report this as a bug");
+    }
+
+    var undefs = this.undefStack.pop();
+
+    for (var undef in undefs) {
+      if (undefs.hasOwnProperty(undef)) {
+        if (undefs[undef] === undefined) {
+          delete this.current[undef];
+        } else {
+          this.current[undef] = undefs[undef];
+        }
+      }
+    }
+  }
+  /**
+   * Detect whether `name` has a definition.  Equivalent to
+   * `get(name) != null`.
+   */
+  ;
+
+  _proto.has = function has(name) {
+    return this.current.hasOwnProperty(name) || this.builtins.hasOwnProperty(name);
+  }
+  /**
+   * Get the current value of a name, or `undefined` if there is no value.
+   *
+   * Note: Do not use `if (namespace.get(...))` to detect whether a macro
+   * is defined, as the definition may be the empty string which evaluates
+   * to `false` in JavaScript.  Use `if (namespace.get(...) != null)` or
+   * `if (namespace.has(...))`.
+   */
+  ;
+
+  _proto.get = function get(name) {
+    if (this.current.hasOwnProperty(name)) {
+      return this.current[name];
+    } else {
+      return this.builtins[name];
+    }
+  }
+  /**
+   * Set the current value of a name, and optionally set it globally too.
+   * Local set() sets the current value and (when appropriate) adds an undo
+   * operation to the undo stack.  Global set() may change the undo
+   * operation at every level, so takes time linear in their number.
+   */
+  ;
+
+  _proto.set = function set(name, value, global) {
+    if (global === void 0) {
+      global = false;
+    }
+
+    if (global) {
+      // Global set is equivalent to setting in all groups.  Simulate this
+      // by destroying any undos currently scheduled for this name,
+      // and adding an undo with the *new* value (in case it later gets
+      // locally reset within this environment).
+      for (var i = 0; i < this.undefStack.length; i++) {
+        delete this.undefStack[i][name];
+      }
+
+      if (this.undefStack.length > 0) {
+        this.undefStack[this.undefStack.length - 1][name] = value;
+      }
+    } else {
+      // Undo this set at end of this group (possibly to `undefined`),
+      // unless an undo is already in place, in which case that older
+      // value is the correct one.
+      var top = this.undefStack[this.undefStack.length - 1];
+
+      if (top && !top.hasOwnProperty(name)) {
+        top[name] = this.current[name];
+      }
+    }
+
+    this.current[name] = value;
+  };
+
+  return Namespace;
+}();
+
+
+// CONCATENATED MODULE: ./src/macros.js
+/**
+ * Predefined macros for KaTeX.
+ * This can be used to define some commands in terms of others.
+ */
+
+
+
+
+
+
+var builtinMacros = {};
+/* harmony default export */ var macros = (builtinMacros); // This function might one day accept an additional argument and do more things.
+
+function defineMacro(name, body) {
+  builtinMacros[name] = body;
+} //////////////////////////////////////////////////////////////////////
+// macro tools
+
+defineMacro("\\noexpand", function (context) {
+  // The expansion is the token itself; but that token is interpreted
+  // as if its meaning were ‘\relax’ if it is a control sequence that
+  // would ordinarily be expanded by TeX’s expansion rules.
+  var t = context.popToken();
+
+  if (context.isExpandable(t.text)) {
+    t.noexpand = true;
+    t.treatAsRelax = true;
+  }
+
+  return {
+    tokens: [t],
+    numArgs: 0
+  };
+});
+defineMacro("\\expandafter", function (context) {
+  // TeX first reads the token that comes immediately after \expandafter,
+  // without expanding it; let’s call this token t. Then TeX reads the
+  // token that comes after t (and possibly more tokens, if that token
+  // has an argument), replacing it by its expansion. Finally TeX puts
+  // t back in front of that expansion.
+  var t = context.popToken();
+  context.expandOnce(true); // expand only an expandable token
+
+  return {
+    tokens: [t],
+    numArgs: 0
+  };
+}); // LaTeX's \@firstoftwo{#1}{#2} expands to #1, skipping #2
+// TeX source: \long\def\@firstoftwo#1#2{#1}
+
+defineMacro("\\@firstoftwo", function (context) {
+  var args = context.consumeArgs(2);
+  return {
+    tokens: args[0],
+    numArgs: 0
+  };
+}); // LaTeX's \@secondoftwo{#1}{#2} expands to #2, skipping #1
+// TeX source: \long\def\@secondoftwo#1#2{#2}
+
+defineMacro("\\@secondoftwo", function (context) {
+  var args = context.consumeArgs(2);
+  return {
+    tokens: args[1],
+    numArgs: 0
+  };
+}); // LaTeX's \@ifnextchar{#1}{#2}{#3} looks ahead to the next (unexpanded)
+// symbol that isn't a space, consuming any spaces but not consuming the
+// first nonspace character.  If that nonspace character matches #1, then
+// the macro expands to #2; otherwise, it expands to #3.
+
+defineMacro("\\@ifnextchar", function (context) {
+  var args = context.consumeArgs(3); // symbol, if, else
+
+  context.consumeSpaces();
+  var nextToken = context.future();
+
+  if (args[0].length === 1 && args[0][0].text === nextToken.text) {
+    return {
+      tokens: args[1],
+      numArgs: 0
+    };
+  } else {
+    return {
+      tokens: args[2],
+      numArgs: 0
+    };
+  }
+}); // LaTeX's \@ifstar{#1}{#2} looks ahead to the next (unexpanded) symbol.
+// If it is `*`, then it consumes the symbol, and the macro expands to #1;
+// otherwise, the macro expands to #2 (without consuming the symbol).
+// TeX source: \def\@ifstar#1{\@ifnextchar *{\@firstoftwo{#1}}}
+
+defineMacro("\\@ifstar", "\\@ifnextchar *{\\@firstoftwo{#1}}"); // LaTeX's \TextOrMath{#1}{#2} expands to #1 in text mode, #2 in math mode
+
+defineMacro("\\TextOrMath", function (context) {
+  var args = context.consumeArgs(2);
+
+  if (context.mode === 'text') {
+    return {
+      tokens: args[0],
+      numArgs: 0
+    };
+  } else {
+    return {
+      tokens: args[1],
+      numArgs: 0
+    };
+  }
+}); // Lookup table for parsing numbers in base 8 through 16
+
+var digitToNumber = {
+  "0": 0,
+  "1": 1,
+  "2": 2,
+  "3": 3,
+  "4": 4,
+  "5": 5,
+  "6": 6,
+  "7": 7,
+  "8": 8,
+  "9": 9,
+  "a": 10,
+  "A": 10,
+  "b": 11,
+  "B": 11,
+  "c": 12,
+  "C": 12,
+  "d": 13,
+  "D": 13,
+  "e": 14,
+  "E": 14,
+  "f": 15,
+  "F": 15
+}; // TeX \char makes a literal character (catcode 12) using the following forms:
+// (see The TeXBook, p. 43)
+//   \char123  -- decimal
+//   \char'123 -- octal
+//   \char"123 -- hex
+//   \char`x   -- character that can be written (i.e. isn't active)
+//   \char`\x  -- character that cannot be written (e.g. %)
+// These all refer to characters from the font, so we turn them into special
+// calls to a function \@char dealt with in the Parser.
+
+defineMacro("\\char", function (context) {
+  var token = context.popToken();
+  var base;
+  var number = '';
+
+  if (token.text === "'") {
+    base = 8;
+    token = context.popToken();
+  } else if (token.text === '"') {
+    base = 16;
+    token = context.popToken();
+  } else if (token.text === "`") {
+    token = context.popToken();
+
+    if (token.text[0] === "\\") {
+      number = token.text.charCodeAt(1);
+    } else if (token.text === "EOF") {
+      throw new src_ParseError("\\char` missing argument");
+    } else {
+      number = token.text.charCodeAt(0);
+    }
+  } else {
+    base = 10;
+  }
+
+  if (base) {
+    // Parse a number in the given base, starting with first `token`.
+    number = digitToNumber[token.text];
+
+    if (number == null || number >= base) {
+      throw new src_ParseError("Invalid base-" + base + " digit " + token.text);
+    }
+
+    var digit;
+
+    while ((digit = digitToNumber[context.future().text]) != null && digit < base) {
+      number *= base;
+      number += digit;
+      context.popToken();
+    }
+  }
+
+  return "\\@char{" + number + "}";
+}); // \newcommand{\macro}[args]{definition}
+// \renewcommand{\macro}[args]{definition}
+// TODO: Optional arguments: \newcommand{\macro}[args][default]{definition}
+
+var macros_newcommand = function newcommand(context, existsOK, nonexistsOK) {
+  var arg = context.consumeArgs(1)[0];
+
+  if (arg.length !== 1) {
+    throw new src_ParseError("\\newcommand's first argument must be a macro name");
+  }
+
+  var name = arg[0].text;
+  var exists = context.isDefined(name);
+
+  if (exists && !existsOK) {
+    throw new src_ParseError("\\newcommand{" + name + "} attempting to redefine " + (name + "; use \\renewcommand"));
+  }
+
+  if (!exists && !nonexistsOK) {
+    throw new src_ParseError("\\renewcommand{" + name + "} when command " + name + " " + "does not yet exist; use \\newcommand");
+  }
+
+  var numArgs = 0;
+  arg = context.consumeArgs(1)[0];
+
+  if (arg.length === 1 && arg[0].text === "[") {
+    var argText = '';
+    var token = context.expandNextToken();
+
+    while (token.text !== "]" && token.text !== "EOF") {
+      // TODO: Should properly expand arg, e.g., ignore {}s
+      argText += token.text;
+      token = context.expandNextToken();
+    }
+
+    if (!argText.match(/^\s*[0-9]+\s*$/)) {
+      throw new src_ParseError("Invalid number of arguments: " + argText);
+    }
+
+    numArgs = parseInt(argText);
+    arg = context.consumeArgs(1)[0];
+  } // Final arg is the expansion of the macro
+
+
+  context.macros.set(name, {
+    tokens: arg,
+    numArgs: numArgs
+  });
+  return '';
+};
+
+defineMacro("\\newcommand", function (context) {
+  return macros_newcommand(context, false, true);
+});
+defineMacro("\\renewcommand", function (context) {
+  return macros_newcommand(context, true, false);
+});
+defineMacro("\\providecommand", function (context) {
+  return macros_newcommand(context, true, true);
+}); // terminal (console) tools
+
+defineMacro("\\message", function (context) {
+  var arg = context.consumeArgs(1)[0]; // eslint-disable-next-line no-console
+
+  console.log(arg.reverse().map(function (token) {
+    return token.text;
+  }).join(""));
+  return '';
+});
+defineMacro("\\errmessage", function (context) {
+  var arg = context.consumeArgs(1)[0]; // eslint-disable-next-line no-console
+
+  console.error(arg.reverse().map(function (token) {
+    return token.text;
+  }).join(""));
+  return '';
+});
+defineMacro("\\show", function (context) {
+  var tok = context.popToken();
+  var name = tok.text; // eslint-disable-next-line no-console
+
+  console.log(tok, context.macros.get(name), src_functions[name], src_symbols.math[name], src_symbols.text[name]);
+  return '';
+}); //////////////////////////////////////////////////////////////////////
+// Grouping
+// \let\bgroup={ \let\egroup=}
+
+defineMacro("\\bgroup", "{");
+defineMacro("\\egroup", "}"); // Symbols from latex.ltx:
+// \def\lq{`}
+// \def\rq{'}
+// \def \aa {\r a}
+// \def \AA {\r A}
+
+defineMacro("\\lq", "`");
+defineMacro("\\rq", "'");
+defineMacro("\\aa", "\\r a");
+defineMacro("\\AA", "\\r A"); // Copyright (C) and registered (R) symbols. Use raw symbol in MathML.
+// \DeclareTextCommandDefault{\textcopyright}{\textcircled{c}}
+// \DeclareTextCommandDefault{\textregistered}{\textcircled{%
+//      \check@mathfonts\fontsize\sf@size\z@\math@fontsfalse\selectfont R}}
+// \DeclareRobustCommand{\copyright}{%
+//    \ifmmode{\nfss@text{\textcopyright}}\else\textcopyright\fi}
+
+defineMacro("\\textcopyright", "\\html@mathml{\\textcircled{c}}{\\char`©}");
+defineMacro("\\copyright", "\\TextOrMath{\\textcopyright}{\\text{\\textcopyright}}");
+defineMacro("\\textregistered", "\\html@mathml{\\textcircled{\\scriptsize R}}{\\char`®}"); // Characters omitted from Unicode range 1D400–1D7FF
+
+defineMacro("\u212C", "\\mathscr{B}"); // script
+
+defineMacro("\u2130", "\\mathscr{E}");
+defineMacro("\u2131", "\\mathscr{F}");
+defineMacro("\u210B", "\\mathscr{H}");
+defineMacro("\u2110", "\\mathscr{I}");
+defineMacro("\u2112", "\\mathscr{L}");
+defineMacro("\u2133", "\\mathscr{M}");
+defineMacro("\u211B", "\\mathscr{R}");
+defineMacro("\u212D", "\\mathfrak{C}"); // Fraktur
+
+defineMacro("\u210C", "\\mathfrak{H}");
+defineMacro("\u2128", "\\mathfrak{Z}"); // Define \Bbbk with a macro that works in both HTML and MathML.
+
+defineMacro("\\Bbbk", "\\Bbb{k}"); // Unicode middle dot
+// The KaTeX fonts do not contain U+00B7. Instead, \cdotp displays
+// the dot at U+22C5 and gives it punct spacing.
+
+defineMacro("\xB7", "\\cdotp"); // \llap and \rlap render their contents in text mode
+
+defineMacro("\\llap", "\\mathllap{\\textrm{#1}}");
+defineMacro("\\rlap", "\\mathrlap{\\textrm{#1}}");
+defineMacro("\\clap", "\\mathclap{\\textrm{#1}}"); // \not is defined by base/fontmath.ltx via
+// \DeclareMathSymbol{\not}{\mathrel}{symbols}{"36}
+// It's thus treated like a \mathrel, but defined by a symbol that has zero
+// width but extends to the right.  We use \rlap to get that spacing.
+// For MathML we write U+0338 here. buildMathML.js will then do the overlay.
+
+defineMacro("\\not", '\\html@mathml{\\mathrel{\\mathrlap\\@not}}{\\char"338}'); // Negated symbols from base/fontmath.ltx:
+// \def\neq{\not=} \let\ne=\neq
+// \DeclareRobustCommand
+//   \notin{\mathrel{\m@th\mathpalette\c@ncel\in}}
+// \def\c@ncel#1#2{\m@th\ooalign{$\hfil#1\mkern1mu/\hfil$\crcr$#1#2$}}
+
+defineMacro("\\neq", "\\html@mathml{\\mathrel{\\not=}}{\\mathrel{\\char`≠}}");
+defineMacro("\\ne", "\\neq");
+defineMacro("\u2260", "\\neq");
+defineMacro("\\notin", "\\html@mathml{\\mathrel{{\\in}\\mathllap{/\\mskip1mu}}}" + "{\\mathrel{\\char`∉}}");
+defineMacro("\u2209", "\\notin"); // Unicode stacked relations
+
+defineMacro("\u2258", "\\html@mathml{" + "\\mathrel{=\\kern{-1em}\\raisebox{0.4em}{$\\scriptsize\\frown$}}" + "}{\\mathrel{\\char`\u2258}}");
+defineMacro("\u2259", "\\html@mathml{\\stackrel{\\tiny\\wedge}{=}}{\\mathrel{\\char`\u2258}}");
+defineMacro("\u225A", "\\html@mathml{\\stackrel{\\tiny\\vee}{=}}{\\mathrel{\\char`\u225A}}");
+defineMacro("\u225B", "\\html@mathml{\\stackrel{\\scriptsize\\star}{=}}" + "{\\mathrel{\\char`\u225B}}");
+defineMacro("\u225D", "\\html@mathml{\\stackrel{\\tiny\\mathrm{def}}{=}}" + "{\\mathrel{\\char`\u225D}}");
+defineMacro("\u225E", "\\html@mathml{\\stackrel{\\tiny\\mathrm{m}}{=}}" + "{\\mathrel{\\char`\u225E}}");
+defineMacro("\u225F", "\\html@mathml{\\stackrel{\\tiny?}{=}}{\\mathrel{\\char`\u225F}}"); // Misc Unicode
+
+defineMacro("\u27C2", "\\perp");
+defineMacro("\u203C", "\\mathclose{!\\mkern-0.8mu!}");
+defineMacro("\u220C", "\\notni");
+defineMacro("\u231C", "\\ulcorner");
+defineMacro("\u231D", "\\urcorner");
+defineMacro("\u231E", "\\llcorner");
+defineMacro("\u231F", "\\lrcorner");
+defineMacro("\xA9", "\\copyright");
+defineMacro("\xAE", "\\textregistered");
+defineMacro("\uFE0F", "\\textregistered"); // The KaTeX fonts have corners at codepoints that don't match Unicode.
+// For MathML purposes, use the Unicode code point.
+
+defineMacro("\\ulcorner", "\\html@mathml{\\@ulcorner}{\\mathop{\\char\"231c}}");
+defineMacro("\\urcorner", "\\html@mathml{\\@urcorner}{\\mathop{\\char\"231d}}");
+defineMacro("\\llcorner", "\\html@mathml{\\@llcorner}{\\mathop{\\char\"231e}}");
+defineMacro("\\lrcorner", "\\html@mathml{\\@lrcorner}{\\mathop{\\char\"231f}}"); //////////////////////////////////////////////////////////////////////
+// LaTeX_2ε
+// \vdots{\vbox{\baselineskip4\p@  \lineskiplimit\z@
+// \kern6\p@\hbox{.}\hbox{.}\hbox{.}}}
+// We'll call \varvdots, which gets a glyph from symbols.js.
+// The zero-width rule gets us an equivalent to the vertical 6pt kern.
+
+defineMacro("\\vdots", "\\mathord{\\varvdots\\rule{0pt}{15pt}}");
+defineMacro("\u22EE", "\\vdots"); //////////////////////////////////////////////////////////////////////
+// amsmath.sty
+// http://mirrors.concertpass.com/tex-archive/macros/latex/required/amsmath/amsmath.pdf
+// Italic Greek capital letters.  AMS defines these with \DeclareMathSymbol,
+// but they are equivalent to \mathit{\Letter}.
+
+defineMacro("\\varGamma", "\\mathit{\\Gamma}");
+defineMacro("\\varDelta", "\\mathit{\\Delta}");
+defineMacro("\\varTheta", "\\mathit{\\Theta}");
+defineMacro("\\varLambda", "\\mathit{\\Lambda}");
+defineMacro("\\varXi", "\\mathit{\\Xi}");
+defineMacro("\\varPi", "\\mathit{\\Pi}");
+defineMacro("\\varSigma", "\\mathit{\\Sigma}");
+defineMacro("\\varUpsilon", "\\mathit{\\Upsilon}");
+defineMacro("\\varPhi", "\\mathit{\\Phi}");
+defineMacro("\\varPsi", "\\mathit{\\Psi}");
+defineMacro("\\varOmega", "\\mathit{\\Omega}"); //\newcommand{\substack}[1]{\subarray{c}#1\endsubarray}
+
+defineMacro("\\substack", "\\begin{subarray}{c}#1\\end{subarray}"); // \renewcommand{\colon}{\nobreak\mskip2mu\mathpunct{}\nonscript
+// \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
+
+defineMacro("\\colon", "\\nobreak\\mskip2mu\\mathpunct{}" + "\\mathchoice{\\mkern-3mu}{\\mkern-3mu}{}{}{:}\\mskip6mu"); // \newcommand{\boxed}[1]{\fbox{\m@th$\displaystyle#1$}}
+
+defineMacro("\\boxed", "\\fbox{$\\displaystyle{#1}$}"); // \def\iff{\DOTSB\;\Longleftrightarrow\;}
+// \def\implies{\DOTSB\;\Longrightarrow\;}
+// \def\impliedby{\DOTSB\;\Longleftarrow\;}
+
+defineMacro("\\iff", "\\DOTSB\\;\\Longleftrightarrow\\;");
+defineMacro("\\implies", "\\DOTSB\\;\\Longrightarrow\\;");
+defineMacro("\\impliedby", "\\DOTSB\\;\\Longleftarrow\\;"); // AMSMath's automatic \dots, based on \mdots@@ macro.
+
+var dotsByToken = {
+  ',': '\\dotsc',
+  '\\not': '\\dotsb',
+  // \keybin@ checks for the following:
+  '+': '\\dotsb',
+  '=': '\\dotsb',
+  '<': '\\dotsb',
+  '>': '\\dotsb',
+  '-': '\\dotsb',
+  '*': '\\dotsb',
+  ':': '\\dotsb',
+  // Symbols whose definition starts with \DOTSB:
+  '\\DOTSB': '\\dotsb',
+  '\\coprod': '\\dotsb',
+  '\\bigvee': '\\dotsb',
+  '\\bigwedge': '\\dotsb',
+  '\\biguplus': '\\dotsb',
+  '\\bigcap': '\\dotsb',
+  '\\bigcup': '\\dotsb',
+  '\\prod': '\\dotsb',
+  '\\sum': '\\dotsb',
+  '\\bigotimes': '\\dotsb',
+  '\\bigoplus': '\\dotsb',
+  '\\bigodot': '\\dotsb',
+  '\\bigsqcup': '\\dotsb',
+  '\\And': '\\dotsb',
+  '\\longrightarrow': '\\dotsb',
+  '\\Longrightarrow': '\\dotsb',
+  '\\longleftarrow': '\\dotsb',
+  '\\Longleftarrow': '\\dotsb',
+  '\\longleftrightarrow': '\\dotsb',
+  '\\Longleftrightarrow': '\\dotsb',
+  '\\mapsto': '\\dotsb',
+  '\\longmapsto': '\\dotsb',
+  '\\hookrightarrow': '\\dotsb',
+  '\\doteq': '\\dotsb',
+  // Symbols whose definition starts with \mathbin:
+  '\\mathbin': '\\dotsb',
+  // Symbols whose definition starts with \mathrel:
+  '\\mathrel': '\\dotsb',
+  '\\relbar': '\\dotsb',
+  '\\Relbar': '\\dotsb',
+  '\\xrightarrow': '\\dotsb',
+  '\\xleftarrow': '\\dotsb',
+  // Symbols whose definition starts with \DOTSI:
+  '\\DOTSI': '\\dotsi',
+  '\\int': '\\dotsi',
+  '\\oint': '\\dotsi',
+  '\\iint': '\\dotsi',
+  '\\iiint': '\\dotsi',
+  '\\iiiint': '\\dotsi',
+  '\\idotsint': '\\dotsi',
+  // Symbols whose definition starts with \DOTSX:
+  '\\DOTSX': '\\dotsx'
+};
+defineMacro("\\dots", function (context) {
+  // TODO: If used in text mode, should expand to \textellipsis.
+  // However, in KaTeX, \textellipsis and \ldots behave the same
+  // (in text mode), and it's unlikely we'd see any of the math commands
+  // that affect the behavior of \dots when in text mode.  So fine for now
+  // (until we support \ifmmode ... \else ... \fi).
+  var thedots = '\\dotso';
+  var next = context.expandAfterFuture().text;
+
+  if (next in dotsByToken) {
+    thedots = dotsByToken[next];
+  } else if (next.substr(0, 4) === '\\not') {
+    thedots = '\\dotsb';
+  } else if (next in src_symbols.math) {
+    if (utils.contains(['bin', 'rel'], src_symbols.math[next].group)) {
+      thedots = '\\dotsb';
+    }
+  }
+
+  return thedots;
+});
+var spaceAfterDots = {
+  // \rightdelim@ checks for the following:
+  ')': true,
+  ']': true,
+  '\\rbrack': true,
+  '\\}': true,
+  '\\rbrace': true,
+  '\\rangle': true,
+  '\\rceil': true,
+  '\\rfloor': true,
+  '\\rgroup': true,
+  '\\rmoustache': true,
+  '\\right': true,
+  '\\bigr': true,
+  '\\biggr': true,
+  '\\Bigr': true,
+  '\\Biggr': true,
+  // \extra@ also tests for the following:
+  '$': true,
+  // \extrap@ checks for the following:
+  ';': true,
+  '.': true,
+  ',': true
+};
+defineMacro("\\dotso", function (context) {
+  var next = context.future().text;
+
+  if (next in spaceAfterDots) {
+    return "\\ldots\\,";
+  } else {
+    return "\\ldots";
+  }
+});
+defineMacro("\\dotsc", function (context) {
+  var next = context.future().text; // \dotsc uses \extra@ but not \extrap@, instead specially checking for
+  // ';' and '.', but doesn't check for ','.
+
+  if (next in spaceAfterDots && next !== ',') {
+    return "\\ldots\\,";
+  } else {
+    return "\\ldots";
+  }
+});
+defineMacro("\\cdots", function (context) {
+  var next = context.future().text;
+
+  if (next in spaceAfterDots) {
+    return "\\@cdots\\,";
+  } else {
+    return "\\@cdots";
+  }
+});
+defineMacro("\\dotsb", "\\cdots");
+defineMacro("\\dotsm", "\\cdots");
+defineMacro("\\dotsi", "\\!\\cdots"); // amsmath doesn't actually define \dotsx, but \dots followed by a macro
+// starting with \DOTSX implies \dotso, and then \extra@ detects this case
+// and forces the added `\,`.
+
+defineMacro("\\dotsx", "\\ldots\\,"); // \let\DOTSI\relax
+// \let\DOTSB\relax
+// \let\DOTSX\relax
+
+defineMacro("\\DOTSI", "\\relax");
+defineMacro("\\DOTSB", "\\relax");
+defineMacro("\\DOTSX", "\\relax"); // Spacing, based on amsmath.sty's override of LaTeX defaults
+// \DeclareRobustCommand{\tmspace}[3]{%
+//   \ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
+
+defineMacro("\\tmspace", "\\TextOrMath{\\kern#1#3}{\\mskip#1#2}\\relax"); // \renewcommand{\,}{\tmspace+\thinmuskip{.1667em}}
+// TODO: math mode should use \thinmuskip
+
+defineMacro("\\,", "\\tmspace+{3mu}{.1667em}"); // \let\thinspace\,
+
+defineMacro("\\thinspace", "\\,"); // \def\>{\mskip\medmuskip}
+// \renewcommand{\:}{\tmspace+\medmuskip{.2222em}}
+// TODO: \> and math mode of \: should use \medmuskip = 4mu plus 2mu minus 4mu
+
+defineMacro("\\>", "\\mskip{4mu}");
+defineMacro("\\:", "\\tmspace+{4mu}{.2222em}"); // \let\medspace\:
+
+defineMacro("\\medspace", "\\:"); // \renewcommand{\;}{\tmspace+\thickmuskip{.2777em}}
+// TODO: math mode should use \thickmuskip = 5mu plus 5mu
+
+defineMacro("\\;", "\\tmspace+{5mu}{.2777em}"); // \let\thickspace\;
+
+defineMacro("\\thickspace", "\\;"); // \renewcommand{\!}{\tmspace-\thinmuskip{.1667em}}
+// TODO: math mode should use \thinmuskip
+
+defineMacro("\\!", "\\tmspace-{3mu}{.1667em}"); // \let\negthinspace\!
+
+defineMacro("\\negthinspace", "\\!"); // \newcommand{\negmedspace}{\tmspace-\medmuskip{.2222em}}
+// TODO: math mode should use \medmuskip
+
+defineMacro("\\negmedspace", "\\tmspace-{4mu}{.2222em}"); // \newcommand{\negthickspace}{\tmspace-\thickmuskip{.2777em}}
+// TODO: math mode should use \thickmuskip
+
+defineMacro("\\negthickspace", "\\tmspace-{5mu}{.277em}"); // \def\enspace{\kern.5em }
+
+defineMacro("\\enspace", "\\kern.5em "); // \def\enskip{\hskip.5em\relax}
+
+defineMacro("\\enskip", "\\hskip.5em\\relax"); // \def\quad{\hskip1em\relax}
+
+defineMacro("\\quad", "\\hskip1em\\relax"); // \def\qquad{\hskip2em\relax}
+
+defineMacro("\\qquad", "\\hskip2em\\relax"); // \tag@in@display form of \tag
+
+defineMacro("\\tag", "\\@ifstar\\tag@literal\\tag@paren");
+defineMacro("\\tag@paren", "\\tag@literal{({#1})}");
+defineMacro("\\tag@literal", function (context) {
+  if (context.macros.get("\\df@tag")) {
+    throw new src_ParseError("Multiple \\tag");
+  }
+
+  return "\\gdef\\df@tag{\\text{#1}}";
+}); // \renewcommand{\bmod}{\nonscript\mskip-\medmuskip\mkern5mu\mathbin
+//   {\operator@font mod}\penalty900
+//   \mkern5mu\nonscript\mskip-\medmuskip}
+// \newcommand{\pod}[1]{\allowbreak
+//   \if@display\mkern18mu\else\mkern8mu\fi(#1)}
+// \renewcommand{\pmod}[1]{\pod{{\operator@font mod}\mkern6mu#1}}
+// \newcommand{\mod}[1]{\allowbreak\if@display\mkern18mu
+//   \else\mkern12mu\fi{\operator@font mod}\,\,#1}
+// TODO: math mode should use \medmuskip = 4mu plus 2mu minus 4mu
+
+defineMacro("\\bmod", "\\mathchoice{\\mskip1mu}{\\mskip1mu}{\\mskip5mu}{\\mskip5mu}" + "\\mathbin{\\rm mod}" + "\\mathchoice{\\mskip1mu}{\\mskip1mu}{\\mskip5mu}{\\mskip5mu}");
+defineMacro("\\pod", "\\allowbreak" + "\\mathchoice{\\mkern18mu}{\\mkern8mu}{\\mkern8mu}{\\mkern8mu}(#1)");
+defineMacro("\\pmod", "\\pod{{\\rm mod}\\mkern6mu#1}");
+defineMacro("\\mod", "\\allowbreak" + "\\mathchoice{\\mkern18mu}{\\mkern12mu}{\\mkern12mu}{\\mkern12mu}" + "{\\rm mod}\\,\\,#1"); // \pmb    --   A simulation of bold.
+// The version in ambsy.sty works by typesetting three copies of the argument
+// with small offsets. We use two copies. We omit the vertical offset because
+// of rendering problems that makeVList encounters in Safari.
+
+defineMacro("\\pmb", "\\html@mathml{" + "\\@binrel{#1}{\\mathrlap{#1}\\kern0.5px#1}}" + "{\\mathbf{#1}}"); //////////////////////////////////////////////////////////////////////
+// LaTeX source2e
+// \\ defaults to \newline, but changes to \cr within array environment
+
+defineMacro("\\\\", "\\newline"); // \def\TeX{T\kern-.1667em\lower.5ex\hbox{E}\kern-.125emX\@}
+// TODO: Doesn't normally work in math mode because \@ fails.  KaTeX doesn't
+// support \@ yet, so that's omitted, and we add \text so that the result
+// doesn't look funny in math mode.
+
+defineMacro("\\TeX", "\\textrm{\\html@mathml{" + "T\\kern-.1667em\\raisebox{-.5ex}{E}\\kern-.125emX" + "}{TeX}}"); // \DeclareRobustCommand{\LaTeX}{L\kern-.36em%
+//         {\sbox\z@ T%
+//          \vbox to\ht\z@{\hbox{\check@mathfonts
+//                               \fontsize\sf@size\z@
+//                               \math@fontsfalse\selectfont
+//                               A}%
+//                         \vss}%
+//         }%
+//         \kern-.15em%
+//         \TeX}
+// This code aligns the top of the A with the T (from the perspective of TeX's
+// boxes, though visually the A appears to extend above slightly).
+// We compute the corresponding \raisebox when A is rendered in \normalsize
+// \scriptstyle, which has a scale factor of 0.7 (see Options.js).
+
+var latexRaiseA = fontMetricsData['Main-Regular']["T".charCodeAt(0)][1] - 0.7 * fontMetricsData['Main-Regular']["A".charCodeAt(0)][1] + "em";
+defineMacro("\\LaTeX", "\\textrm{\\html@mathml{" + ("L\\kern-.36em\\raisebox{" + latexRaiseA + "}{\\scriptstyle A}") + "\\kern-.15em\\TeX}{LaTeX}}"); // New KaTeX logo based on tweaking LaTeX logo
+
+defineMacro("\\KaTeX", "\\textrm{\\html@mathml{" + ("K\\kern-.17em\\raisebox{" + latexRaiseA + "}{\\scriptstyle A}") + "\\kern-.15em\\TeX}{KaTeX}}"); // \DeclareRobustCommand\hspace{\@ifstar\@hspacer\@hspace}
+// \def\@hspace#1{\hskip  #1\relax}
+// \def\@hspacer#1{\vrule \@width\z@\nobreak
+//                 \hskip #1\hskip \z@skip}
+
+defineMacro("\\hspace", "\\@ifstar\\@hspacer\\@hspace");
+defineMacro("\\@hspace", "\\hskip #1\\relax");
+defineMacro("\\@hspacer", "\\rule{0pt}{0pt}\\hskip #1\\relax"); //////////////////////////////////////////////////////////////////////
+// mathtools.sty
+//\providecommand\ordinarycolon{:}
+
+defineMacro("\\ordinarycolon", ":"); //\def\vcentcolon{\mathrel{\mathop\ordinarycolon}}
+//TODO(edemaine): Not yet centered. Fix via \raisebox or #726
+
+defineMacro("\\vcentcolon", "\\mathrel{\\mathop\\ordinarycolon}"); // \providecommand*\dblcolon{\vcentcolon\mathrel{\mkern-.9mu}\vcentcolon}
+
+defineMacro("\\dblcolon", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-.9mu}\\vcentcolon}}" + "{\\mathop{\\char\"2237}}"); // \providecommand*\coloneqq{\vcentcolon\mathrel{\mkern-1.2mu}=}
+
+defineMacro("\\coloneqq", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}=}}" + "{\\mathop{\\char\"2254}}"); // ≔
+// \providecommand*\Coloneqq{\dblcolon\mathrel{\mkern-1.2mu}=}
+
+defineMacro("\\Coloneqq", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}=}}" + "{\\mathop{\\char\"2237\\char\"3d}}"); // \providecommand*\coloneq{\vcentcolon\mathrel{\mkern-1.2mu}\mathrel{-}}
+
+defineMacro("\\coloneq", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\mathrel{-}}}" + "{\\mathop{\\char\"3a\\char\"2212}}"); // \providecommand*\Coloneq{\dblcolon\mathrel{\mkern-1.2mu}\mathrel{-}}
+
+defineMacro("\\Coloneq", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\mathrel{-}}}" + "{\\mathop{\\char\"2237\\char\"2212}}"); // \providecommand*\eqqcolon{=\mathrel{\mkern-1.2mu}\vcentcolon}
+
+defineMacro("\\eqqcolon", "\\html@mathml{" + "\\mathrel{=\\mathrel{\\mkern-1.2mu}\\vcentcolon}}" + "{\\mathop{\\char\"2255}}"); // ≕
+// \providecommand*\Eqqcolon{=\mathrel{\mkern-1.2mu}\dblcolon}
+
+defineMacro("\\Eqqcolon", "\\html@mathml{" + "\\mathrel{=\\mathrel{\\mkern-1.2mu}\\dblcolon}}" + "{\\mathop{\\char\"3d\\char\"2237}}"); // \providecommand*\eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\vcentcolon}
+
+defineMacro("\\eqcolon", "\\html@mathml{" + "\\mathrel{\\mathrel{-}\\mathrel{\\mkern-1.2mu}\\vcentcolon}}" + "{\\mathop{\\char\"2239}}"); // \providecommand*\Eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\dblcolon}
+
+defineMacro("\\Eqcolon", "\\html@mathml{" + "\\mathrel{\\mathrel{-}\\mathrel{\\mkern-1.2mu}\\dblcolon}}" + "{\\mathop{\\char\"2212\\char\"2237}}"); // \providecommand*\colonapprox{\vcentcolon\mathrel{\mkern-1.2mu}\approx}
+
+defineMacro("\\colonapprox", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\approx}}" + "{\\mathop{\\char\"3a\\char\"2248}}"); // \providecommand*\Colonapprox{\dblcolon\mathrel{\mkern-1.2mu}\approx}
+
+defineMacro("\\Colonapprox", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\approx}}" + "{\\mathop{\\char\"2237\\char\"2248}}"); // \providecommand*\colonsim{\vcentcolon\mathrel{\mkern-1.2mu}\sim}
+
+defineMacro("\\colonsim", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\sim}}" + "{\\mathop{\\char\"3a\\char\"223c}}"); // \providecommand*\Colonsim{\dblcolon\mathrel{\mkern-1.2mu}\sim}
+
+defineMacro("\\Colonsim", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\sim}}" + "{\\mathop{\\char\"2237\\char\"223c}}"); // Some Unicode characters are implemented with macros to mathtools functions.
+
+defineMacro("\u2237", "\\dblcolon"); // ::
+
+defineMacro("\u2239", "\\eqcolon"); // -:
+
+defineMacro("\u2254", "\\coloneqq"); // :=
+
+defineMacro("\u2255", "\\eqqcolon"); // =:
+
+defineMacro("\u2A74", "\\Coloneqq"); // ::=
+//////////////////////////////////////////////////////////////////////
+// colonequals.sty
+// Alternate names for mathtools's macros:
+
+defineMacro("\\ratio", "\\vcentcolon");
+defineMacro("\\coloncolon", "\\dblcolon");
+defineMacro("\\colonequals", "\\coloneqq");
+defineMacro("\\coloncolonequals", "\\Coloneqq");
+defineMacro("\\equalscolon", "\\eqqcolon");
+defineMacro("\\equalscoloncolon", "\\Eqqcolon");
+defineMacro("\\colonminus", "\\coloneq");
+defineMacro("\\coloncolonminus", "\\Coloneq");
+defineMacro("\\minuscolon", "\\eqcolon");
+defineMacro("\\minuscoloncolon", "\\Eqcolon"); // \colonapprox name is same in mathtools and colonequals.
+
+defineMacro("\\coloncolonapprox", "\\Colonapprox"); // \colonsim name is same in mathtools and colonequals.
+
+defineMacro("\\coloncolonsim", "\\Colonsim"); // Additional macros, implemented by analogy with mathtools definitions:
+
+defineMacro("\\simcolon", "\\mathrel{\\sim\\mathrel{\\mkern-1.2mu}\\vcentcolon}");
+defineMacro("\\simcoloncolon", "\\mathrel{\\sim\\mathrel{\\mkern-1.2mu}\\dblcolon}");
+defineMacro("\\approxcolon", "\\mathrel{\\approx\\mathrel{\\mkern-1.2mu}\\vcentcolon}");
+defineMacro("\\approxcoloncolon", "\\mathrel{\\approx\\mathrel{\\mkern-1.2mu}\\dblcolon}"); // Present in newtxmath, pxfonts and txfonts
+
+defineMacro("\\notni", "\\html@mathml{\\not\\ni}{\\mathrel{\\char`\u220C}}");
+defineMacro("\\limsup", "\\DOTSB\\operatorname*{lim\\,sup}");
+defineMacro("\\liminf", "\\DOTSB\\operatorname*{lim\\,inf}"); //////////////////////////////////////////////////////////////////////
+// MathML alternates for KaTeX glyphs in the Unicode private area
+
+defineMacro("\\gvertneqq", "\\html@mathml{\\@gvertneqq}{\u2269}");
+defineMacro("\\lvertneqq", "\\html@mathml{\\@lvertneqq}{\u2268}");
+defineMacro("\\ngeqq", "\\html@mathml{\\@ngeqq}{\u2271}");
+defineMacro("\\ngeqslant", "\\html@mathml{\\@ngeqslant}{\u2271}");
+defineMacro("\\nleqq", "\\html@mathml{\\@nleqq}{\u2270}");
+defineMacro("\\nleqslant", "\\html@mathml{\\@nleqslant}{\u2270}");
+defineMacro("\\nshortmid", "\\html@mathml{\\@nshortmid}{∤}");
+defineMacro("\\nshortparallel", "\\html@mathml{\\@nshortparallel}{∦}");
+defineMacro("\\nsubseteqq", "\\html@mathml{\\@nsubseteqq}{\u2288}");
+defineMacro("\\nsupseteqq", "\\html@mathml{\\@nsupseteqq}{\u2289}");
+defineMacro("\\varsubsetneq", "\\html@mathml{\\@varsubsetneq}{⊊}");
+defineMacro("\\varsubsetneqq", "\\html@mathml{\\@varsubsetneqq}{⫋}");
+defineMacro("\\varsupsetneq", "\\html@mathml{\\@varsupsetneq}{⊋}");
+defineMacro("\\varsupsetneqq", "\\html@mathml{\\@varsupsetneqq}{⫌}");
+defineMacro("\\imath", "\\html@mathml{\\@imath}{\u0131}");
+defineMacro("\\jmath", "\\html@mathml{\\@jmath}{\u0237}"); //////////////////////////////////////////////////////////////////////
+// stmaryrd and semantic
+// The stmaryrd and semantic packages render the next four items by calling a
+// glyph. Those glyphs do not exist in the KaTeX fonts. Hence the macros.
+
+defineMacro("\\llbracket", "\\html@mathml{" + "\\mathopen{[\\mkern-3.2mu[}}" + "{\\mathopen{\\char`\u27E6}}");
+defineMacro("\\rrbracket", "\\html@mathml{" + "\\mathclose{]\\mkern-3.2mu]}}" + "{\\mathclose{\\char`\u27E7}}");
+defineMacro("\u27E6", "\\llbracket"); // blackboard bold [
+
+defineMacro("\u27E7", "\\rrbracket"); // blackboard bold ]
+
+defineMacro("\\lBrace", "\\html@mathml{" + "\\mathopen{\\{\\mkern-3.2mu[}}" + "{\\mathopen{\\char`\u2983}}");
+defineMacro("\\rBrace", "\\html@mathml{" + "\\mathclose{]\\mkern-3.2mu\\}}}" + "{\\mathclose{\\char`\u2984}}");
+defineMacro("\u2983", "\\lBrace"); // blackboard bold {
+
+defineMacro("\u2984", "\\rBrace"); // blackboard bold }
+// TODO: Create variable sized versions of the last two items. I believe that
+// will require new font glyphs.
+// The stmaryrd function `\minuso` provides a "Plimsoll" symbol that
+// superimposes the characters \circ and \mathminus. Used in chemistry.
+
+defineMacro("\\minuso", "\\mathbin{\\html@mathml{" + "{\\mathrlap{\\mathchoice{\\kern{0.145em}}{\\kern{0.145em}}" + "{\\kern{0.1015em}}{\\kern{0.0725em}}\\circ}{-}}}" + "{\\char`⦵}}");
+defineMacro("⦵", "\\minuso"); //////////////////////////////////////////////////////////////////////
+// texvc.sty
+// The texvc package contains macros available in mediawiki pages.
+// We omit the functions deprecated at
+// https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax
+// We also omit texvc's \O, which conflicts with \text{\O}
+
+defineMacro("\\darr", "\\downarrow");
+defineMacro("\\dArr", "\\Downarrow");
+defineMacro("\\Darr", "\\Downarrow");
+defineMacro("\\lang", "\\langle");
+defineMacro("\\rang", "\\rangle");
+defineMacro("\\uarr", "\\uparrow");
+defineMacro("\\uArr", "\\Uparrow");
+defineMacro("\\Uarr", "\\Uparrow");
+defineMacro("\\N", "\\mathbb{N}");
+defineMacro("\\R", "\\mathbb{R}");
+defineMacro("\\Z", "\\mathbb{Z}");
+defineMacro("\\alef", "\\aleph");
+defineMacro("\\alefsym", "\\aleph");
+defineMacro("\\Alpha", "\\mathrm{A}");
+defineMacro("\\Beta", "\\mathrm{B}");
+defineMacro("\\bull", "\\bullet");
+defineMacro("\\Chi", "\\mathrm{X}");
+defineMacro("\\clubs", "\\clubsuit");
+defineMacro("\\cnums", "\\mathbb{C}");
+defineMacro("\\Complex", "\\mathbb{C}");
+defineMacro("\\Dagger", "\\ddagger");
+defineMacro("\\diamonds", "\\diamondsuit");
+defineMacro("\\empty", "\\emptyset");
+defineMacro("\\Epsilon", "\\mathrm{E}");
+defineMacro("\\Eta", "\\mathrm{H}");
+defineMacro("\\exist", "\\exists");
+defineMacro("\\harr", "\\leftrightarrow");
+defineMacro("\\hArr", "\\Leftrightarrow");
+defineMacro("\\Harr", "\\Leftrightarrow");
+defineMacro("\\hearts", "\\heartsuit");
+defineMacro("\\image", "\\Im");
+defineMacro("\\infin", "\\infty");
+defineMacro("\\Iota", "\\mathrm{I}");
+defineMacro("\\isin", "\\in");
+defineMacro("\\Kappa", "\\mathrm{K}");
+defineMacro("\\larr", "\\leftarrow");
+defineMacro("\\lArr", "\\Leftarrow");
+defineMacro("\\Larr", "\\Leftarrow");
+defineMacro("\\lrarr", "\\leftrightarrow");
+defineMacro("\\lrArr", "\\Leftrightarrow");
+defineMacro("\\Lrarr", "\\Leftrightarrow");
+defineMacro("\\Mu", "\\mathrm{M}");
+defineMacro("\\natnums", "\\mathbb{N}");
+defineMacro("\\Nu", "\\mathrm{N}");
+defineMacro("\\Omicron", "\\mathrm{O}");
+defineMacro("\\plusmn", "\\pm");
+defineMacro("\\rarr", "\\rightarrow");
+defineMacro("\\rArr", "\\Rightarrow");
+defineMacro("\\Rarr", "\\Rightarrow");
+defineMacro("\\real", "\\Re");
+defineMacro("\\reals", "\\mathbb{R}");
+defineMacro("\\Reals", "\\mathbb{R}");
+defineMacro("\\Rho", "\\mathrm{P}");
+defineMacro("\\sdot", "\\cdot");
+defineMacro("\\sect", "\\S");
+defineMacro("\\spades", "\\spadesuit");
+defineMacro("\\sub", "\\subset");
+defineMacro("\\sube", "\\subseteq");
+defineMacro("\\supe", "\\supseteq");
+defineMacro("\\Tau", "\\mathrm{T}");
+defineMacro("\\thetasym", "\\vartheta"); // TODO: defineMacro("\\varcoppa", "\\\mbox{\\coppa}");
+
+defineMacro("\\weierp", "\\wp");
+defineMacro("\\Zeta", "\\mathrm{Z}"); //////////////////////////////////////////////////////////////////////
+// statmath.sty
+// https://ctan.math.illinois.edu/macros/latex/contrib/statmath/statmath.pdf
+
+defineMacro("\\argmin", "\\DOTSB\\operatorname*{arg\\,min}");
+defineMacro("\\argmax", "\\DOTSB\\operatorname*{arg\\,max}");
+defineMacro("\\plim", "\\DOTSB\\mathop{\\operatorname{plim}}\\limits"); //////////////////////////////////////////////////////////////////////
+// braket.sty
+// http://ctan.math.washington.edu/tex-archive/macros/latex/contrib/braket/braket.pdf
+
+defineMacro("\\bra", "\\mathinner{\\langle{#1}|}");
+defineMacro("\\ket", "\\mathinner{|{#1}\\rangle}");
+defineMacro("\\braket", "\\mathinner{\\langle{#1}\\rangle}");
+defineMacro("\\Bra", "\\left\\langle#1\\right|");
+defineMacro("\\Ket", "\\left|#1\\right\\rangle"); // Custom Khan Academy colors, should be moved to an optional package
+
+defineMacro("\\blue", "\\textcolor{##6495ed}{#1}");
+defineMacro("\\orange", "\\textcolor{##ffa500}{#1}");
+defineMacro("\\pink", "\\textcolor{##ff00af}{#1}");
+defineMacro("\\red", "\\textcolor{##df0030}{#1}");
+defineMacro("\\green", "\\textcolor{##28ae7b}{#1}");
+defineMacro("\\gray", "\\textcolor{gray}{#1}");
+defineMacro("\\purple", "\\textcolor{##9d38bd}{#1}");
+defineMacro("\\blueA", "\\textcolor{##ccfaff}{#1}");
+defineMacro("\\blueB", "\\textcolor{##80f6ff}{#1}");
+defineMacro("\\blueC", "\\textcolor{##63d9ea}{#1}");
+defineMacro("\\blueD", "\\textcolor{##11accd}{#1}");
+defineMacro("\\blueE", "\\textcolor{##0c7f99}{#1}");
+defineMacro("\\tealA", "\\textcolor{##94fff5}{#1}");
+defineMacro("\\tealB", "\\textcolor{##26edd5}{#1}");
+defineMacro("\\tealC", "\\textcolor{##01d1c1}{#1}");
+defineMacro("\\tealD", "\\textcolor{##01a995}{#1}");
+defineMacro("\\tealE", "\\textcolor{##208170}{#1}");
+defineMacro("\\greenA", "\\textcolor{##b6ffb0}{#1}");
+defineMacro("\\greenB", "\\textcolor{##8af281}{#1}");
+defineMacro("\\greenC", "\\textcolor{##74cf70}{#1}");
+defineMacro("\\greenD", "\\textcolor{##1fab54}{#1}");
+defineMacro("\\greenE", "\\textcolor{##0d923f}{#1}");
+defineMacro("\\goldA", "\\textcolor{##ffd0a9}{#1}");
+defineMacro("\\goldB", "\\textcolor{##ffbb71}{#1}");
+defineMacro("\\goldC", "\\textcolor{##ff9c39}{#1}");
+defineMacro("\\goldD", "\\textcolor{##e07d10}{#1}");
+defineMacro("\\goldE", "\\textcolor{##a75a05}{#1}");
+defineMacro("\\redA", "\\textcolor{##fca9a9}{#1}");
+defineMacro("\\redB", "\\textcolor{##ff8482}{#1}");
+defineMacro("\\redC", "\\textcolor{##f9685d}{#1}");
+defineMacro("\\redD", "\\textcolor{##e84d39}{#1}");
+defineMacro("\\redE", "\\textcolor{##bc2612}{#1}");
+defineMacro("\\maroonA", "\\textcolor{##ffbde0}{#1}");
+defineMacro("\\maroonB", "\\textcolor{##ff92c6}{#1}");
+defineMacro("\\maroonC", "\\textcolor{##ed5fa6}{#1}");
+defineMacro("\\maroonD", "\\textcolor{##ca337c}{#1}");
+defineMacro("\\maroonE", "\\textcolor{##9e034e}{#1}");
+defineMacro("\\purpleA", "\\textcolor{##ddd7ff}{#1}");
+defineMacro("\\purpleB", "\\textcolor{##c6b9fc}{#1}");
+defineMacro("\\purpleC", "\\textcolor{##aa87ff}{#1}");
+defineMacro("\\purpleD", "\\textcolor{##7854ab}{#1}");
+defineMacro("\\purpleE", "\\textcolor{##543b78}{#1}");
+defineMacro("\\mintA", "\\textcolor{##f5f9e8}{#1}");
+defineMacro("\\mintB", "\\textcolor{##edf2df}{#1}");
+defineMacro("\\mintC", "\\textcolor{##e0e5cc}{#1}");
+defineMacro("\\grayA", "\\textcolor{##f6f7f7}{#1}");
+defineMacro("\\grayB", "\\textcolor{##f0f1f2}{#1}");
+defineMacro("\\grayC", "\\textcolor{##e3e5e6}{#1}");
+defineMacro("\\grayD", "\\textcolor{##d6d8da}{#1}");
+defineMacro("\\grayE", "\\textcolor{##babec2}{#1}");
+defineMacro("\\grayF", "\\textcolor{##888d93}{#1}");
+defineMacro("\\grayG", "\\textcolor{##626569}{#1}");
+defineMacro("\\grayH", "\\textcolor{##3b3e40}{#1}");
+defineMacro("\\grayI", "\\textcolor{##21242c}{#1}");
+defineMacro("\\kaBlue", "\\textcolor{##314453}{#1}");
+defineMacro("\\kaGreen", "\\textcolor{##71B307}{#1}");
+// CONCATENATED MODULE: ./src/MacroExpander.js
+/**
+ * This file contains the “gullet” where macros are expanded
+ * until only non-macro tokens remain.
+ */
+
+
+
+
+
+
+
+// List of commands that act like macros but aren't defined as a macro,
+// function, or symbol.  Used in `isDefined`.
+var implicitCommands = {
+  "\\relax": true,
+  // MacroExpander.js
+  "^": true,
+  // Parser.js
+  "_": true,
+  // Parser.js
+  "\\limits": true,
+  // Parser.js
+  "\\nolimits": true // Parser.js
+
+};
+
+var MacroExpander_MacroExpander =
+/*#__PURE__*/
+function () {
+  function MacroExpander(input, settings, mode) {
+    this.settings = void 0;
+    this.expansionCount = void 0;
+    this.lexer = void 0;
+    this.macros = void 0;
+    this.stack = void 0;
+    this.mode = void 0;
+    this.settings = settings;
+    this.expansionCount = 0;
+    this.feed(input); // Make new global namespace
+
+    this.macros = new Namespace_Namespace(macros, settings.macros);
+    this.mode = mode;
+    this.stack = []; // contains tokens in REVERSE order
+  }
+  /**
+   * Feed a new input string to the same MacroExpander
+   * (with existing macros etc.).
+   */
+
+
+  var _proto = MacroExpander.prototype;
+
+  _proto.feed = function feed(input) {
+    this.lexer = new Lexer_Lexer(input, this.settings);
+  }
+  /**
+   * Switches between "text" and "math" modes.
+   */
+  ;
+
+  _proto.switchMode = function switchMode(newMode) {
+    this.mode = newMode;
+  }
+  /**
+   * Start a new group nesting within all namespaces.
+   */
+  ;
+
+  _proto.beginGroup = function beginGroup() {
+    this.macros.beginGroup();
+  }
+  /**
+   * End current group nesting within all namespaces.
+   */
+  ;
+
+  _proto.endGroup = function endGroup() {
+    this.macros.endGroup();
+  }
+  /**
+   * Returns the topmost token on the stack, without expanding it.
+   * Similar in behavior to TeX's `\futurelet`.
+   */
+  ;
+
+  _proto.future = function future() {
+    if (this.stack.length === 0) {
+      this.pushToken(this.lexer.lex());
+    }
+
+    return this.stack[this.stack.length - 1];
+  }
+  /**
+   * Remove and return the next unexpanded token.
+   */
+  ;
+
+  _proto.popToken = function popToken() {
+    this.future(); // ensure non-empty stack
+
+    return this.stack.pop();
+  }
+  /**
+   * Add a given token to the token stack.  In particular, this get be used
+   * to put back a token returned from one of the other methods.
+   */
+  ;
+
+  _proto.pushToken = function pushToken(token) {
+    this.stack.push(token);
+  }
+  /**
+   * Append an array of tokens to the token stack.
+   */
+  ;
+
+  _proto.pushTokens = function pushTokens(tokens) {
+    var _this$stack;
+
+    (_this$stack = this.stack).push.apply(_this$stack, tokens);
+  }
+  /**
+   * Consume all following space tokens, without expansion.
+   */
+  ;
+
+  _proto.consumeSpaces = function consumeSpaces() {
+    for (;;) {
+      var token = this.future();
+
+      if (token.text === " ") {
+        this.stack.pop();
+      } else {
+        break;
+      }
+    }
+  }
+  /**
+   * Consume the specified number of arguments from the token stream,
+   * and return the resulting array of arguments.
+   */
+  ;
+
+  _proto.consumeArgs = function consumeArgs(numArgs) {
+    var args = []; // obtain arguments, either single token or balanced {…} group
+
+    for (var i = 0; i < numArgs; ++i) {
+      this.consumeSpaces(); // ignore spaces before each argument
+
+      var startOfArg = this.popToken();
+
+      if (startOfArg.text === "{") {
+        var arg = [];
+        var depth = 1;
+
+        while (depth !== 0) {
+          var tok = this.popToken();
+          arg.push(tok);
+
+          if (tok.text === "{") {
+            ++depth;
+          } else if (tok.text === "}") {
+            --depth;
+          } else if (tok.text === "EOF") {
+            throw new src_ParseError("End of input in macro argument", startOfArg);
+          }
+        }
+
+        arg.pop(); // remove last }
+
+        arg.reverse(); // like above, to fit in with stack order
+
+        args[i] = arg;
+      } else if (startOfArg.text === "EOF") {
+        throw new src_ParseError("End of input expecting macro argument");
+      } else {
+        args[i] = [startOfArg];
+      }
+    }
+
+    return args;
+  }
+  /**
+   * Expand the next token only once if possible.
+   *
+   * If the token is expanded, the resulting tokens will be pushed onto
+   * the stack in reverse order and will be returned as an array,
+   * also in reverse order.
+   *
+   * If not, the next token will be returned without removing it
+   * from the stack.  This case can be detected by a `Token` return value
+   * instead of an `Array` return value.
+   *
+   * In either case, the next token will be on the top of the stack,
+   * or the stack will be empty.
+   *
+   * Used to implement `expandAfterFuture` and `expandNextToken`.
+   *
+   * At the moment, macro expansion doesn't handle delimited macros,
+   * i.e. things like those defined by \def\foo#1\end{…}.
+   * See the TeX book page 202ff. for details on how those should behave.
+   *
+   * If expandableOnly, only expandable tokens are expanded and
+   * an undefined control sequence results in an error.
+   */
+  ;
+
+  _proto.expandOnce = function expandOnce(expandableOnly) {
+    var topToken = this.popToken();
+    var name = topToken.text;
+    var expansion = !topToken.noexpand ? this._getExpansion(name) : null;
+
+    if (expansion == null || expandableOnly && expansion.unexpandable) {
+      if (expandableOnly && expansion == null && name[0] === "\\" && !this.isDefined(name)) {
+        throw new src_ParseError("Undefined control sequence: " + name);
+      }
+
+      this.pushToken(topToken);
+      return topToken;
+    }
+
+    this.expansionCount++;
+
+    if (this.expansionCount > this.settings.maxExpand) {
+      throw new src_ParseError("Too many expansions: infinite loop or " + "need to increase maxExpand setting");
+    }
+
+    var tokens = expansion.tokens;
+
+    if (expansion.numArgs) {
+      var args = this.consumeArgs(expansion.numArgs); // paste arguments in place of the placeholders
+
+      tokens = tokens.slice(); // make a shallow copy
+
+      for (var i = tokens.length - 1; i >= 0; --i) {
+        var tok = tokens[i];
+
+        if (tok.text === "#") {
+          if (i === 0) {
+            throw new src_ParseError("Incomplete placeholder at end of macro body", tok);
+          }
+
+          tok = tokens[--i]; // next token on stack
+
+          if (tok.text === "#") {
+            // ## → #
+            tokens.splice(i + 1, 1); // drop first #
+          } else if (/^[1-9]$/.test(tok.text)) {
+            var _tokens;
+
+            // replace the placeholder with the indicated argument
+            (_tokens = tokens).splice.apply(_tokens, [i, 2].concat(args[+tok.text - 1]));
+          } else {
+            throw new src_ParseError("Not a valid argument number", tok);
+          }
+        }
+      }
+    } // Concatenate expansion onto top of stack.
+
+
+    this.pushTokens(tokens);
+    return tokens;
+  }
+  /**
+   * Expand the next token only once (if possible), and return the resulting
+   * top token on the stack (without removing anything from the stack).
+   * Similar in behavior to TeX's `\expandafter\futurelet`.
+   * Equivalent to expandOnce() followed by future().
+   */
+  ;
+
+  _proto.expandAfterFuture = function expandAfterFuture() {
+    this.expandOnce();
+    return this.future();
+  }
+  /**
+   * Recursively expand first token, then return first non-expandable token.
+   */
+  ;
+
+  _proto.expandNextToken = function expandNextToken() {
+    for (;;) {
+      var expanded = this.expandOnce(); // expandOnce returns Token if and only if it's fully expanded.
+
+      if (expanded instanceof Token_Token) {
+        // \relax stops the expansion, but shouldn't get returned (a
+        // null return value couldn't get implemented as a function).
+        // the token after \noexpand is interpreted as if its meaning
+        // were ‘\relax’
+        if (expanded.text === "\\relax" || expanded.treatAsRelax) {
+          this.stack.pop();
+        } else {
+          return this.stack.pop(); // === expanded
+        }
+      }
+    } // Flow unable to figure out that this pathway is impossible.
+    // https://github.com/facebook/flow/issues/4808
+
+
+    throw new Error(); // eslint-disable-line no-unreachable
+  }
+  /**
+   * Fully expand the given macro name and return the resulting list of
+   * tokens, or return `undefined` if no such macro is defined.
+   */
+  ;
+
+  _proto.expandMacro = function expandMacro(name) {
+    return this.macros.has(name) ? this.expandTokens([new Token_Token(name)]) : undefined;
+  }
+  /**
+   * Fully expand the given token stream and return the resulting list of tokens
+   */
+  ;
+
+  _proto.expandTokens = function expandTokens(tokens) {
+    var output = [];
+    var oldStackLength = this.stack.length;
+    this.pushTokens(tokens);
+
+    while (this.stack.length > oldStackLength) {
+      var expanded = this.expandOnce(true); // expand only expandable tokens
+      // expandOnce returns Token if and only if it's fully expanded.
+
+      if (expanded instanceof Token_Token) {
+        if (expanded.treatAsRelax) {
+          // the expansion of \noexpand is the token itself
+          expanded.noexpand = false;
+          expanded.treatAsRelax = false;
+        }
+
+        output.push(this.stack.pop());
+      }
+    }
+
+    return output;
+  }
+  /**
+   * Fully expand the given macro name and return the result as a string,
+   * or return `undefined` if no such macro is defined.
+   */
+  ;
+
+  _proto.expandMacroAsText = function expandMacroAsText(name) {
+    var tokens = this.expandMacro(name);
+
+    if (tokens) {
+      return tokens.map(function (token) {
+        return token.text;
+      }).join("");
+    } else {
+      return tokens;
+    }
+  }
+  /**
+   * Returns the expanded macro as a reversed array of tokens and a macro
+   * argument count.  Or returns `null` if no such macro.
+   */
+  ;
+
+  _proto._getExpansion = function _getExpansion(name) {
+    var definition = this.macros.get(name);
+
+    if (definition == null) {
+      // mainly checking for undefined here
+      return definition;
+    }
+
+    var expansion = typeof definition === "function" ? definition(this) : definition;
+
+    if (typeof expansion === "string") {
+      var numArgs = 0;
+
+      if (expansion.indexOf("#") !== -1) {
+        var stripped = expansion.replace(/##/g, "");
+
+        while (stripped.indexOf("#" + (numArgs + 1)) !== -1) {
+          ++numArgs;
+        }
+      }
+
+      var bodyLexer = new Lexer_Lexer(expansion, this.settings);
+      var tokens = [];
+      var tok = bodyLexer.lex();
+
+      while (tok.text !== "EOF") {
+        tokens.push(tok);
+        tok = bodyLexer.lex();
+      }
+
+      tokens.reverse(); // to fit in with stack using push and pop
+
+      var expanded = {
+        tokens: tokens,
+        numArgs: numArgs
+      };
+      return expanded;
+    }
+
+    return expansion;
+  }
+  /**
+   * Determine whether a command is currently "defined" (has some
+   * functionality), meaning that it's a macro (in the current group),
+   * a function, a symbol, or one of the special commands listed in
+   * `implicitCommands`.
+   */
+  ;
+
+  _proto.isDefined = function isDefined(name) {
+    return this.macros.has(name) || src_functions.hasOwnProperty(name) || src_symbols.math.hasOwnProperty(name) || src_symbols.text.hasOwnProperty(name) || implicitCommands.hasOwnProperty(name);
+  }
+  /**
+   * Determine whether a command is expandable.
+   */
+  ;
+
+  _proto.isExpandable = function isExpandable(name) {
+    var macro = this.macros.get(name);
+    return macro != null ? typeof macro === "string" || typeof macro === "function" || !macro.unexpandable // TODO(ylem): #2085
+    : src_functions.hasOwnProperty(name)
+    /* && !functions[name].primitive*/
+    ;
+  };
+
+  return MacroExpander;
+}();
+
+
+// CONCATENATED MODULE: ./src/Parser.js
+/* eslint no-constant-condition:0 */
+
+
+
+
+
+
+
+
+
+ // Pre-evaluate both modules as unicodeSymbols require String.normalize()
+
+var unicodeAccents = {
+  "́": {
+    "text": "\\'",
+    "math": "\\acute"
+  },
+  "̀": {
+    "text": "\\`",
+    "math": "\\grave"
+  },
+  "̈": {
+    "text": "\\\"",
+    "math": "\\ddot"
+  },
+  "̃": {
+    "text": "\\~",
+    "math": "\\tilde"
+  },
+  "̄": {
+    "text": "\\=",
+    "math": "\\bar"
+  },
+  "̆": {
+    "text": "\\u",
+    "math": "\\breve"
+  },
+  "̌": {
+    "text": "\\v",
+    "math": "\\check"
+  },
+  "̂": {
+    "text": "\\^",
+    "math": "\\hat"
+  },
+  "̇": {
+    "text": "\\.",
+    "math": "\\dot"
+  },
+  "̊": {
+    "text": "\\r",
+    "math": "\\mathring"
+  },
+  "̋": {
+    "text": "\\H"
+  }
+};
+var unicodeSymbols = {
+  "á": "á",
+  "à": "à",
+  "ä": "ä",
+  "ǟ": "ǟ",
+  "ã": "ã",
+  "ā": "ā",
+  "ă": "ă",
+  "ắ": "ắ",
+  "ằ": "ằ",
+  "ẵ": "ẵ",
+  "ǎ": "ǎ",
+  "â": "â",
+  "ấ": "ấ",
+  "ầ": "ầ",
+  "ẫ": "ẫ",
+  "ȧ": "ȧ",
+  "ǡ": "ǡ",
+  "å": "å",
+  "ǻ": "ǻ",
+  "ḃ": "ḃ",
+  "ć": "ć",
+  "č": "č",
+  "ĉ": "ĉ",
+  "ċ": "ċ",
+  "ď": "ď",
+  "ḋ": "ḋ",
+  "é": "é",
+  "è": "è",
+  "ë": "ë",
+  "ẽ": "ẽ",
+  "ē": "ē",
+  "ḗ": "ḗ",
+  "ḕ": "ḕ",
+  "ĕ": "ĕ",
+  "ě": "ě",
+  "ê": "ê",
+  "ế": "ế",
+  "ề": "ề",
+  "ễ": "ễ",
+  "ė": "ė",
+  "ḟ": "ḟ",
+  "ǵ": "ǵ",
+  "ḡ": "ḡ",
+  "ğ": "ğ",
+  "ǧ": "ǧ",
+  "ĝ": "ĝ",
+  "ġ": "ġ",
+  "ḧ": "ḧ",
+  "ȟ": "ȟ",
+  "ĥ": "ĥ",
+  "ḣ": "ḣ",
+  "í": "í",
+  "ì": "ì",
+  "ï": "ï",
+  "ḯ": "ḯ",
+  "ĩ": "ĩ",
+  "ī": "ī",
+  "ĭ": "ĭ",
+  "ǐ": "ǐ",
+  "î": "î",
+  "ǰ": "ǰ",
+  "ĵ": "ĵ",
+  "ḱ": "ḱ",
+  "ǩ": "ǩ",
+  "ĺ": "ĺ",
+  "ľ": "ľ",
+  "ḿ": "ḿ",
+  "ṁ": "ṁ",
+  "ń": "ń",
+  "ǹ": "ǹ",
+  "ñ": "ñ",
+  "ň": "ň",
+  "ṅ": "ṅ",
+  "ó": "ó",
+  "ò": "ò",
+  "ö": "ö",
+  "ȫ": "ȫ",
+  "õ": "õ",
+  "ṍ": "ṍ",
+  "ṏ": "ṏ",
+  "ȭ": "ȭ",
+  "ō": "ō",
+  "ṓ": "ṓ",
+  "ṑ": "ṑ",
+  "ŏ": "ŏ",
+  "ǒ": "ǒ",
+  "ô": "ô",
+  "ố": "ố",
+  "ồ": "ồ",
+  "ỗ": "ỗ",
+  "ȯ": "ȯ",
+  "ȱ": "ȱ",
+  "ő": "ő",
+  "ṕ": "ṕ",
+  "ṗ": "ṗ",
+  "ŕ": "ŕ",
+  "ř": "ř",
+  "ṙ": "ṙ",
+  "ś": "ś",
+  "ṥ": "ṥ",
+  "š": "š",
+  "ṧ": "ṧ",
+  "ŝ": "ŝ",
+  "ṡ": "ṡ",
+  "ẗ": "ẗ",
+  "ť": "ť",
+  "ṫ": "ṫ",
+  "ú": "ú",
+  "ù": "ù",
+  "ü": "ü",
+  "ǘ": "ǘ",
+  "ǜ": "ǜ",
+  "ǖ": "ǖ",
+  "ǚ": "ǚ",
+  "ũ": "ũ",
+  "ṹ": "ṹ",
+  "ū": "ū",
+  "ṻ": "ṻ",
+  "ŭ": "ŭ",
+  "ǔ": "ǔ",
+  "û": "û",
+  "ů": "ů",
+  "ű": "ű",
+  "ṽ": "ṽ",
+  "ẃ": "ẃ",
+  "ẁ": "ẁ",
+  "ẅ": "ẅ",
+  "ŵ": "ŵ",
+  "ẇ": "ẇ",
+  "ẘ": "ẘ",
+  "ẍ": "ẍ",
+  "ẋ": "ẋ",
+  "ý": "ý",
+  "ỳ": "ỳ",
+  "ÿ": "ÿ",
+  "ỹ": "ỹ",
+  "ȳ": "ȳ",
+  "ŷ": "ŷ",
+  "ẏ": "ẏ",
+  "ẙ": "ẙ",
+  "ź": "ź",
+  "ž": "ž",
+  "ẑ": "ẑ",
+  "ż": "ż",
+  "Á": "Á",
+  "À": "À",
+  "Ä": "Ä",
+  "Ǟ": "Ǟ",
+  "Ã": "Ã",
+  "Ā": "Ā",
+  "Ă": "Ă",
+  "Ắ": "Ắ",
+  "Ằ": "Ằ",
+  "Ẵ": "Ẵ",
+  "Ǎ": "Ǎ",
+  "Â": "Â",
+  "Ấ": "Ấ",
+  "Ầ": "Ầ",
+  "Ẫ": "Ẫ",
+  "Ȧ": "Ȧ",
+  "Ǡ": "Ǡ",
+  "Å": "Å",
+  "Ǻ": "Ǻ",
+  "Ḃ": "Ḃ",
+  "Ć": "Ć",
+  "Č": "Č",
+  "Ĉ": "Ĉ",
+  "Ċ": "Ċ",
+  "Ď": "Ď",
+  "Ḋ": "Ḋ",
+  "É": "É",
+  "È": "È",
+  "Ë": "Ë",
+  "Ẽ": "Ẽ",
+  "Ē": "Ē",
+  "Ḗ": "Ḗ",
+  "Ḕ": "Ḕ",
+  "Ĕ": "Ĕ",
+  "Ě": "Ě",
+  "Ê": "Ê",
+  "Ế": "Ế",
+  "Ề": "Ề",
+  "Ễ": "Ễ",
+  "Ė": "Ė",
+  "Ḟ": "Ḟ",
+  "Ǵ": "Ǵ",
+  "Ḡ": "Ḡ",
+  "Ğ": "Ğ",
+  "Ǧ": "Ǧ",
+  "Ĝ": "Ĝ",
+  "Ġ": "Ġ",
+  "Ḧ": "Ḧ",
+  "Ȟ": "Ȟ",
+  "Ĥ": "Ĥ",
+  "Ḣ": "Ḣ",
+  "Í": "Í",
+  "Ì": "Ì",
+  "Ï": "Ï",
+  "Ḯ": "Ḯ",
+  "Ĩ": "Ĩ",
+  "Ī": "Ī",
+  "Ĭ": "Ĭ",
+  "Ǐ": "Ǐ",
+  "Î": "Î",
+  "İ": "İ",
+  "Ĵ": "Ĵ",
+  "Ḱ": "Ḱ",
+  "Ǩ": "Ǩ",
+  "Ĺ": "Ĺ",
+  "Ľ": "Ľ",
+  "Ḿ": "Ḿ",
+  "Ṁ": "Ṁ",
+  "Ń": "Ń",
+  "Ǹ": "Ǹ",
+  "Ñ": "Ñ",
+  "Ň": "Ň",
+  "Ṅ": "Ṅ",
+  "Ó": "Ó",
+  "Ò": "Ò",
+  "Ö": "Ö",
+  "Ȫ": "Ȫ",
+  "Õ": "Õ",
+  "Ṍ": "Ṍ",
+  "Ṏ": "Ṏ",
+  "Ȭ": "Ȭ",
+  "Ō": "Ō",
+  "Ṓ": "Ṓ",
+  "Ṑ": "Ṑ",
+  "Ŏ": "Ŏ",
+  "Ǒ": "Ǒ",
+  "Ô": "Ô",
+  "Ố": "Ố",
+  "Ồ": "Ồ",
+  "Ỗ": "Ỗ",
+  "Ȯ": "Ȯ",
+  "Ȱ": "Ȱ",
+  "Ő": "Ő",
+  "Ṕ": "Ṕ",
+  "Ṗ": "Ṗ",
+  "Ŕ": "Ŕ",
+  "Ř": "Ř",
+  "Ṙ": "Ṙ",
+  "Ś": "Ś",
+  "Ṥ": "Ṥ",
+  "Š": "Š",
+  "Ṧ": "Ṧ",
+  "Ŝ": "Ŝ",
+  "Ṡ": "Ṡ",
+  "Ť": "Ť",
+  "Ṫ": "Ṫ",
+  "Ú": "Ú",
+  "Ù": "Ù",
+  "Ü": "Ü",
+  "Ǘ": "Ǘ",
+  "Ǜ": "Ǜ",
+  "Ǖ": "Ǖ",
+  "Ǚ": "Ǚ",
+  "Ũ": "Ũ",
+  "Ṹ": "Ṹ",
+  "Ū": "Ū",
+  "Ṻ": "Ṻ",
+  "Ŭ": "Ŭ",
+  "Ǔ": "Ǔ",
+  "Û": "Û",
+  "Ů": "Ů",
+  "Ű": "Ű",
+  "Ṽ": "Ṽ",
+  "Ẃ": "Ẃ",
+  "Ẁ": "Ẁ",
+  "Ẅ": "Ẅ",
+  "Ŵ": "Ŵ",
+  "Ẇ": "Ẇ",
+  "Ẍ": "Ẍ",
+  "Ẋ": "Ẋ",
+  "Ý": "Ý",
+  "Ỳ": "Ỳ",
+  "Ÿ": "Ÿ",
+  "Ỹ": "Ỹ",
+  "Ȳ": "Ȳ",
+  "Ŷ": "Ŷ",
+  "Ẏ": "Ẏ",
+  "Ź": "Ź",
+  "Ž": "Ž",
+  "Ẑ": "Ẑ",
+  "Ż": "Ż",
+  "ά": "ά",
+  "ὰ": "ὰ",
+  "ᾱ": "ᾱ",
+  "ᾰ": "ᾰ",
+  "έ": "έ",
+  "ὲ": "ὲ",
+  "ή": "ή",
+  "ὴ": "ὴ",
+  "ί": "ί",
+  "ὶ": "ὶ",
+  "ϊ": "ϊ",
+  "ΐ": "ΐ",
+  "ῒ": "ῒ",
+  "ῑ": "ῑ",
+  "ῐ": "ῐ",
+  "ό": "ό",
+  "ὸ": "ὸ",
+  "ύ": "ύ",
+  "ὺ": "ὺ",
+  "ϋ": "ϋ",
+  "ΰ": "ΰ",
+  "ῢ": "ῢ",
+  "ῡ": "ῡ",
+  "ῠ": "ῠ",
+  "ώ": "ώ",
+  "ὼ": "ὼ",
+  "Ύ": "Ύ",
+  "Ὺ": "Ὺ",
+  "Ϋ": "Ϋ",
+  "Ῡ": "Ῡ",
+  "Ῠ": "Ῠ",
+  "Ώ": "Ώ",
+  "Ὼ": "Ὼ"
+};
+
+/**
+ * This file contains the parser used to parse out a TeX expression from the
+ * input. Since TeX isn't context-free, standard parsers don't work particularly
+ * well.
+ *
+ * The strategy of this parser is as such:
+ *
+ * The main functions (the `.parse...` ones) take a position in the current
+ * parse string to parse tokens from. The lexer (found in Lexer.js, stored at
+ * this.gullet.lexer) also supports pulling out tokens at arbitrary places. When
+ * individual tokens are needed at a position, the lexer is called to pull out a
+ * token, which is then used.
+ *
+ * The parser has a property called "mode" indicating the mode that
+ * the parser is currently in. Currently it has to be one of "math" or
+ * "text", which denotes whether the current environment is a math-y
+ * one or a text-y one (e.g. inside \text). Currently, this serves to
+ * limit the functions which can be used in text mode.
+ *
+ * The main functions then return an object which contains the useful data that
+ * was parsed at its given point, and a new position at the end of the parsed
+ * data. The main functions can call each other and continue the parsing by
+ * using the returned position as a new starting point.
+ *
+ * There are also extra `.handle...` functions, which pull out some reused
+ * functionality into self-contained functions.
+ *
+ * The functions return ParseNodes.
+ */
+var Parser_Parser =
+/*#__PURE__*/
+function () {
+  function Parser(input, settings) {
+    this.mode = void 0;
+    this.gullet = void 0;
+    this.settings = void 0;
+    this.leftrightDepth = void 0;
+    this.nextToken = void 0;
+    // Start in math mode
+    this.mode = "math"; // Create a new macro expander (gullet) and (indirectly via that) also a
+    // new lexer (mouth) for this parser (stomach, in the language of TeX)
+
+    this.gullet = new MacroExpander_MacroExpander(input, settings, this.mode); // Store the settings for use in parsing
+
+    this.settings = settings; // Count leftright depth (for \middle errors)
+
+    this.leftrightDepth = 0;
+  }
+  /**
+   * Checks a result to make sure it has the right type, and throws an
+   * appropriate error otherwise.
+   */
+
+
+  var _proto = Parser.prototype;
+
+  _proto.expect = function expect(text, consume) {
+    if (consume === void 0) {
+      consume = true;
+    }
+
+    if (this.fetch().text !== text) {
+      throw new src_ParseError("Expected '" + text + "', got '" + this.fetch().text + "'", this.fetch());
+    }
+
+    if (consume) {
+      this.consume();
+    }
+  }
+  /**
+   * Discards the current lookahead token, considering it consumed.
+   */
+  ;
+
+  _proto.consume = function consume() {
+    this.nextToken = null;
+  }
+  /**
+   * Return the current lookahead token, or if there isn't one (at the
+   * beginning, or if the previous lookahead token was consume()d),
+   * fetch the next token as the new lookahead token and return it.
+   */
+  ;
+
+  _proto.fetch = function fetch() {
+    if (this.nextToken == null) {
+      this.nextToken = this.gullet.expandNextToken();
+    }
+
+    return this.nextToken;
+  }
+  /**
+   * Switches between "text" and "math" modes.
+   */
+  ;
+
+  _proto.switchMode = function switchMode(newMode) {
+    this.mode = newMode;
+    this.gullet.switchMode(newMode);
+  }
+  /**
+   * Main parsing function, which parses an entire input.
+   */
+  ;
+
+  _proto.parse = function parse() {
+    if (!this.settings.globalGroup) {
+      // Create a group namespace for the math expression.
+      // (LaTeX creates a new group for every $...$, $$...$$, \[...\].)
+      this.gullet.beginGroup();
+    } // Use old \color behavior (same as LaTeX's \textcolor) if requested.
+    // We do this within the group for the math expression, so it doesn't
+    // pollute settings.macros.
+
+
+    if (this.settings.colorIsTextColor) {
+      this.gullet.macros.set("\\color", "\\textcolor");
+    } // Try to parse the input
+
+
+    var parse = this.parseExpression(false); // If we succeeded, make sure there's an EOF at the end
+
+    this.expect("EOF"); // End the group namespace for the expression
+
+    if (!this.settings.globalGroup) {
+      this.gullet.endGroup();
+    }
+
+    return parse;
+  };
+
+  _proto.parseExpression = function parseExpression(breakOnInfix, breakOnTokenText) {
+    var body = []; // Keep adding atoms to the body until we can't parse any more atoms (either
+    // we reached the end, a }, or a \right)
+
+    while (true) {
+      // Ignore spaces in math mode
+      if (this.mode === "math") {
+        this.consumeSpaces();
+      }
+
+      var lex = this.fetch();
+
+      if (Parser.endOfExpression.indexOf(lex.text) !== -1) {
+        break;
+      }
+
+      if (breakOnTokenText && lex.text === breakOnTokenText) {
+        break;
+      }
+
+      if (breakOnInfix && src_functions[lex.text] && src_functions[lex.text].infix) {
+        break;
+      }
+
+      var atom = this.parseAtom(breakOnTokenText);
+
+      if (!atom) {
+        break;
+      } else if (atom.type === "internal") {
+        continue;
+      }
+
+      body.push(atom);
+    }
+
+    if (this.mode === "text") {
+      this.formLigatures(body);
+    }
+
+    return this.handleInfixNodes(body);
+  }
+  /**
+   * Rewrites infix operators such as \over with corresponding commands such
+   * as \frac.
+   *
+   * There can only be one infix operator per group.  If there's more than one
+   * then the expression is ambiguous.  This can be resolved by adding {}.
+   */
+  ;
+
+  _proto.handleInfixNodes = function handleInfixNodes(body) {
+    var overIndex = -1;
+    var funcName;
+
+    for (var i = 0; i < body.length; i++) {
+      if (body[i].type === "infix") {
+        if (overIndex !== -1) {
+          throw new src_ParseError("only one infix operator per group", body[i].token);
+        }
+
+        overIndex = i;
+        funcName = body[i].replaceWith;
+      }
+    }
+
+    if (overIndex !== -1 && funcName) {
+      var numerNode;
+      var denomNode;
+      var numerBody = body.slice(0, overIndex);
+      var denomBody = body.slice(overIndex + 1);
+
+      if (numerBody.length === 1 && numerBody[0].type === "ordgroup") {
+        numerNode = numerBody[0];
+      } else {
+        numerNode = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: numerBody
+        };
+      }
+
+      if (denomBody.length === 1 && denomBody[0].type === "ordgroup") {
+        denomNode = denomBody[0];
+      } else {
+        denomNode = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: denomBody
+        };
+      }
+
+      var node;
+
+      if (funcName === "\\\\abovefrac") {
+        node = this.callFunction(funcName, [numerNode, body[overIndex], denomNode], []);
+      } else {
+        node = this.callFunction(funcName, [numerNode, denomNode], []);
+      }
+
+      return [node];
+    } else {
+      return body;
+    }
+  } // The greediness of a superscript or subscript
+  ;
+
+  /**
+   * Handle a subscript or superscript with nice errors.
+   */
+  _proto.handleSupSubscript = function handleSupSubscript(name) {
+    var symbolToken = this.fetch();
+    var symbol = symbolToken.text;
+    this.consume();
+    var group = this.parseGroup(name, false, Parser.SUPSUB_GREEDINESS, undefined, undefined, true); // ignore spaces before sup/subscript argument
+
+    if (!group) {
+      throw new src_ParseError("Expected group after '" + symbol + "'", symbolToken);
+    }
+
+    return group;
+  }
+  /**
+   * Converts the textual input of an unsupported command into a text node
+   * contained within a color node whose color is determined by errorColor
+   */
+  ;
+
+  _proto.formatUnsupportedCmd = function formatUnsupportedCmd(text) {
+    var textordArray = [];
+
+    for (var i = 0; i < text.length; i++) {
+      textordArray.push({
+        type: "textord",
+        mode: "text",
+        text: text[i]
+      });
+    }
+
+    var textNode = {
+      type: "text",
+      mode: this.mode,
+      body: textordArray
+    };
+    var colorNode = {
+      type: "color",
+      mode: this.mode,
+      color: this.settings.errorColor,
+      body: [textNode]
+    };
+    return colorNode;
+  }
+  /**
+   * Parses a group with optional super/subscripts.
+   */
+  ;
+
+  _proto.parseAtom = function parseAtom(breakOnTokenText) {
+    // The body of an atom is an implicit group, so that things like
+    // \left(x\right)^2 work correctly.
+    var base = this.parseGroup("atom", false, null, breakOnTokenText); // In text mode, we don't have superscripts or subscripts
+
+    if (this.mode === "text") {
+      return base;
+    } // Note that base may be empty (i.e. null) at this point.
+
+
+    var superscript;
+    var subscript;
+
+    while (true) {
+      // Guaranteed in math mode, so eat any spaces first.
+      this.consumeSpaces(); // Lex the first token
+
+      var lex = this.fetch();
+
+      if (lex.text === "\\limits" || lex.text === "\\nolimits") {
+        // We got a limit control
+        if (base && base.type === "op") {
+          var limits = lex.text === "\\limits";
+          base.limits = limits;
+          base.alwaysHandleSupSub = true;
+        } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub) {
+          var _limits = lex.text === "\\limits";
+
+          base.limits = _limits;
+        } else {
+          throw new src_ParseError("Limit controls must follow a math operator", lex);
+        }
+
+        this.consume();
+      } else if (lex.text === "^") {
+        // We got a superscript start
+        if (superscript) {
+          throw new src_ParseError("Double superscript", lex);
+        }
+
+        superscript = this.handleSupSubscript("superscript");
+      } else if (lex.text === "_") {
+        // We got a subscript start
+        if (subscript) {
+          throw new src_ParseError("Double subscript", lex);
+        }
+
+        subscript = this.handleSupSubscript("subscript");
+      } else if (lex.text === "'") {
+        // We got a prime
+        if (superscript) {
+          throw new src_ParseError("Double superscript", lex);
+        }
+
+        var prime = {
+          type: "textord",
+          mode: this.mode,
+          text: "\\prime"
+        }; // Many primes can be grouped together, so we handle this here
+
+        var primes = [prime];
+        this.consume(); // Keep lexing tokens until we get something that's not a prime
+
+        while (this.fetch().text === "'") {
+          // For each one, add another prime to the list
+          primes.push(prime);
+          this.consume();
+        } // If there's a superscript following the primes, combine that
+        // superscript in with the primes.
+
+
+        if (this.fetch().text === "^") {
+          primes.push(this.handleSupSubscript("superscript"));
+        } // Put everything into an ordgroup as the superscript
+
+
+        superscript = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: primes
+        };
+      } else {
+        // If it wasn't ^, _, or ', stop parsing super/subscripts
+        break;
+      }
+    } // Base must be set if superscript or subscript are set per logic above,
+    // but need to check here for type check to pass.
+
+
+    if (superscript || subscript) {
+      // If we got either a superscript or subscript, create a supsub
+      return {
+        type: "supsub",
+        mode: this.mode,
+        base: base,
+        sup: superscript,
+        sub: subscript
+      };
+    } else {
+      // Otherwise return the original body
+      return base;
+    }
+  }
+  /**
+   * Parses an entire function, including its base and all of its arguments.
+   */
+  ;
+
+  _proto.parseFunction = function parseFunction(breakOnTokenText, name, // For error reporting.
+  greediness) {
+    var token = this.fetch();
+    var func = token.text;
+    var funcData = src_functions[func];
+
+    if (!funcData) {
+      return null;
+    }
+
+    this.consume(); // consume command token
+
+    if (greediness != null && funcData.greediness <= greediness) {
+      throw new src_ParseError("Got function '" + func + "' with no arguments" + (name ? " as " + name : ""), token);
+    } else if (this.mode === "text" && !funcData.allowedInText) {
+      throw new src_ParseError("Can't use function '" + func + "' in text mode", token);
+    } else if (this.mode === "math" && funcData.allowedInMath === false) {
+      throw new src_ParseError("Can't use function '" + func + "' in math mode", token);
+    }
+
+    var _this$parseArguments = this.parseArguments(func, funcData),
+        args = _this$parseArguments.args,
+        optArgs = _this$parseArguments.optArgs;
+
+    return this.callFunction(func, args, optArgs, token, breakOnTokenText);
+  }
+  /**
+   * Call a function handler with a suitable context and arguments.
+   */
+  ;
+
+  _proto.callFunction = function callFunction(name, args, optArgs, token, breakOnTokenText) {
+    var context = {
+      funcName: name,
+      parser: this,
+      token: token,
+      breakOnTokenText: breakOnTokenText
+    };
+    var func = src_functions[name];
+
+    if (func && func.handler) {
+      return func.handler(context, args, optArgs);
+    } else {
+      throw new src_ParseError("No function handler for " + name);
+    }
+  }
+  /**
+   * Parses the arguments of a function or environment
+   */
+  ;
+
+  _proto.parseArguments = function parseArguments(func, // Should look like "\name" or "\begin{name}".
+  funcData) {
+    var totalArgs = funcData.numArgs + funcData.numOptionalArgs;
+
+    if (totalArgs === 0) {
+      return {
+        args: [],
+        optArgs: []
+      };
+    }
+
+    var baseGreediness = funcData.greediness;
+    var args = [];
+    var optArgs = [];
+
+    for (var i = 0; i < totalArgs; i++) {
+      var argType = funcData.argTypes && funcData.argTypes[i];
+      var isOptional = i < funcData.numOptionalArgs; // Ignore spaces between arguments.  As the TeXbook says:
+      // "After you have said ‘\def\row#1#2{...}’, you are allowed to
+      //  put spaces between the arguments (e.g., ‘\row x n’), because
+      //  TeX doesn’t use single spaces as undelimited arguments."
+
+      var consumeSpaces = i > 0 && !isOptional || // Also consume leading spaces in math mode, as parseSymbol
+      // won't know what to do with them.  This can only happen with
+      // macros, e.g. \frac\foo\foo where \foo expands to a space symbol.
+      // In LaTeX, the \foo's get treated as (blank) arguments.
+      // In KaTeX, for now, both spaces will get consumed.
+      // TODO(edemaine)
+      i === 0 && !isOptional && this.mode === "math";
+      var arg = this.parseGroupOfType("argument to '" + func + "'", argType, isOptional, baseGreediness, consumeSpaces);
+
+      if (!arg) {
+        if (isOptional) {
+          optArgs.push(null);
+          continue;
+        }
+
+        throw new src_ParseError("Expected group after '" + func + "'", this.fetch());
+      }
+
+      (isOptional ? optArgs : args).push(arg);
+    }
+
+    return {
+      args: args,
+      optArgs: optArgs
+    };
+  }
+  /**
+   * Parses a group when the mode is changing.
+   */
+  ;
+
+  _proto.parseGroupOfType = function parseGroupOfType(name, type, optional, greediness, consumeSpaces) {
+    switch (type) {
+      case "color":
+        if (consumeSpaces) {
+          this.consumeSpaces();
+        }
+
+        return this.parseColorGroup(optional);
+
+      case "size":
+        if (consumeSpaces) {
+          this.consumeSpaces();
+        }
+
+        return this.parseSizeGroup(optional);
+
+      case "url":
+        return this.parseUrlGroup(optional, consumeSpaces);
+
+      case "math":
+      case "text":
+        return this.parseGroup(name, optional, greediness, undefined, type, consumeSpaces);
+
+      case "hbox":
+        {
+          // hbox argument type wraps the argument in the equivalent of
+          // \hbox, which is like \text but switching to \textstyle size.
+          var group = this.parseGroup(name, optional, greediness, undefined, "text", consumeSpaces);
+
+          if (!group) {
+            return group;
+          }
+
+          var styledGroup = {
+            type: "styling",
+            mode: group.mode,
+            body: [group],
+            style: "text" // simulate \textstyle
+
+          };
+          return styledGroup;
+        }
+
+      case "raw":
+        {
+          if (consumeSpaces) {
+            this.consumeSpaces();
+          }
+
+          if (optional && this.fetch().text === "{") {
+            return null;
+          }
+
+          var token = this.parseStringGroup("raw", optional, true);
+
+          if (token) {
+            return {
+              type: "raw",
+              mode: "text",
+              string: token.text
+            };
+          } else {
+            throw new src_ParseError("Expected raw group", this.fetch());
+          }
+        }
+
+      case "original":
+      case null:
+      case undefined:
+        return this.parseGroup(name, optional, greediness, undefined, undefined, consumeSpaces);
+
+      default:
+        throw new src_ParseError("Unknown group type as " + name, this.fetch());
+    }
+  }
+  /**
+   * Discard any space tokens, fetching the next non-space token.
+   */
+  ;
+
+  _proto.consumeSpaces = function consumeSpaces() {
+    while (this.fetch().text === " ") {
+      this.consume();
+    }
+  }
+  /**
+   * Parses a group, essentially returning the string formed by the
+   * brace-enclosed tokens plus some position information.
+   */
+  ;
+
+  _proto.parseStringGroup = function parseStringGroup(modeName, // Used to describe the mode in error messages.
+  optional, raw) {
+    var groupBegin = optional ? "[" : "{";
+    var groupEnd = optional ? "]" : "}";
+    var beginToken = this.fetch();
+
+    if (beginToken.text !== groupBegin) {
+      if (optional) {
+        return null;
+      } else if (raw && beginToken.text !== "EOF" && /[^{}[\]]/.test(beginToken.text)) {
+        this.consume();
+        return beginToken;
+      }
+    }
+
+    var outerMode = this.mode;
+    this.mode = "text";
+    this.expect(groupBegin);
+    var str = "";
+    var firstToken = this.fetch();
+    var nested = 0; // allow nested braces in raw string group
+
+    var lastToken = firstToken;
+    var nextToken;
+
+    while ((nextToken = this.fetch()).text !== groupEnd || raw && nested > 0) {
+      switch (nextToken.text) {
+        case "EOF":
+          throw new src_ParseError("Unexpected end of input in " + modeName, firstToken.range(lastToken, str));
+
+        case groupBegin:
+          nested++;
+          break;
+
+        case groupEnd:
+          nested--;
+          break;
+      }
+
+      lastToken = nextToken;
+      str += lastToken.text;
+      this.consume();
+    }
+
+    this.expect(groupEnd);
+    this.mode = outerMode;
+    return firstToken.range(lastToken, str);
+  }
+  /**
+   * Parses a regex-delimited group: the largest sequence of tokens
+   * whose concatenated strings match `regex`. Returns the string
+   * formed by the tokens plus some position information.
+   */
+  ;
+
+  _proto.parseRegexGroup = function parseRegexGroup(regex, modeName) {
+    var outerMode = this.mode;
+    this.mode = "text";
+    var firstToken = this.fetch();
+    var lastToken = firstToken;
+    var str = "";
+    var nextToken;
+
+    while ((nextToken = this.fetch()).text !== "EOF" && regex.test(str + nextToken.text)) {
+      lastToken = nextToken;
+      str += lastToken.text;
+      this.consume();
+    }
+
+    if (str === "") {
+      throw new src_ParseError("Invalid " + modeName + ": '" + firstToken.text + "'", firstToken);
+    }
+
+    this.mode = outerMode;
+    return firstToken.range(lastToken, str);
+  }
+  /**
+   * Parses a color description.
+   */
+  ;
+
+  _proto.parseColorGroup = function parseColorGroup(optional) {
+    var res = this.parseStringGroup("color", optional);
+
+    if (!res) {
+      return null;
+    }
+
+    var match = /^(#[a-f0-9]{3}|#?[a-f0-9]{6}|[a-z]+)$/i.exec(res.text);
+
+    if (!match) {
+      throw new src_ParseError("Invalid color: '" + res.text + "'", res);
+    }
+
+    var color = match[0];
+
+    if (/^[0-9a-f]{6}$/i.test(color)) {
+      // We allow a 6-digit HTML color spec without a leading "#".
+      // This follows the xcolor package's HTML color model.
+      // Predefined color names are all missed by this RegEx pattern.
+      color = "#" + color;
+    }
+
+    return {
+      type: "color-token",
+      mode: this.mode,
+      color: color
+    };
+  }
+  /**
+   * Parses a size specification, consisting of magnitude and unit.
+   */
+  ;
+
+  _proto.parseSizeGroup = function parseSizeGroup(optional) {
+    var res;
+    var isBlank = false;
+
+    if (!optional && this.fetch().text !== "{") {
+      res = this.parseRegexGroup(/^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2} *$/, "size");
+    } else {
+      res = this.parseStringGroup("size", optional);
+    }
+
+    if (!res) {
+      return null;
+    }
+
+    if (!optional && res.text.length === 0) {
+      // Because we've tested for what is !optional, this block won't
+      // affect \kern, \hspace, etc. It will capture the mandatory arguments
+      // to \genfrac and \above.
+      res.text = "0pt"; // Enable \above{}
+
+      isBlank = true; // This is here specifically for \genfrac
+    }
+
+    var match = /([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(res.text);
+
+    if (!match) {
+      throw new src_ParseError("Invalid size: '" + res.text + "'", res);
+    }
+
+    var data = {
+      number: +(match[1] + match[2]),
+      // sign + magnitude, cast to number
+      unit: match[3]
+    };
+
+    if (!validUnit(data)) {
+      throw new src_ParseError("Invalid unit: '" + data.unit + "'", res);
+    }
+
+    return {
+      type: "size",
+      mode: this.mode,
+      value: data,
+      isBlank: isBlank
+    };
+  }
+  /**
+   * Parses an URL, checking escaped letters and allowed protocols,
+   * and setting the catcode of % as an active character (as in \hyperref).
+   */
+  ;
+
+  _proto.parseUrlGroup = function parseUrlGroup(optional, consumeSpaces) {
+    this.gullet.lexer.setCatcode("%", 13); // active character
+
+    var res = this.parseStringGroup("url", optional, true); // get raw string
+
+    this.gullet.lexer.setCatcode("%", 14); // comment character
+
+    if (!res) {
+      return null;
+    } // hyperref package allows backslashes alone in href, but doesn't
+    // generate valid links in such cases; we interpret this as
+    // "undefined" behaviour, and keep them as-is. Some browser will
+    // replace backslashes with forward slashes.
+
+
+    var url = res.text.replace(/\\([#$%&~_^{}])/g, '$1');
+    return {
+      type: "url",
+      mode: this.mode,
+      url: url
+    };
+  }
+  /**
+   * If `optional` is false or absent, this parses an ordinary group,
+   * which is either a single nucleus (like "x") or an expression
+   * in braces (like "{x+y}") or an implicit group, a group that starts
+   * at the current position, and ends right before a higher explicit
+   * group ends, or at EOF.
+   * If `optional` is true, it parses either a bracket-delimited expression
+   * (like "[x+y]") or returns null to indicate the absence of a
+   * bracket-enclosed group.
+   * If `mode` is present, switches to that mode while parsing the group,
+   * and switches back after.
+   */
+  ;
+
+  _proto.parseGroup = function parseGroup(name, // For error reporting.
+  optional, greediness, breakOnTokenText, mode, consumeSpaces) {
+    // Switch to specified mode
+    var outerMode = this.mode;
+
+    if (mode) {
+      this.switchMode(mode);
+    } // Consume spaces if requested, crucially *after* we switch modes,
+    // so that the next non-space token is parsed in the correct mode.
+
+
+    if (consumeSpaces) {
+      this.consumeSpaces();
+    } // Get first token
+
+
+    var firstToken = this.fetch();
+    var text = firstToken.text;
+    var result; // Try to parse an open brace or \begingroup
+
+    if (optional ? text === "[" : text === "{" || text === "\\begingroup") {
+      this.consume();
+      var groupEnd = Parser.endOfGroup[text]; // Start a new group namespace
+
+      this.gullet.beginGroup(); // If we get a brace, parse an expression
+
+      var expression = this.parseExpression(false, groupEnd);
+      var lastToken = this.fetch(); // Check that we got a matching closing brace
+
+      this.expect(groupEnd); // End group namespace
+
+      this.gullet.endGroup();
+      result = {
+        type: "ordgroup",
+        mode: this.mode,
+        loc: SourceLocation.range(firstToken, lastToken),
+        body: expression,
+        // A group formed by \begingroup...\endgroup is a semi-simple group
+        // which doesn't affect spacing in math mode, i.e., is transparent.
+        // https://tex.stackexchange.com/questions/1930/when-should-one-
+        // use-begingroup-instead-of-bgroup
+        semisimple: text === "\\begingroup" || undefined
+      };
+    } else if (optional) {
+      // Return nothing for an optional group
+      result = null;
+    } else {
+      // If there exists a function with this name, parse the function.
+      // Otherwise, just return a nucleus
+      result = this.parseFunction(breakOnTokenText, name, greediness) || this.parseSymbol();
+
+      if (result == null && text[0] === "\\" && !implicitCommands.hasOwnProperty(text)) {
+        if (this.settings.throwOnError) {
+          throw new src_ParseError("Undefined control sequence: " + text, firstToken);
+        }
+
+        result = this.formatUnsupportedCmd(text);
+        this.consume();
+      }
+    } // Switch mode back
+
+
+    if (mode) {
+      this.switchMode(outerMode);
+    }
+
+    return result;
+  }
+  /**
+   * Form ligature-like combinations of characters for text mode.
+   * This includes inputs like "--", "---", "``" and "''".
+   * The result will simply replace multiple textord nodes with a single
+   * character in each value by a single textord node having multiple
+   * characters in its value.  The representation is still ASCII source.
+   * The group will be modified in place.
+   */
+  ;
+
+  _proto.formLigatures = function formLigatures(group) {
+    var n = group.length - 1;
+
+    for (var i = 0; i < n; ++i) {
+      var a = group[i]; // $FlowFixMe: Not every node type has a `text` property.
+
+      var v = a.text;
+
+      if (v === "-" && group[i + 1].text === "-") {
+        if (i + 1 < n && group[i + 2].text === "-") {
+          group.splice(i, 3, {
+            type: "textord",
+            mode: "text",
+            loc: SourceLocation.range(a, group[i + 2]),
+            text: "---"
+          });
+          n -= 2;
+        } else {
+          group.splice(i, 2, {
+            type: "textord",
+            mode: "text",
+            loc: SourceLocation.range(a, group[i + 1]),
+            text: "--"
+          });
+          n -= 1;
+        }
+      }
+
+      if ((v === "'" || v === "`") && group[i + 1].text === v) {
+        group.splice(i, 2, {
+          type: "textord",
+          mode: "text",
+          loc: SourceLocation.range(a, group[i + 1]),
+          text: v + v
+        });
+        n -= 1;
+      }
+    }
+  }
+  /**
+   * Parse a single symbol out of the string. Here, we handle single character
+   * symbols and special functions like \verb.
+   */
+  ;
+
+  _proto.parseSymbol = function parseSymbol() {
+    var nucleus = this.fetch();
+    var text = nucleus.text;
+
+    if (/^\\verb[^a-zA-Z]/.test(text)) {
+      this.consume();
+      var arg = text.slice(5);
+      var star = arg.charAt(0) === "*";
+
+      if (star) {
+        arg = arg.slice(1);
+      } // Lexer's tokenRegex is constructed to always have matching
+      // first/last characters.
+
+
+      if (arg.length < 2 || arg.charAt(0) !== arg.slice(-1)) {
+        throw new src_ParseError("\\verb assertion failed --\n                    please report what input caused this bug");
+      }
+
+      arg = arg.slice(1, -1); // remove first and last char
+
+      return {
+        type: "verb",
+        mode: "text",
+        body: arg,
+        star: star
+      };
+    } // At this point, we should have a symbol, possibly with accents.
+    // First expand any accented base symbol according to unicodeSymbols.
+
+
+    if (unicodeSymbols.hasOwnProperty(text[0]) && !src_symbols[this.mode][text[0]]) {
+      // This behavior is not strict (XeTeX-compatible) in math mode.
+      if (this.settings.strict && this.mode === "math") {
+        this.settings.reportNonstrict("unicodeTextInMathMode", "Accented Unicode text character \"" + text[0] + "\" used in " + "math mode", nucleus);
+      }
+
+      text = unicodeSymbols[text[0]] + text.substr(1);
+    } // Strip off any combining characters
+
+
+    var match = combiningDiacriticalMarksEndRegex.exec(text);
+
+    if (match) {
+      text = text.substring(0, match.index);
+
+      if (text === 'i') {
+        text = "\u0131"; // dotless i, in math and text mode
+      } else if (text === 'j') {
+        text = "\u0237"; // dotless j, in math and text mode
+      }
+    } // Recognize base symbol
+
+
+    var symbol;
+
+    if (src_symbols[this.mode][text]) {
+      if (this.settings.strict && this.mode === 'math' && extraLatin.indexOf(text) >= 0) {
+        this.settings.reportNonstrict("unicodeTextInMathMode", "Latin-1/Unicode text character \"" + text[0] + "\" used in " + "math mode", nucleus);
+      }
+
+      var group = src_symbols[this.mode][text].group;
+      var loc = SourceLocation.range(nucleus);
+      var s;
+
+      if (ATOMS.hasOwnProperty(group)) {
+        // $FlowFixMe
+        var family = group;
+        s = {
+          type: "atom",
+          mode: this.mode,
+          family: family,
+          loc: loc,
+          text: text
+        };
+      } else {
+        // $FlowFixMe
+        s = {
+          type: group,
+          mode: this.mode,
+          loc: loc,
+          text: text
+        };
+      }
+
+      symbol = s;
+    } else if (text.charCodeAt(0) >= 0x80) {
+      // no symbol for e.g. ^
+      if (this.settings.strict) {
+        if (!supportedCodepoint(text.charCodeAt(0))) {
+          this.settings.reportNonstrict("unknownSymbol", "Unrecognized Unicode character \"" + text[0] + "\"" + (" (" + text.charCodeAt(0) + ")"), nucleus);
+        } else if (this.mode === "math") {
+          this.settings.reportNonstrict("unicodeTextInMathMode", "Unicode text character \"" + text[0] + "\" used in math mode", nucleus);
+        }
+      } // All nonmathematical Unicode characters are rendered as if they
+      // are in text mode (wrapped in \text) because that's what it
+      // takes to render them in LaTeX.  Setting `mode: this.mode` is
+      // another natural choice (the user requested math mode), but
+      // this makes it more difficult for getCharacterMetrics() to
+      // distinguish Unicode characters without metrics and those for
+      // which we want to simulate the letter M.
+
+
+      symbol = {
+        type: "textord",
+        mode: "text",
+        loc: SourceLocation.range(nucleus),
+        text: text
+      };
+    } else {
+      return null; // EOF, ^, _, {, }, etc.
+    }
+
+    this.consume(); // Transform combining characters into accents
+
+    if (match) {
+      for (var i = 0; i < match[0].length; i++) {
+        var accent = match[0][i];
+
+        if (!unicodeAccents[accent]) {
+          throw new src_ParseError("Unknown accent ' " + accent + "'", nucleus);
+        }
+
+        var command = unicodeAccents[accent][this.mode];
+
+        if (!command) {
+          throw new src_ParseError("Accent " + accent + " unsupported in " + this.mode + " mode", nucleus);
+        }
+
+        symbol = {
+          type: "accent",
+          mode: this.mode,
+          loc: SourceLocation.range(nucleus),
+          label: command,
+          isStretchy: false,
+          isShifty: true,
+          base: symbol
+        };
+      }
+    }
+
+    return symbol;
+  };
+
+  return Parser;
+}();
+
+Parser_Parser.endOfExpression = ["}", "\\endgroup", "\\end", "\\right", "&"];
+Parser_Parser.endOfGroup = {
+  "[": "]",
+  "{": "}",
+  "\\begingroup": "\\endgroup"
+  /**
+   * Parses an "expression", which is a list of atoms.
+   *
+   * `breakOnInfix`: Should the parsing stop when we hit infix nodes? This
+   *                 happens when functions have higher precendence han infix
+   *                 nodes in implicit parses.
+   *
+   * `breakOnTokenText`: The text of the token that the expression should end
+   *                     with, or `null` if something else should end the
+   *                     expression.
+   */
+
+};
+Parser_Parser.SUPSUB_GREEDINESS = 1;
+
+// CONCATENATED MODULE: ./src/parseTree.js
+/**
+ * Provides a single function for parsing an expression using a Parser
+ * TODO(emily): Remove this
+ */
+
+
+
+/**
+ * Parses an expression using a Parser, then returns the parsed result.
+ */
+var parseTree_parseTree = function parseTree(toParse, settings) {
+  if (!(typeof toParse === 'string' || toParse instanceof String)) {
+    throw new TypeError('KaTeX can only parse string typed expression');
+  }
+
+  var parser = new Parser_Parser(toParse, settings); // Blank out any \df@tag to avoid spurious "Duplicate \tag" errors
+
+  delete parser.gullet.macros.current["\\df@tag"];
+  var tree = parser.parse(); // If the input used \tag, it will set the \df@tag macro to the tag.
+  // In this case, we separately parse the tag and wrap the tree.
+
+  if (parser.gullet.macros.get("\\df@tag")) {
+    if (!settings.displayMode) {
+      throw new src_ParseError("\\tag works only in display equations");
+    }
+
+    parser.gullet.feed("\\df@tag");
+    tree = [{
+      type: "tag",
+      mode: "text",
+      body: tree,
+      tag: parser.parse()
+    }];
+  }
+
+  return tree;
+};
+
+/* harmony default export */ var src_parseTree = (parseTree_parseTree);
+// CONCATENATED MODULE: ./katex.js
+/* eslint no-console:0 */
+
+/**
+ * This is the main entry point for KaTeX. Here, we expose functions for
+ * rendering expressions either to DOM nodes or to markup strings.
+ *
+ * We also expose the ParseError class to check if errors thrown from KaTeX are
+ * errors in the expression, or errors in javascript handling.
+ */
+
+
+
+
+
+
+
+
+
+
+/**
+ * Parse and build an expression, and place that expression in the DOM node
+ * given.
+ */
+var katex_render = function render(expression, baseNode, options) {
+  baseNode.textContent = "";
+  var node = katex_renderToDomTree(expression, options).toNode();
+  baseNode.appendChild(node);
+}; // KaTeX's styles don't work properly in quirks mode. Print out an error, and
+// disable rendering.
+
+
+if (typeof document !== "undefined") {
+  if (document.compatMode !== "CSS1Compat") {
+    typeof console !== "undefined" && console.warn("Warning: KaTeX doesn't work in quirks mode. Make sure your " + "website has a suitable doctype.");
+
+    katex_render = function render() {
+      throw new src_ParseError("KaTeX doesn't work in quirks mode.");
+    };
+  }
+}
+/**
+ * Parse and build an expression, and return the markup for that.
+ */
+
+
+var renderToString = function renderToString(expression, options) {
+  var markup = katex_renderToDomTree(expression, options).toMarkup();
+  return markup;
+};
+/**
+ * Parse an expression and return the parse tree.
+ */
+
+
+var katex_generateParseTree = function generateParseTree(expression, options) {
+  var settings = new Settings_Settings(options);
+  return src_parseTree(expression, settings);
+};
+/**
+ * If the given error is a KaTeX ParseError and options.throwOnError is false,
+ * renders the invalid LaTeX as a span with hover title giving the KaTeX
+ * error message.  Otherwise, simply throws the error.
+ */
+
+
+var katex_renderError = function renderError(error, expression, options) {
+  if (options.throwOnError || !(error instanceof src_ParseError)) {
+    throw error;
+  }
+
+  var node = buildCommon.makeSpan(["katex-error"], [new domTree_SymbolNode(expression)]);
+  node.setAttribute("title", error.toString());
+  node.setAttribute("style", "color:" + options.errorColor);
+  return node;
+};
+/**
+ * Generates and returns the katex build tree. This is used for advanced
+ * use cases (like rendering to custom output).
+ */
+
+
+var katex_renderToDomTree = function renderToDomTree(expression, options) {
+  var settings = new Settings_Settings(options);
+
+  try {
+    var tree = src_parseTree(expression, settings);
+    return buildTree_buildTree(tree, expression, settings);
+  } catch (error) {
+    return katex_renderError(error, expression, settings);
+  }
+};
+/**
+ * Generates and returns the katex build tree, with just HTML (no MathML).
+ * This is used for advanced use cases (like rendering to custom output).
+ */
+
+
+var katex_renderToHTMLTree = function renderToHTMLTree(expression, options) {
+  var settings = new Settings_Settings(options);
+
+  try {
+    var tree = src_parseTree(expression, settings);
+    return buildTree_buildHTMLTree(tree, expression, settings);
+  } catch (error) {
+    return katex_renderError(error, expression, settings);
+  }
+};
+
+/* harmony default export */ var katex_0 = ({
+  /**
+   * Current KaTeX version
+   */
+  version: "0.12.0",
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML combination, and adds
+   * it as a child to the specified DOM node.
+   */
+  render: katex_render,
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML combination string,
+   * for sending to the client.
+   */
+  renderToString: renderToString,
+
+  /**
+   * KaTeX error, usually during parsing.
+   */
+  ParseError: src_ParseError,
+
+  /**
+   * Parses the given LaTeX into KaTeX's internal parse tree structure,
+   * without rendering to HTML or MathML.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __parse: katex_generateParseTree,
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML internal DOM tree
+   * representation, without flattening that representation to a string.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __renderToDomTree: katex_renderToDomTree,
+
+  /**
+   * Renders the given LaTeX into an HTML internal DOM tree representation,
+   * without MathML and without flattening that representation to a string.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __renderToHTMLTree: katex_renderToHTMLTree,
+
+  /**
+   * extends internal font metrics object with a new object
+   * each key in the new object represents a font name
+  */
+  __setFontMetrics: setFontMetrics,
+
+  /**
+   * adds a new symbol to builtin symbols table
+   */
+  __defineSymbol: defineSymbol,
+
+  /**
+   * adds a new macro to builtin macro list
+   */
+  __defineMacro: defineMacro,
+
+  /**
+   * Expose the dom tree node types, which can be useful for type checking nodes.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __domTree: {
+    Span: domTree_Span,
+    Anchor: domTree_Anchor,
+    SymbolNode: domTree_SymbolNode,
+    SvgNode: SvgNode,
+    PathNode: domTree_PathNode,
+    LineNode: LineNode
+  }
+});
+// CONCATENATED MODULE: ./katex.webpack.js
+/**
+ * This is the webpack entry point for KaTeX. As ECMAScript, flow[1] and jest[2]
+ * doesn't support CSS modules natively, a separate entry point is used and
+ * it is not flowtyped.
+ *
+ * [1] https://gist.github.com/lambdahands/d19e0da96285b749f0ef
+ * [2] https://facebook.github.io/jest/docs/en/webpack.html
+ */
+
+
+/* harmony default export */ var katex_webpack = __webpack_exports__["default"] = (katex_0);
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/docs/pt/katex/katex.min.css b/docs/pt/katex/katex.min.css
new file mode 100644
index 0000000..98b7c7b
--- /dev/null
+++ b/docs/pt/katex/katex.min.css
@@ -0,0 +1 @@
+@font-face{font-family:KaTeX_AMS;src:url(fonts/KaTeX_AMS-Regular.woff2) format("woff2"),url(fonts/KaTeX_AMS-Regular.woff) format("woff"),url(fonts/KaTeX_AMS-Regular.ttf) format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Caligraphic;src:url(fonts/KaTeX_Caligraphic-Bold.woff2) format("woff2"),url(fonts/KaTeX_Caligraphic-Bold.woff) format("woff"),url(fonts/KaTeX_Caligraphic-Bold.ttf) format("truetype");font-weight:700;font-style:normal}@font-face{font-family:KaTeX_Caligraphic;src:url(fonts/KaTeX_Caligraphic-Regular.woff2) format("woff2"),url(fonts/KaTeX_Caligraphic-Regular.woff) format("woff"),url(fonts/KaTeX_Caligraphic-Regular.ttf) format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Fraktur;src:url(fonts/KaTeX_Fraktur-Bold.woff2) format("woff2"),url(fonts/KaTeX_Fraktur-Bold.woff) format("woff"),url(fonts/KaTeX_Fraktur-Bold.ttf) 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n=this.fetch();if(-1!==t.endOfExpression.indexOf(n.text))break;if(r&&n.text===r)break;if(e&&ia[n.text]&&ia[n.text].infix)break;var i=this.parseAtom(r);if(!i)break;"internal"!==i.type&&a.push(i)}return"text"===this.mode&&this.formLigatures(a),this.handleInfixNodes(a)},e.handleInfixNodes=function(t){for(var e,r=-1,a=0;a<t.length;a++)if("infix"===t[a].type){if(-1!==r)throw new o("only one infix operator per group",t[a].token);r=a,e=t[a].replaceWith}if(-1!==r&&e){var n,i,s=t.slice(0,r),l=t.slice(r+1);return n=1===s.length&&"ordgroup"===s[0].type?s[0]:{type:"ordgroup",mode:this.mode,body:s},i=1===l.length&&"ordgroup"===l[0].type?l[0]:{type:"ordgroup",mode:this.mode,body:l},["\\\\abovefrac"===e?this.callFunction(e,[n,t[r],i],[]):this.callFunction(e,[n,i],[])]}return t},e.handleSupSubscript=function(e){var r=this.fetch(),a=r.text;this.consume();var n=this.parseGroup(e,!1,t.SUPSUB_GREEDINESS,void 0,void 0,!0);if(!n)throw new o("Expected group after '"+a+"'",r);return n},e.formatUnsupportedCmd=function(t){for(var e=[],r=0;r<t.length;r++)e.push({type:"textord",mode:"text",text:t[r]});var a={type:"text",mode:this.mode,body:e};return{type:"color",mode:this.mode,color:this.settings.errorColor,body:[a]}},e.parseAtom=function(t){var e,r,a=this.parseGroup("atom",!1,null,t);if("text"===this.mode)return a;for(;;){this.consumeSpaces();var n=this.fetch();if("\\limits"===n.text||"\\nolimits"===n.text){if(a&&"op"===a.type){var i="\\limits"===n.text;a.limits=i,a.alwaysHandleSupSub=!0}else{if(!a||"operatorname"!==a.type||!a.alwaysHandleSupSub)throw new o("Limit controls must follow a math operator",n);var s="\\limits"===n.text;a.limits=s}this.consume()}else if("^"===n.text){if(e)throw new o("Double superscript",n);e=this.handleSupSubscript("superscript")}else if("_"===n.text){if(r)throw new o("Double subscript",n);r=this.handleSupSubscript("subscript")}else{if("'"!==n.text)break;if(e)throw new o("Double superscript",n);var l={type:"textord",mode:this.mode,text:"\\prime"},h=[l];for(this.consume();"'"===this.fetch().text;)h.push(l),this.consume();"^"===this.fetch().text&&h.push(this.handleSupSubscript("superscript")),e={type:"ordgroup",mode:this.mode,body:h}}}return e||r?{type:"supsub",mode:this.mode,base:a,sup:e,sub:r}:a},e.parseFunction=function(t,e,r){var a=this.fetch(),n=a.text,i=ia[n];if(!i)return null;if(this.consume(),null!=r&&i.greediness<=r)throw new o("Got function '"+n+"' with no arguments"+(e?" as "+e:""),a);if("text"===this.mode&&!i.allowedInText)throw new o("Can't use function '"+n+"' in text mode",a);if("math"===this.mode&&!1===i.allowedInMath)throw new o("Can't use function '"+n+"' in math mode",a);var s=this.parseArguments(n,i),l=s.args,h=s.optArgs;return this.callFunction(n,l,h,a,t)},e.callFunction=function(t,e,r,a,n){var i={funcName:t,parser:this,token:a,breakOnTokenText:n},s=ia[t];if(s&&s.handler)return s.handler(i,e,r);throw new o("No function handler for "+t)},e.parseArguments=function(t,e){var r=e.numArgs+e.numOptionalArgs;if(0===r)return{args:[],optArgs:[]};for(var a=e.greediness,n=[],i=[],s=0;s<r;s++){var l=e.argTypes&&e.argTypes[s],h=s<e.numOptionalArgs,m=s>0&&!h||0===s&&!h&&"math"===this.mode,c=this.parseGroupOfType("argument to '"+t+"'",l,h,a,m);if(!c){if(h){i.push(null);continue}throw new o("Expected group after '"+t+"'",this.fetch())}(h?i:n).push(c)}return{args:n,optArgs:i}},e.parseGroupOfType=function(t,e,r,a,n){switch(e){case"color":return n&&this.consumeSpaces(),this.parseColorGroup(r);case"size":return n&&this.consumeSpaces(),this.parseSizeGroup(r);case"url":return this.parseUrlGroup(r,n);case"math":case"text":return this.parseGroup(t,r,a,void 0,e,n);case"hbox":var i=this.parseGroup(t,r,a,void 0,"text",n);return i?{type:"styling",mode:i.mode,body:[i],style:"text"}:i;case"raw":if(n&&this.consumeSpaces(),r&&"{"===this.fetch().text)return null;var s=this.parseStringGroup("raw",r,!0);if(s)return{type:"raw",mode:"text",string:s.text};throw new o("Expected raw group",this.fetch());case"original":case null:case void 0:return this.parseGroup(t,r,a,void 0,void 0,n);default:throw new o("Unknown group type as "+t,this.fetch())}},e.consumeSpaces=function(){for(;" "===this.fetch().text;)this.consume()},e.parseStringGroup=function(t,e,r){var a=e?"[":"{",n=e?"]":"}",i=this.fetch();if(i.text!==a){if(e)return null;if(r&&"EOF"!==i.text&&/[^{}[\]]/.test(i.text))return this.consume(),i}var s=this.mode;this.mode="text",this.expect(a);for(var l,h="",m=this.fetch(),c=0,u=m;(l=this.fetch()).text!==n||r&&c>0;){switch(l.text){case"EOF":throw new o("Unexpected end of input in "+t,m.range(u,h));case a:c++;break;case n:c--}h+=(u=l).text,this.consume()}return this.expect(n),this.mode=s,m.range(u,h)},e.parseRegexGroup=function(t,e){var r=this.mode;this.mode="text";for(var a,n=this.fetch(),i=n,s="";"EOF"!==(a=this.fetch()).text&&t.test(s+a.text);)s+=(i=a).text,this.consume();if(""===s)throw new o("Invalid "+e+": '"+n.text+"'",n);return this.mode=r,n.range(i,s)},e.parseColorGroup=function(t){var e=this.parseStringGroup("color",t);if(!e)return null;var r=/^(#[a-f0-9]{3}|#?[a-f0-9]{6}|[a-z]+)$/i.exec(e.text);if(!r)throw new o("Invalid color: '"+e.text+"'",e);var a=r[0];return/^[0-9a-f]{6}$/i.test(a)&&(a="#"+a),{type:"color-token",mode:this.mode,color:a}},e.parseSizeGroup=function(t){var e,r=!1;if(!(e=t||"{"===this.fetch().text?this.parseStringGroup("size",t):this.parseRegexGroup(/^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2} *$/,"size")))return null;t||0!==e.text.length||(e.text="0pt",r=!0);var a=/([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(e.text);if(!a)throw new o("Invalid size: '"+e.text+"'",e);var n={number:+(a[1]+a[2]),unit:a[3]};if(!At(n))throw new o("Invalid unit: '"+n.unit+"'",e);return{type:"size",mode:this.mode,value:n,isBlank:r}},e.parseUrlGroup=function(t,e){this.gullet.lexer.setCatcode("%",13);var r=this.parseStringGroup("url",t,!0);if(this.gullet.lexer.setCatcode("%",14),!r)return null;var a=r.text.replace(/\\([#$%&~_^{}])/g,"$1");return{type:"url",mode:this.mode,url:a}},e.parseGroup=function(e,r,n,i,s,l){var h=this.mode;s&&this.switchMode(s),l&&this.consumeSpaces();var m,c=this.fetch(),u=c.text;if(r?"["===u:"{"===u||"\\begingroup"===u){this.consume();var p=t.endOfGroup[u];this.gullet.beginGroup();var d=this.parseExpression(!1,p),f=this.fetch();this.expect(p),this.gullet.endGroup(),m={type:"ordgroup",mode:this.mode,loc:a.range(c,f),body:d,semisimple:"\\begingroup"===u||void 0}}else if(r)m=null;else if(null==(m=this.parseFunction(i,e,n)||this.parseSymbol())&&"\\"===u[0]&&!ba.hasOwnProperty(u)){if(this.settings.throwOnError)throw new o("Undefined control sequence: "+u,c);m=this.formatUnsupportedCmd(u),this.consume()}return s&&this.switchMode(h),m},e.formLigatures=function(t){for(var e=t.length-1,r=0;r<e;++r){var n=t[r],i=n.text;"-"===i&&"-"===t[r+1].text&&(r+1<e&&"-"===t[r+2].text?(t.splice(r,3,{type:"textord",mode:"text",loc:a.range(n,t[r+2]),text:"---"}),e-=2):(t.splice(r,2,{type:"textord",mode:"text",loc:a.range(n,t[r+1]),text:"--"}),e-=1)),"'"!==i&&"`"!==i||t[r+1].text!==i||(t.splice(r,2,{type:"textord",mode:"text",loc:a.range(n,t[r+1]),text:i+i}),e-=1)}},e.parseSymbol=function(){var t=this.fetch(),e=t.text;if(/^\\verb[^a-zA-Z]/.test(e)){this.consume();var r=e.slice(5),n="*"===r.charAt(0);if(n&&(r=r.slice(1)),r.length<2||r.charAt(0)!==r.slice(-1))throw new o("\\verb assertion failed --\n                    please report what input caused this bug");return{type:"verb",mode:"text",body:r=r.slice(1,-1),star:n}}ka.hasOwnProperty(e[0])&&!j[this.mode][e[0]]&&(this.settings.strict&&"math"===this.mode&&this.settings.reportNonstrict("unicodeTextInMathMode",'Accented Unicode text character "'+e[0]+'" used in math mode',t),e=ka[e[0]]+e.substr(1));var i,s=sa.exec(e);if(s&&("i"===(e=e.substring(0,s.index))?e="\u0131":"j"===e&&(e="\u0237")),j[this.mode][e]){this.settings.strict&&"math"===this.mode&&"\xc7\xd0\xde\xe7\xfe".indexOf(e)>=0&&this.settings.reportNonstrict("unicodeTextInMathMode",'Latin-1/Unicode text character "'+e[0]+'" used in math mode',t);var l,h=j[this.mode][e].group,m=a.range(t);if(W.hasOwnProperty(h)){var c=h;l={type:"atom",mode:this.mode,family:c,loc:m,text:e}}else l={type:h,mode:this.mode,loc:m,text:e};i=l}else{if(!(e.charCodeAt(0)>=128))return null;this.settings.strict&&(M(e.charCodeAt(0))?"math"===this.mode&&this.settings.reportNonstrict("unicodeTextInMathMode",'Unicode text character "'+e[0]+'" used in math mode',t):this.settings.reportNonstrict("unknownSymbol",'Unrecognized Unicode character "'+e[0]+'" ('+e.charCodeAt(0)+")",t)),i={type:"textord",mode:"text",loc:a.range(t),text:e}}if(this.consume(),s)for(var u=0;u<s[0].length;u++){var p=s[0][u];if(!wa[p])throw new o("Unknown accent ' "+p+"'",t);var d=wa[p][this.mode];if(!d)throw new o("Accent "+p+" unsupported in "+this.mode+" mode",t);i={type:"accent",mode:this.mode,loc:a.range(t),label:d,isStretchy:!1,isShifty:!0,base:i}}return i},t}();Sa.endOfExpression=["}","\\endgroup","\\end","\\right","&"],Sa.endOfGroup={"[":"]","{":"}","\\begingroup":"\\endgroup"},Sa.SUPSUB_GREEDINESS=1;var Ma=function(t,e){if(!("string"==typeof t||t instanceof String))throw new TypeError("KaTeX can only parse string typed expression");var r=new Sa(t,e);delete r.gullet.macros.current["\\df@tag"];var a=r.parse();if(r.gullet.macros.get("\\df@tag")){if(!e.displayMode)throw new o("\\tag works only in display equations");r.gullet.feed("\\df@tag"),a=[{type:"tag",mode:"text",body:a,tag:r.parse()}]}return a},za=function(t,e,r){e.textContent="";var a=Ta(t,r).toNode();e.appendChild(a)};"undefined"!=typeof document&&"CSS1Compat"!==document.compatMode&&("undefined"!=typeof console&&console.warn("Warning: KaTeX doesn't work in quirks mode. Make sure your website has a suitable doctype."),za=function(){throw new o("KaTeX doesn't work in quirks mode.")});var Aa=function(t,e,r){if(r.throwOnError||!(t instanceof o))throw t;var a=Dt.makeSpan(["katex-error"],[new E(e)]);return a.setAttribute("title",t.toString()),a.setAttribute("style","color:"+r.errorColor),a},Ta=function(t,e){var r=new u(e);try{var a=Ma(t,r);return ke(a,t,r)}catch(e){return Aa(e,t,r)}},Ba={version:"0.12.0",render:za,renderToString:function(t,e){return Ta(t,e).toMarkup()},ParseError:o,__parse:function(t,e){var r=new u(e);return Ma(t,r)},__renderToDomTree:Ta,__renderToHTMLTree:function(t,e){var r=new u(e);try{return function(t,e,r){var a=le(t,ye(r)),n=Dt.makeSpan(["katex"],[a]);return we(n,r)}(Ma(t,r),0,r)}catch(e){return Aa(e,t,r)}},__setFontMetrics:function(t,e){F[t]=e},__defineSymbol:$,__defineMacro:pa,__domTree:{Span:N,Anchor:I,SymbolNode:E,SvgNode:L,PathNode:P,LineNode:D}};e.default=Ba}]).default});
\ No newline at end of file
diff --git a/docs/pt/katex/katex.mjs b/docs/pt/katex/katex.mjs
new file mode 100644
index 0000000..7693822
--- /dev/null
+++ b/docs/pt/katex/katex.mjs
@@ -0,0 +1,16911 @@
+/**
+ * Lexing or parsing positional information for error reporting.
+ * This object is immutable.
+ */
+class SourceLocation {
+  // The + prefix indicates that these fields aren't writeable
+  // Lexer holding the input string.
+  // Start offset, zero-based inclusive.
+  // End offset, zero-based exclusive.
+  constructor(lexer, start, end) {
+    this.lexer = void 0;
+    this.start = void 0;
+    this.end = void 0;
+    this.lexer = lexer;
+    this.start = start;
+    this.end = end;
+  }
+  /**
+   * Merges two `SourceLocation`s from location providers, given they are
+   * provided in order of appearance.
+   * - Returns the first one's location if only the first is provided.
+   * - Returns a merged range of the first and the last if both are provided
+   *   and their lexers match.
+   * - Otherwise, returns null.
+   */
+
+
+  static range(first, second) {
+    if (!second) {
+      return first && first.loc;
+    } else if (!first || !first.loc || !second.loc || first.loc.lexer !== second.loc.lexer) {
+      return null;
+    } else {
+      return new SourceLocation(first.loc.lexer, first.loc.start, second.loc.end);
+    }
+  }
+
+}
+
+/**
+ * Interface required to break circular dependency between Token, Lexer, and
+ * ParseError.
+ */
+
+/**
+ * The resulting token returned from `lex`.
+ *
+ * It consists of the token text plus some position information.
+ * The position information is essentially a range in an input string,
+ * but instead of referencing the bare input string, we refer to the lexer.
+ * That way it is possible to attach extra metadata to the input string,
+ * like for example a file name or similar.
+ *
+ * The position information is optional, so it is OK to construct synthetic
+ * tokens if appropriate. Not providing available position information may
+ * lead to degraded error reporting, though.
+ */
+class Token {
+  // don't expand the token
+  // used in \noexpand
+  constructor(text, // the text of this token
+  loc) {
+    this.text = void 0;
+    this.loc = void 0;
+    this.noexpand = void 0;
+    this.treatAsRelax = void 0;
+    this.text = text;
+    this.loc = loc;
+  }
+  /**
+   * Given a pair of tokens (this and endToken), compute a `Token` encompassing
+   * the whole input range enclosed by these two.
+   */
+
+
+  range(endToken, // last token of the range, inclusive
+  text) // the text of the newly constructed token
+  {
+    return new Token(text, SourceLocation.range(this, endToken));
+  }
+
+}
+
+/**
+ * This is the ParseError class, which is the main error thrown by KaTeX
+ * functions when something has gone wrong. This is used to distinguish internal
+ * errors from errors in the expression that the user provided.
+ *
+ * If possible, a caller should provide a Token or ParseNode with information
+ * about where in the source string the problem occurred.
+ */
+class ParseError {
+  // Error position based on passed-in Token or ParseNode.
+  constructor(message, // The error message
+  token) // An object providing position information
+  {
+    this.position = void 0;
+    let error = "KaTeX parse error: " + message;
+    let start;
+    const loc = token && token.loc;
+
+    if (loc && loc.start <= loc.end) {
+      // If we have the input and a position, make the error a bit fancier
+      // Get the input
+      const input = loc.lexer.input; // Prepend some information
+
+      start = loc.start;
+      const end = loc.end;
+
+      if (start === input.length) {
+        error += " at end of input: ";
+      } else {
+        error += " at position " + (start + 1) + ": ";
+      } // Underline token in question using combining underscores
+
+
+      const underlined = input.slice(start, end).replace(/[^]/g, "$&\u0332"); // Extract some context from the input and add it to the error
+
+      let left;
+
+      if (start > 15) {
+        left = "…" + input.slice(start - 15, start);
+      } else {
+        left = input.slice(0, start);
+      }
+
+      let right;
+
+      if (end + 15 < input.length) {
+        right = input.slice(end, end + 15) + "…";
+      } else {
+        right = input.slice(end);
+      }
+
+      error += left + underlined + right;
+    } // Some hackery to make ParseError a prototype of Error
+    // See http://stackoverflow.com/a/8460753
+
+
+    const self = new Error(error);
+    self.name = "ParseError"; // $FlowFixMe
+
+    self.__proto__ = ParseError.prototype; // $FlowFixMe
+
+    self.position = start;
+    return self;
+  }
+
+} // $FlowFixMe More hackery
+
+
+ParseError.prototype.__proto__ = Error.prototype;
+
+/**
+ * This file contains a list of utility functions which are useful in other
+ * files.
+ */
+
+/**
+ * Return whether an element is contained in a list
+ */
+const contains = function contains(list, elem) {
+  return list.indexOf(elem) !== -1;
+};
+/**
+ * Provide a default value if a setting is undefined
+ * NOTE: Couldn't use `T` as the output type due to facebook/flow#5022.
+ */
+
+
+const deflt = function deflt(setting, defaultIfUndefined) {
+  return setting === undefined ? defaultIfUndefined : setting;
+}; // hyphenate and escape adapted from Facebook's React under Apache 2 license
+
+
+const uppercase = /([A-Z])/g;
+
+const hyphenate = function hyphenate(str) {
+  return str.replace(uppercase, "-$1").toLowerCase();
+};
+
+const ESCAPE_LOOKUP = {
+  "&": "&amp;",
+  ">": "&gt;",
+  "<": "&lt;",
+  "\"": "&quot;",
+  "'": "&#x27;"
+};
+const ESCAPE_REGEX = /[&><"']/g;
+/**
+ * Escapes text to prevent scripting attacks.
+ */
+
+function escape(text) {
+  return String(text).replace(ESCAPE_REGEX, match => ESCAPE_LOOKUP[match]);
+}
+/**
+ * Sometimes we want to pull out the innermost element of a group. In most
+ * cases, this will just be the group itself, but when ordgroups and colors have
+ * a single element, we want to pull that out.
+ */
+
+
+const getBaseElem = function getBaseElem(group) {
+  if (group.type === "ordgroup") {
+    if (group.body.length === 1) {
+      return getBaseElem(group.body[0]);
+    } else {
+      return group;
+    }
+  } else if (group.type === "color") {
+    if (group.body.length === 1) {
+      return getBaseElem(group.body[0]);
+    } else {
+      return group;
+    }
+  } else if (group.type === "font") {
+    return getBaseElem(group.body);
+  } else {
+    return group;
+  }
+};
+/**
+ * TeXbook algorithms often reference "character boxes", which are simply groups
+ * with a single character in them. To decide if something is a character box,
+ * we find its innermost group, and see if it is a single character.
+ */
+
+
+const isCharacterBox = function isCharacterBox(group) {
+  const baseElem = getBaseElem(group); // These are all they types of groups which hold single characters
+
+  return baseElem.type === "mathord" || baseElem.type === "textord" || baseElem.type === "atom";
+};
+
+const assert = function assert(value) {
+  if (!value) {
+    throw new Error('Expected non-null, but got ' + String(value));
+  }
+
+  return value;
+};
+/**
+ * Return the protocol of a URL, or "_relative" if the URL does not specify a
+ * protocol (and thus is relative).
+ */
+
+const protocolFromUrl = function protocolFromUrl(url) {
+  const protocol = /^\s*([^\\/#]*?)(?::|&#0*58|&#x0*3a)/i.exec(url);
+  return protocol != null ? protocol[1] : "_relative";
+};
+var utils = {
+  contains,
+  deflt,
+  escape,
+  hyphenate,
+  getBaseElem,
+  isCharacterBox,
+  protocolFromUrl
+};
+
+/* eslint no-console:0 */
+
+/**
+ * The main Settings object
+ *
+ * The current options stored are:
+ *  - displayMode: Whether the expression should be typeset as inline math
+ *                 (false, the default), meaning that the math starts in
+ *                 \textstyle and is placed in an inline-block); or as display
+ *                 math (true), meaning that the math starts in \displaystyle
+ *                 and is placed in a block with vertical margin.
+ */
+class Settings {
+  constructor(options) {
+    this.displayMode = void 0;
+    this.output = void 0;
+    this.leqno = void 0;
+    this.fleqn = void 0;
+    this.throwOnError = void 0;
+    this.errorColor = void 0;
+    this.macros = void 0;
+    this.minRuleThickness = void 0;
+    this.colorIsTextColor = void 0;
+    this.strict = void 0;
+    this.trust = void 0;
+    this.maxSize = void 0;
+    this.maxExpand = void 0;
+    this.globalGroup = void 0;
+    // allow null options
+    options = options || {};
+    this.displayMode = utils.deflt(options.displayMode, false);
+    this.output = utils.deflt(options.output, "htmlAndMathml");
+    this.leqno = utils.deflt(options.leqno, false);
+    this.fleqn = utils.deflt(options.fleqn, false);
+    this.throwOnError = utils.deflt(options.throwOnError, true);
+    this.errorColor = utils.deflt(options.errorColor, "#cc0000");
+    this.macros = options.macros || {};
+    this.minRuleThickness = Math.max(0, utils.deflt(options.minRuleThickness, 0));
+    this.colorIsTextColor = utils.deflt(options.colorIsTextColor, false);
+    this.strict = utils.deflt(options.strict, "warn");
+    this.trust = utils.deflt(options.trust, false);
+    this.maxSize = Math.max(0, utils.deflt(options.maxSize, Infinity));
+    this.maxExpand = Math.max(0, utils.deflt(options.maxExpand, 1000));
+    this.globalGroup = utils.deflt(options.globalGroup, false);
+  }
+  /**
+   * Report nonstrict (non-LaTeX-compatible) input.
+   * Can safely not be called if `this.strict` is false in JavaScript.
+   */
+
+
+  reportNonstrict(errorCode, errorMsg, token) {
+    let strict = this.strict;
+
+    if (typeof strict === "function") {
+      // Allow return value of strict function to be boolean or string
+      // (or null/undefined, meaning no further processing).
+      strict = strict(errorCode, errorMsg, token);
+    }
+
+    if (!strict || strict === "ignore") {
+      return;
+    } else if (strict === true || strict === "error") {
+      throw new ParseError("LaTeX-incompatible input and strict mode is set to 'error': " + `${errorMsg} [${errorCode}]`, token);
+    } else if (strict === "warn") {
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to 'warn': " + `${errorMsg} [${errorCode}]`);
+    } else {
+      // won't happen in type-safe code
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to " + `unrecognized '${strict}': ${errorMsg} [${errorCode}]`);
+    }
+  }
+  /**
+   * Check whether to apply strict (LaTeX-adhering) behavior for unusual
+   * input (like `\\`).  Unlike `nonstrict`, will not throw an error;
+   * instead, "error" translates to a return value of `true`, while "ignore"
+   * translates to a return value of `false`.  May still print a warning:
+   * "warn" prints a warning and returns `false`.
+   * This is for the second category of `errorCode`s listed in the README.
+   */
+
+
+  useStrictBehavior(errorCode, errorMsg, token) {
+    let strict = this.strict;
+
+    if (typeof strict === "function") {
+      // Allow return value of strict function to be boolean or string
+      // (or null/undefined, meaning no further processing).
+      // But catch any exceptions thrown by function, treating them
+      // like "error".
+      try {
+        strict = strict(errorCode, errorMsg, token);
+      } catch (error) {
+        strict = "error";
+      }
+    }
+
+    if (!strict || strict === "ignore") {
+      return false;
+    } else if (strict === true || strict === "error") {
+      return true;
+    } else if (strict === "warn") {
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to 'warn': " + `${errorMsg} [${errorCode}]`);
+      return false;
+    } else {
+      // won't happen in type-safe code
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to " + `unrecognized '${strict}': ${errorMsg} [${errorCode}]`);
+      return false;
+    }
+  }
+  /**
+   * Check whether to test potentially dangerous input, and return
+   * `true` (trusted) or `false` (untrusted).  The sole argument `context`
+   * should be an object with `command` field specifying the relevant LaTeX
+   * command (as a string starting with `\`), and any other arguments, etc.
+   * If `context` has a `url` field, a `protocol` field will automatically
+   * get added by this function (changing the specified object).
+   */
+
+
+  isTrusted(context) {
+    if (context.url && !context.protocol) {
+      context.protocol = utils.protocolFromUrl(context.url);
+    }
+
+    const trust = typeof this.trust === "function" ? this.trust(context) : this.trust;
+    return Boolean(trust);
+  }
+
+}
+
+/**
+ * This file contains information and classes for the various kinds of styles
+ * used in TeX. It provides a generic `Style` class, which holds information
+ * about a specific style. It then provides instances of all the different kinds
+ * of styles possible, and provides functions to move between them and get
+ * information about them.
+ */
+
+/**
+ * The main style class. Contains a unique id for the style, a size (which is
+ * the same for cramped and uncramped version of a style), and a cramped flag.
+ */
+class Style {
+  constructor(id, size, cramped) {
+    this.id = void 0;
+    this.size = void 0;
+    this.cramped = void 0;
+    this.id = id;
+    this.size = size;
+    this.cramped = cramped;
+  }
+  /**
+   * Get the style of a superscript given a base in the current style.
+   */
+
+
+  sup() {
+    return styles[sup[this.id]];
+  }
+  /**
+   * Get the style of a subscript given a base in the current style.
+   */
+
+
+  sub() {
+    return styles[sub[this.id]];
+  }
+  /**
+   * Get the style of a fraction numerator given the fraction in the current
+   * style.
+   */
+
+
+  fracNum() {
+    return styles[fracNum[this.id]];
+  }
+  /**
+   * Get the style of a fraction denominator given the fraction in the current
+   * style.
+   */
+
+
+  fracDen() {
+    return styles[fracDen[this.id]];
+  }
+  /**
+   * Get the cramped version of a style (in particular, cramping a cramped style
+   * doesn't change the style).
+   */
+
+
+  cramp() {
+    return styles[cramp[this.id]];
+  }
+  /**
+   * Get a text or display version of this style.
+   */
+
+
+  text() {
+    return styles[text[this.id]];
+  }
+  /**
+   * Return true if this style is tightly spaced (scriptstyle/scriptscriptstyle)
+   */
+
+
+  isTight() {
+    return this.size >= 2;
+  }
+
+} // Export an interface for type checking, but don't expose the implementation.
+// This way, no more styles can be generated.
+
+
+// IDs of the different styles
+const D = 0;
+const Dc = 1;
+const T = 2;
+const Tc = 3;
+const S = 4;
+const Sc = 5;
+const SS = 6;
+const SSc = 7; // Instances of the different styles
+
+const styles = [new Style(D, 0, false), new Style(Dc, 0, true), new Style(T, 1, false), new Style(Tc, 1, true), new Style(S, 2, false), new Style(Sc, 2, true), new Style(SS, 3, false), new Style(SSc, 3, true)]; // Lookup tables for switching from one style to another
+
+const sup = [S, Sc, S, Sc, SS, SSc, SS, SSc];
+const sub = [Sc, Sc, Sc, Sc, SSc, SSc, SSc, SSc];
+const fracNum = [T, Tc, S, Sc, SS, SSc, SS, SSc];
+const fracDen = [Tc, Tc, Sc, Sc, SSc, SSc, SSc, SSc];
+const cramp = [Dc, Dc, Tc, Tc, Sc, Sc, SSc, SSc];
+const text = [D, Dc, T, Tc, T, Tc, T, Tc]; // We only export some of the styles.
+
+var Style$1 = {
+  DISPLAY: styles[D],
+  TEXT: styles[T],
+  SCRIPT: styles[S],
+  SCRIPTSCRIPT: styles[SS]
+};
+
+/*
+ * This file defines the Unicode scripts and script families that we
+ * support. To add new scripts or families, just add a new entry to the
+ * scriptData array below. Adding scripts to the scriptData array allows
+ * characters from that script to appear in \text{} environments.
+ */
+
+/**
+ * Each script or script family has a name and an array of blocks.
+ * Each block is an array of two numbers which specify the start and
+ * end points (inclusive) of a block of Unicode codepoints.
+ */
+
+/**
+ * Unicode block data for the families of scripts we support in \text{}.
+ * Scripts only need to appear here if they do not have font metrics.
+ */
+const scriptData = [{
+  // Latin characters beyond the Latin-1 characters we have metrics for.
+  // Needed for Czech, Hungarian and Turkish text, for example.
+  name: 'latin',
+  blocks: [[0x0100, 0x024f], // Latin Extended-A and Latin Extended-B
+  [0x0300, 0x036f]]
+}, {
+  // The Cyrillic script used by Russian and related languages.
+  // A Cyrillic subset used to be supported as explicitly defined
+  // symbols in symbols.js
+  name: 'cyrillic',
+  blocks: [[0x0400, 0x04ff]]
+}, {
+  // The Brahmic scripts of South and Southeast Asia
+  // Devanagari (0900–097F)
+  // Bengali (0980–09FF)
+  // Gurmukhi (0A00–0A7F)
+  // Gujarati (0A80–0AFF)
+  // Oriya (0B00–0B7F)
+  // Tamil (0B80–0BFF)
+  // Telugu (0C00–0C7F)
+  // Kannada (0C80–0CFF)
+  // Malayalam (0D00–0D7F)
+  // Sinhala (0D80–0DFF)
+  // Thai (0E00–0E7F)
+  // Lao (0E80–0EFF)
+  // Tibetan (0F00–0FFF)
+  // Myanmar (1000–109F)
+  name: 'brahmic',
+  blocks: [[0x0900, 0x109F]]
+}, {
+  name: 'georgian',
+  blocks: [[0x10A0, 0x10ff]]
+}, {
+  // Chinese and Japanese.
+  // The "k" in cjk is for Korean, but we've separated Korean out
+  name: "cjk",
+  blocks: [[0x3000, 0x30FF], // CJK symbols and punctuation, Hiragana, Katakana
+  [0x4E00, 0x9FAF], // CJK ideograms
+  [0xFF00, 0xFF60]]
+}, {
+  // Korean
+  name: 'hangul',
+  blocks: [[0xAC00, 0xD7AF]]
+}];
+/**
+ * Given a codepoint, return the name of the script or script family
+ * it is from, or null if it is not part of a known block
+ */
+
+function scriptFromCodepoint(codepoint) {
+  for (let i = 0; i < scriptData.length; i++) {
+    const script = scriptData[i];
+
+    for (let i = 0; i < script.blocks.length; i++) {
+      const block = script.blocks[i];
+
+      if (codepoint >= block[0] && codepoint <= block[1]) {
+        return script.name;
+      }
+    }
+  }
+
+  return null;
+}
+/**
+ * A flattened version of all the supported blocks in a single array.
+ * This is an optimization to make supportedCodepoint() fast.
+ */
+
+const allBlocks = [];
+scriptData.forEach(s => s.blocks.forEach(b => allBlocks.push(...b)));
+/**
+ * Given a codepoint, return true if it falls within one of the
+ * scripts or script families defined above and false otherwise.
+ *
+ * Micro benchmarks shows that this is faster than
+ * /[\u3000-\u30FF\u4E00-\u9FAF\uFF00-\uFF60\uAC00-\uD7AF\u0900-\u109F]/.test()
+ * in Firefox, Chrome and Node.
+ */
+
+function supportedCodepoint(codepoint) {
+  for (let i = 0; i < allBlocks.length; i += 2) {
+    if (codepoint >= allBlocks[i] && codepoint <= allBlocks[i + 1]) {
+      return true;
+    }
+  }
+
+  return false;
+}
+
+/**
+ * This file provides support to domTree.js and delimiter.js.
+ * It's a storehouse of path geometry for SVG images.
+ */
+// In all paths below, the viewBox-to-em scale is 1000:1.
+const hLinePad = 80; // padding above a sqrt viniculum. Prevents image cropping.
+// The viniculum of a \sqrt can be made thicker by a KaTeX rendering option.
+// Think of variable extraViniculum as two detours in the SVG path.
+// The detour begins at the lower left of the area labeled extraViniculum below.
+// The detour proceeds one extraViniculum distance up and slightly to the right,
+// displacing the radiused corner between surd and viniculum. The radius is
+// traversed as usual, then the detour resumes. It goes right, to the end of
+// the very long viniculumn, then down one extraViniculum distance,
+// after which it resumes regular path geometry for the radical.
+
+/*                                                  viniculum
+                                                   /
+         /▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒←extraViniculum
+        / █████████████████████←0.04em (40 unit) std viniculum thickness
+       / /
+      / /
+     / /\
+    / / surd
+*/
+
+const sqrtMain = function sqrtMain(extraViniculum, hLinePad) {
+  // sqrtMain path geometry is from glyph U221A in the font KaTeX Main
+  return `M95,${622 + extraViniculum + hLinePad}
+c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
+c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
+c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
+s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
+c69,-144,104.5,-217.7,106.5,-221
+l${extraViniculum / 2.075} -${extraViniculum}
+c5.3,-9.3,12,-14,20,-14
+H400000v${40 + extraViniculum}H845.2724
+s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
+c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
+M${834 + extraViniculum} ${hLinePad}h400000v${40 + extraViniculum}h-400000z`;
+};
+
+const sqrtSize1 = function sqrtSize1(extraViniculum, hLinePad) {
+  // size1 is from glyph U221A in the font KaTeX_Size1-Regular
+  return `M263,${601 + extraViniculum + hLinePad}c0.7,0,18,39.7,52,119
+c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
+c340,-704.7,510.7,-1060.3,512,-1067
+l${extraViniculum / 2.084} -${extraViniculum}
+c4.7,-7.3,11,-11,19,-11
+H40000v${40 + extraViniculum}H1012.3
+s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
+c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
+s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
+c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z
+M${1001 + extraViniculum} ${hLinePad}h400000v${40 + extraViniculum}h-400000z`;
+};
+
+const sqrtSize2 = function sqrtSize2(extraViniculum, hLinePad) {
+  // size2 is from glyph U221A in the font KaTeX_Size2-Regular
+  return `M983 ${10 + extraViniculum + hLinePad}
+l${extraViniculum / 3.13} -${extraViniculum}
+c4,-6.7,10,-10,18,-10 H400000v${40 + extraViniculum}
+H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7
+s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744
+c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30
+c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722
+c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5
+c53.7,-170.3,84.5,-266.8,92.5,-289.5z
+M${1001 + extraViniculum} ${hLinePad}h400000v${40 + extraViniculum}h-400000z`;
+};
+
+const sqrtSize3 = function sqrtSize3(extraViniculum, hLinePad) {
+  // size3 is from glyph U221A in the font KaTeX_Size3-Regular
+  return `M424,${2398 + extraViniculum + hLinePad}
+c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514
+c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20
+s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121
+s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081
+l${extraViniculum / 4.223} -${extraViniculum}c4,-6.7,10,-10,18,-10 H400000
+v${40 + extraViniculum}H1014.6
+s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185
+c-2,6,-10,9,-24,9
+c-8,0,-12,-0.7,-12,-2z M${1001 + extraViniculum} ${hLinePad}
+h400000v${40 + extraViniculum}h-400000z`;
+};
+
+const sqrtSize4 = function sqrtSize4(extraViniculum, hLinePad) {
+  // size4 is from glyph U221A in the font KaTeX_Size4-Regular
+  return `M473,${2713 + extraViniculum + hLinePad}
+c339.3,-1799.3,509.3,-2700,510,-2702 l${extraViniculum / 5.298} -${extraViniculum}
+c3.3,-7.3,9.3,-11,18,-11 H400000v${40 + extraViniculum}H1017.7
+s-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9
+c-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200
+c0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26
+s76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,
+606zM${1001 + extraViniculum} ${hLinePad}h400000v${40 + extraViniculum}H1017.7z`;
+};
+
+const sqrtTall = function sqrtTall(extraViniculum, hLinePad, viewBoxHeight) {
+  // sqrtTall is from glyph U23B7 in the font KaTeX_Size4-Regular
+  // One path edge has a variable length. It runs vertically from the viniculumn
+  // to a point near (14 units) the bottom of the surd. The viniculum
+  // is normally 40 units thick. So the length of the line in question is:
+  const vertSegment = viewBoxHeight - 54 - hLinePad - extraViniculum;
+  return `M702 ${extraViniculum + hLinePad}H400000${40 + extraViniculum}
+H742v${vertSegment}l-4 4-4 4c-.667.7 -2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1
+h-12l-28-84c-16.667-52-96.667 -294.333-240-727l-212 -643 -85 170
+c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667
+219 661 l218 661zM702 ${hLinePad}H400000v${40 + extraViniculum}H742z`;
+};
+
+const sqrtPath = function sqrtPath(size, extraViniculum, viewBoxHeight) {
+  extraViniculum = 1000 * extraViniculum; // Convert from document ems to viewBox.
+
+  let path = "";
+
+  switch (size) {
+    case "sqrtMain":
+      path = sqrtMain(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize1":
+      path = sqrtSize1(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize2":
+      path = sqrtSize2(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize3":
+      path = sqrtSize3(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize4":
+      path = sqrtSize4(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtTall":
+      path = sqrtTall(extraViniculum, hLinePad, viewBoxHeight);
+  }
+
+  return path;
+};
+const path = {
+  // Two paths that cover gaps in built-up parentheses.
+  leftParenInner: `M291 0 H417 V300 H291 z`,
+  rightParenInner: `M457 0 H583 V300 H457 z`,
+  // The doubleleftarrow geometry is from glyph U+21D0 in the font KaTeX Main
+  doubleleftarrow: `M262 157
+l10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3
+ 0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28
+ 14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5
+c2 1.7 6.3 3.5 13 5.5 68 17.3 128.2 47.8 180.5 91.5 52.3 43.7 93.8 96.2 124.5
+ 157.5 9.3 8 15.3 12.3 18 13h6c12-.7 18-4 18-10 0-2-1.7-7-5-15-23.3-46-52-87
+-86-123l-10-10h399738v-40H218c328 0 0 0 0 0l-10-8c-26.7-20-65.7-43-117-69 2.7
+-2 6-3.7 10-5 36.7-16 72.3-37.3 107-64l10-8h399782v-40z
+m8 0v40h399730v-40zm0 194v40h399730v-40z`,
+  // doublerightarrow is from glyph U+21D2 in font KaTeX Main
+  doublerightarrow: `M399738 392l
+-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
+ 14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
+-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
+-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
+-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
+c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
+-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z`,
+  // leftarrow is from glyph U+2190 in font KaTeX Main
+  leftarrow: `M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z`,
+  // overbrace is from glyphs U+23A9/23A8/23A7 in font KaTeX_Size4-Regular
+  leftbrace: `M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
+-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7
+ 5-6 9-10 13-.7 1-7.3 1-20 1H6z`,
+  leftbraceunder: `M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
+ 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
+ 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
+-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z`,
+  // overgroup is from the MnSymbol package (public domain)
+  leftgroup: `M400000 80
+H435C64 80 168.3 229.4 21 260c-5.9 1.2-18 0-18 0-2 0-3-1-3-3v-38C76 61 257 0
+ 435 0h399565z`,
+  leftgroupunder: `M400000 262
+H435C64 262 168.3 112.6 21 82c-5.9-1.2-18 0-18 0-2 0-3 1-3 3v38c76 158 257 219
+ 435 219h399565z`,
+  // Harpoons are from glyph U+21BD in font KaTeX Main
+  leftharpoon: `M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3
+-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5
+-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7
+-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z`,
+  leftharpoonplus: `M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3-3.3 10.2-9.5
+ 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5-18.3 3-21-1.3
+-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7-196 228-6.7 4.7
+-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40zM0 435v40h400000v-40z
+m0 0v40h400000v-40z`,
+  leftharpoondown: `M7 241c-4 4-6.333 8.667-7 14 0 5.333.667 9 2 11s5.333
+ 5.333 12 10c90.667 54 156 130 196 228 3.333 10.667 6.333 16.333 9 17 2 .667 5
+ 1 9 1h5c10.667 0 16.667-2 18-6 2-2.667 1-9.667-3-21-32-87.333-82.667-157.667
+-152-211l-3-3h399907v-40zM93 281 H400000 v-40L7 241z`,
+  leftharpoondownplus: `M7 435c-4 4-6.3 8.7-7 14 0 5.3.7 9 2 11s5.3 5.3 12
+ 10c90.7 54 156 130 196 228 3.3 10.7 6.3 16.3 9 17 2 .7 5 1 9 1h5c10.7 0 16.7
+-2 18-6 2-2.7 1-9.7-3-21-32-87.3-82.7-157.7-152-211l-3-3h399907v-40H7zm93 0
+v40h399900v-40zM0 241v40h399900v-40zm0 0v40h399900v-40z`,
+  // hook is from glyph U+21A9 in font KaTeX Main
+  lefthook: `M400000 281 H103s-33-11.2-61-33.5S0 197.3 0 164s14.2-61.2 42.5
+-83.5C70.8 58.2 104 47 142 47 c16.7 0 25 6.7 25 20 0 12-8.7 18.7-26 20-40 3.3
+-68.7 15.7-86 37-10 12-15 25.3-15 40 0 22.7 9.8 40.7 29.5 54 19.7 13.3 43.5 21
+ 71.5 23h399859zM103 281v-40h399897v40z`,
+  leftlinesegment: `M40 281 V428 H0 V94 H40 V241 H400000 v40z
+M40 281 V428 H0 V94 H40 V241 H400000 v40z`,
+  leftmapsto: `M40 281 V448H0V74H40V241H400000v40z
+M40 281 V448H0V74H40V241H400000v40z`,
+  // tofrom is from glyph U+21C4 in font KaTeX AMS Regular
+  leftToFrom: `M0 147h400000v40H0zm0 214c68 40 115.7 95.7 143 167h22c15.3 0 23
+-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69-70-101l-7-8h399905v-40H95l7-8
+c28.7-32 52-65.7 70-101 10.7-23.3 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 265.3
+ 68 321 0 361zm0-174v-40h399900v40zm100 154v40h399900v-40z`,
+  longequal: `M0 50 h400000 v40H0z m0 194h40000v40H0z
+M0 50 h400000 v40H0z m0 194h40000v40H0z`,
+  midbrace: `M200428 334
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+ 311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11
+ 12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z`,
+  midbraceunder: `M199572 214
+c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14
+ 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3
+ 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0
+-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z`,
+  oiintSize1: `M512.6 71.6c272.6 0 320.3 106.8 320.3 178.2 0 70.8-47.7 177.6
+-320.3 177.6S193.1 320.6 193.1 249.8c0-71.4 46.9-178.2 319.5-178.2z
+m368.1 178.2c0-86.4-60.9-215.4-368.1-215.4-306.4 0-367.3 129-367.3 215.4 0 85.8
+60.9 214.8 367.3 214.8 307.2 0 368.1-129 368.1-214.8z`,
+  oiintSize2: `M757.8 100.1c384.7 0 451.1 137.6 451.1 230 0 91.3-66.4 228.8
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+m502.4 230c0-111.2-82.4-277.2-502.4-277.2s-504 166-504 277.2
+c0 110 84 276 504 276s502.4-166 502.4-276z`,
+  oiiintSize1: `M681.4 71.6c408.9 0 480.5 106.8 480.5 178.2 0 70.8-71.6 177.6
+-480.5 177.6S202.1 320.6 202.1 249.8c0-71.4 70.5-178.2 479.3-178.2z
+m525.8 178.2c0-86.4-86.8-215.4-525.7-215.4-437.9 0-524.7 129-524.7 215.4 0
+85.8 86.8 214.8 524.7 214.8 438.9 0 525.7-129 525.7-214.8z`,
+  oiiintSize2: `M1021.2 53c603.6 0 707.8 165.8 707.8 277.2 0 110-104.2 275.8
+-707.8 275.8-606 0-710.2-165.8-710.2-275.8C311 218.8 415.2 53 1021.2 53z
+m770.4 277.1c0-131.2-126.4-327.6-770.5-327.6S248.4 198.9 248.4 330.1
+c0 130 128.8 326.4 772.7 326.4s770.5-196.4 770.5-326.4z`,
+  rightarrow: `M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z`,
+  rightbrace: `M400000 542l
+-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
+s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
+c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z`,
+  rightbraceunder: `M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3
+ 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237
+-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z`,
+  rightgroup: `M0 80h399565c371 0 266.7 149.4 414 180 5.9 1.2 18 0 18 0 2 0
+ 3-1 3-3v-38c-76-158-257-219-435-219H0z`,
+  rightgroupunder: `M0 262h399565c371 0 266.7-149.4 414-180 5.9-1.2 18 0 18
+ 0 2 0 3 1 3 3v38c-76 158-257 219-435 219H0z`,
+  rightharpoon: `M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3
+-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2
+-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58
+ 69.2 92 94.5zm0 0v40h399900v-40z`,
+  rightharpoonplus: `M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3-3.7-15.3-11
+-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2-10.7 0-16.7
+ 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58 69.2 92 94.5z
+m0 0v40h399900v-40z m100 194v40h399900v-40zm0 0v40h399900v-40z`,
+  rightharpoondown: `M399747 511c0 7.3 6.7 11 20 11 8 0 13-.8 15-2.5s4.7-6.8
+ 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3 8.5-5.8 9.5
+-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3-64.7 57-92 95
+-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 241v40h399900v-40z`,
+  rightharpoondownplus: `M399747 705c0 7.3 6.7 11 20 11 8 0 13-.8
+ 15-2.5s4.7-6.8 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3
+ 8.5-5.8 9.5-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3
+-64.7 57-92 95-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 435v40h399900v-40z
+m0-194v40h400000v-40zm0 0v40h400000v-40z`,
+  righthook: `M399859 241c-764 0 0 0 0 0 40-3.3 68.7-15.7 86-37 10-12 15-25.3
+ 15-40 0-22.7-9.8-40.7-29.5-54-19.7-13.3-43.5-21-71.5-23-17.3-1.3-26-8-26-20 0
+-13.3 8.7-20 26-20 38 0 71 11.2 99 33.5 0 0 7 5.6 21 16.7 14 11.2 21 33.5 21
+ 66.8s-14 61.2-42 83.5c-28 22.3-61 33.5-99 33.5L0 241z M0 281v-40h399859v40z`,
+  rightlinesegment: `M399960 241 V94 h40 V428 h-40 V281 H0 v-40z
+M399960 241 V94 h40 V428 h-40 V281 H0 v-40z`,
+  rightToFrom: `M400000 167c-70.7-42-118-97.7-142-167h-23c-15.3 0-23 .3-23
+ 1 0 1.3 5.3 13.7 16 37 18 35.3 41.3 69 70 101l7 8H0v40h399905l-7 8c-28.7 32
+-52 65.7-70 101-10.7 23.3-16 35.7-16 37 0 .7 7.7 1 23 1h23c24-69.3 71.3-125 142
+-167z M100 147v40h399900v-40zM0 341v40h399900v-40z`,
+  // twoheadleftarrow is from glyph U+219E in font KaTeX AMS Regular
+  twoheadleftarrow: `M0 167c68 40
+ 115.7 95.7 143 167h22c15.3 0 23-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69
+-70-101l-7-8h125l9 7c50.7 39.3 85 86 103 140h46c0-4.7-6.3-18.7-19-42-18-35.3
+-40-67.3-66-96l-9-9h399716v-40H284l9-9c26-28.7 48-60.7 66-96 12.7-23.333 19
+-37.333 19-42h-46c-18 54-52.3 100.7-103 140l-9 7H95l7-8c28.7-32 52-65.7 70-101
+ 10.7-23.333 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 71.3 68 127 0 167z`,
+  twoheadrightarrow: `M400000 167
+c-68-40-115.7-95.7-143-167h-22c-15.3 0-23 .3-23 1 0 1.3 5.3 13.7 16 37 18 35.3
+ 41.3 69 70 101l7 8h-125l-9-7c-50.7-39.3-85-86-103-140h-46c0 4.7 6.3 18.7 19 42
+ 18 35.3 40 67.3 66 96l9 9H0v40h399716l-9 9c-26 28.7-48 60.7-66 96-12.7 23.333
+-19 37.333-19 42h46c18-54 52.3-100.7 103-140l9-7h125l-7 8c-28.7 32-52 65.7-70
+ 101-10.7 23.333-16 35.7-16 37 0 .7 7.7 1 23 1h22c27.3-71.3 75-127 143-167z`,
+  // tilde1 is a modified version of a glyph from the MnSymbol package
+  tilde1: `M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
+-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
+ 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
+ 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
+-68.267.847-113-73.952-191-73.952z`,
+  // ditto tilde2, tilde3, & tilde4
+  tilde2: `M344 55.266c-142 0-300.638 81.316-311.5 86.418
+-8.01 3.762-22.5 10.91-23.5 5.562L1 120c-1-2-1-3-1-4 0-5 3-9 8-10l18.4-9C160.9
+ 31.9 283 0 358 0c148 0 188 122 331 122s314-97 326-97c4 0 8 2 10 7l7 21.114
+c1 2.14 1 3.21 1 4.28 0 5.347-3 9.626-7 10.696l-22.3 12.622C852.6 158.372 751
+ 181.476 676 181.476c-149 0-189-126.21-332-126.21z`,
+  tilde3: `M786 59C457 59 32 175.242 13 175.242c-6 0-10-3.457
+-11-10.37L.15 138c-1-7 3-12 10-13l19.2-6.4C378.4 40.7 634.3 0 804.3 0c337 0
+ 411.8 157 746.8 157 328 0 754-112 773-112 5 0 10 3 11 9l1 14.075c1 8.066-.697
+ 16.595-6.697 17.492l-21.052 7.31c-367.9 98.146-609.15 122.696-778.15 122.696
+ -338 0-409-156.573-744-156.573z`,
+  tilde4: `M786 58C457 58 32 177.487 13 177.487c-6 0-10-3.345
+-11-10.035L.15 143c-1-7 3-12 10-13l22-6.7C381.2 35 637.15 0 807.15 0c337 0 409
+ 177 744 177 328 0 754-127 773-127 5 0 10 3 11 9l1 14.794c1 7.805-3 13.38-9
+ 14.495l-20.7 5.574c-366.85 99.79-607.3 139.372-776.3 139.372-338 0-409
+ -175.236-744-175.236z`,
+  // vec is from glyph U+20D7 in font KaTeX Main
+  vec: `M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
+3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
+10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
+-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
+-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
+H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
+c-16-25.333-24-45-24-59z`,
+  // widehat1 is a modified version of a glyph from the MnSymbol package
+  widehat1: `M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22
+c-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z`,
+  // ditto widehat2, widehat3, & widehat4
+  widehat2: `M1181 0h2l1171 176c6 0 10 5 10 11l-2 23c-1 6-5 10
+-11 10h-1L1182 67 15 220h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z`,
+  widehat3: `M1181 0h2l1171 236c6 0 10 5 10 11l-2 23c-1 6-5 10
+-11 10h-1L1182 67 15 280h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z`,
+  widehat4: `M1181 0h2l1171 296c6 0 10 5 10 11l-2 23c-1 6-5 10
+-11 10h-1L1182 67 15 340h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z`,
+  // widecheck paths are all inverted versions of widehat
+  widecheck1: `M529,159h5l519,-115c5,-1,9,-5,9,-10c0,-1,-1,-2,-1,-3l-4,-22c-1,
+-5,-5,-9,-11,-9h-2l-512,92l-513,-92h-2c-5,0,-9,4,-11,9l-5,22c-1,6,2,12,8,13z`,
+  widecheck2: `M1181,220h2l1171,-176c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,
+-11,-10h-1l-1168,153l-1167,-153h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z`,
+  widecheck3: `M1181,280h2l1171,-236c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,
+-11,-10h-1l-1168,213l-1167,-213h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z`,
+  widecheck4: `M1181,340h2l1171,-296c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,
+-11,-10h-1l-1168,273l-1167,-273h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z`,
+  // The next ten paths support reaction arrows from the mhchem package.
+  // Arrows for \ce{<-->} are offset from xAxis by 0.22ex, per mhchem in LaTeX
+  // baraboveleftarrow is mostly from from glyph U+2190 in font KaTeX Main
+  baraboveleftarrow: `M400000 620h-399890l3 -3c68.7 -52.7 113.7 -120 135 -202
+c4 -14.7 6 -23 6 -25c0 -7.3 -7 -11 -21 -11c-8 0 -13.2 0.8 -15.5 2.5
+c-2.3 1.7 -4.2 5.8 -5.5 12.5c-1.3 4.7 -2.7 10.3 -4 17c-12 48.7 -34.8 92 -68.5 130
+s-74.2 66.3 -121.5 85c-10 4 -16 7.7 -18 11c0 8.7 6 14.3 18 17c47.3 18.7 87.8 47
+121.5 85s56.5 81.3 68.5 130c0.7 2 1.3 5 2 9s1.2 6.7 1.5 8c0.3 1.3 1 3.3 2 6
+s2.2 4.5 3.5 5.5c1.3 1 3.3 1.8 6 2.5s6 1 10 1c14 0 21 -3.7 21 -11
+c0 -2 -2 -10.3 -6 -25c-20 -79.3 -65 -146.7 -135 -202l-3 -3h399890z
+M100 620v40h399900v-40z M0 241v40h399900v-40zM0 241v40h399900v-40z`,
+  // rightarrowabovebar is mostly from glyph U+2192, KaTeX Main
+  rightarrowabovebar: `M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32
+-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0
+13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39
+-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5
+-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+151.7 139 205zm96 379h399894v40H0zm0 0h399904v40H0z`,
+  // The short left harpoon has 0.5em (i.e. 500 units) kern on the left end.
+  // Ref from mhchem.sty: \rlap{\raisebox{-.22ex}{$\kern0.5em
+  baraboveshortleftharpoon: `M507,435c-4,4,-6.3,8.7,-7,14c0,5.3,0.7,9,2,11
+c1.3,2,5.3,5.3,12,10c90.7,54,156,130,196,228c3.3,10.7,6.3,16.3,9,17
+c2,0.7,5,1,9,1c0,0,5,0,5,0c10.7,0,16.7,-2,18,-6c2,-2.7,1,-9.7,-3,-21
+c-32,-87.3,-82.7,-157.7,-152,-211c0,0,-3,-3,-3,-3l399351,0l0,-40
+c-398570,0,-399437,0,-399437,0z M593 435 v40 H399500 v-40z
+M0 281 v-40 H399908 v40z M0 281 v-40 H399908 v40z`,
+  rightharpoonaboveshortbar: `M0,241 l0,40c399126,0,399993,0,399993,0
+c4.7,-4.7,7,-9.3,7,-14c0,-9.3,-3.7,-15.3,-11,-18c-92.7,-56.7,-159,-133.7,-199,
+-231c-3.3,-9.3,-6,-14.7,-8,-16c-2,-1.3,-7,-2,-15,-2c-10.7,0,-16.7,2,-18,6
+c-2,2.7,-1,9.7,3,21c15.3,42,36.7,81.8,64,119.5c27.3,37.7,58,69.2,92,94.5z
+M0 241 v40 H399908 v-40z M0 475 v-40 H399500 v40z M0 475 v-40 H399500 v40z`,
+  shortbaraboveleftharpoon: `M7,435c-4,4,-6.3,8.7,-7,14c0,5.3,0.7,9,2,11
+c1.3,2,5.3,5.3,12,10c90.7,54,156,130,196,228c3.3,10.7,6.3,16.3,9,17c2,0.7,5,1,9,
+1c0,0,5,0,5,0c10.7,0,16.7,-2,18,-6c2,-2.7,1,-9.7,-3,-21c-32,-87.3,-82.7,-157.7,
+-152,-211c0,0,-3,-3,-3,-3l399907,0l0,-40c-399126,0,-399993,0,-399993,0z
+M93 435 v40 H400000 v-40z M500 241 v40 H400000 v-40z M500 241 v40 H400000 v-40z`,
+  shortrightharpoonabovebar: `M53,241l0,40c398570,0,399437,0,399437,0
+c4.7,-4.7,7,-9.3,7,-14c0,-9.3,-3.7,-15.3,-11,-18c-92.7,-56.7,-159,-133.7,-199,
+-231c-3.3,-9.3,-6,-14.7,-8,-16c-2,-1.3,-7,-2,-15,-2c-10.7,0,-16.7,2,-18,6
+c-2,2.7,-1,9.7,3,21c15.3,42,36.7,81.8,64,119.5c27.3,37.7,58,69.2,92,94.5z
+M500 241 v40 H399408 v-40z M500 435 v40 H400000 v-40z`
+};
+
+/**
+ * This node represents a document fragment, which contains elements, but when
+ * placed into the DOM doesn't have any representation itself. It only contains
+ * children and doesn't have any DOM node properties.
+ */
+class DocumentFragment {
+  // HtmlDomNode
+  // Never used; needed for satisfying interface.
+  constructor(children) {
+    this.children = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    this.children = children;
+    this.classes = [];
+    this.height = 0;
+    this.depth = 0;
+    this.maxFontSize = 0;
+    this.style = {};
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+  /** Convert the fragment into a node. */
+
+
+  toNode() {
+    const frag = document.createDocumentFragment();
+
+    for (let i = 0; i < this.children.length; i++) {
+      frag.appendChild(this.children[i].toNode());
+    }
+
+    return frag;
+  }
+  /** Convert the fragment into HTML markup. */
+
+
+  toMarkup() {
+    let markup = ""; // Simply concatenate the markup for the children together.
+
+    for (let i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    return markup;
+  }
+  /**
+   * Converts the math node into a string, similar to innerText. Applies to
+   * MathDomNode's only.
+   */
+
+
+  toText() {
+    // To avoid this, we would subclass documentFragment separately for
+    // MathML, but polyfills for subclassing is expensive per PR 1469.
+    // $FlowFixMe: Only works for ChildType = MathDomNode.
+    const toText = child => child.toText();
+
+    return this.children.map(toText).join("");
+  }
+
+}
+
+/**
+ * These objects store the data about the DOM nodes we create, as well as some
+ * extra data. They can then be transformed into real DOM nodes with the
+ * `toNode` function or HTML markup using `toMarkup`. They are useful for both
+ * storing extra properties on the nodes, as well as providing a way to easily
+ * work with the DOM.
+ *
+ * Similar functions for working with MathML nodes exist in mathMLTree.js.
+ *
+ * TODO: refactor `span` and `anchor` into common superclass when
+ * target environments support class inheritance
+ */
+
+/**
+ * Create an HTML className based on a list of classes. In addition to joining
+ * with spaces, we also remove empty classes.
+ */
+const createClass = function createClass(classes) {
+  return classes.filter(cls => cls).join(" ");
+};
+
+const initNode = function initNode(classes, options, style) {
+  this.classes = classes || [];
+  this.attributes = {};
+  this.height = 0;
+  this.depth = 0;
+  this.maxFontSize = 0;
+  this.style = style || {};
+
+  if (options) {
+    if (options.style.isTight()) {
+      this.classes.push("mtight");
+    }
+
+    const color = options.getColor();
+
+    if (color) {
+      this.style.color = color;
+    }
+  }
+};
+/**
+ * Convert into an HTML node
+ */
+
+
+const toNode = function toNode(tagName) {
+  const node = document.createElement(tagName); // Apply the class
+
+  node.className = createClass(this.classes); // Apply inline styles
+
+  for (const style in this.style) {
+    if (this.style.hasOwnProperty(style)) {
+      // $FlowFixMe Flow doesn't seem to understand span.style's type.
+      node.style[style] = this.style[style];
+    }
+  } // Apply attributes
+
+
+  for (const attr in this.attributes) {
+    if (this.attributes.hasOwnProperty(attr)) {
+      node.setAttribute(attr, this.attributes[attr]);
+    }
+  } // Append the children, also as HTML nodes
+
+
+  for (let i = 0; i < this.children.length; i++) {
+    node.appendChild(this.children[i].toNode());
+  }
+
+  return node;
+};
+/**
+ * Convert into an HTML markup string
+ */
+
+
+const toMarkup = function toMarkup(tagName) {
+  let markup = `<${tagName}`; // Add the class
+
+  if (this.classes.length) {
+    markup += ` class="${utils.escape(createClass(this.classes))}"`;
+  }
+
+  let styles = ""; // Add the styles, after hyphenation
+
+  for (const style in this.style) {
+    if (this.style.hasOwnProperty(style)) {
+      styles += `${utils.hyphenate(style)}:${this.style[style]};`;
+    }
+  }
+
+  if (styles) {
+    markup += ` style="${utils.escape(styles)}"`;
+  } // Add the attributes
+
+
+  for (const attr in this.attributes) {
+    if (this.attributes.hasOwnProperty(attr)) {
+      markup += ` ${attr}="${utils.escape(this.attributes[attr])}"`;
+    }
+  }
+
+  markup += ">"; // Add the markup of the children, also as markup
+
+  for (let i = 0; i < this.children.length; i++) {
+    markup += this.children[i].toMarkup();
+  }
+
+  markup += `</${tagName}>`;
+  return markup;
+}; // Making the type below exact with all optional fields doesn't work due to
+// - https://github.com/facebook/flow/issues/4582
+// - https://github.com/facebook/flow/issues/5688
+// However, since *all* fields are optional, $Shape<> works as suggested in 5688
+// above.
+// This type does not include all CSS properties. Additional properties should
+// be added as needed.
+
+
+/**
+ * This node represents a span node, with a className, a list of children, and
+ * an inline style. It also contains information about its height, depth, and
+ * maxFontSize.
+ *
+ * Represents two types with different uses: SvgSpan to wrap an SVG and DomSpan
+ * otherwise. This typesafety is important when HTML builders access a span's
+ * children.
+ */
+class Span {
+  constructor(classes, children, options, style) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.width = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    initNode.call(this, classes, options, style);
+    this.children = children || [];
+  }
+  /**
+   * Sets an arbitrary attribute on the span. Warning: use this wisely. Not
+   * all browsers support attributes the same, and having too many custom
+   * attributes is probably bad.
+   */
+
+
+  setAttribute(attribute, value) {
+    this.attributes[attribute] = value;
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+
+  toNode() {
+    return toNode.call(this, "span");
+  }
+
+  toMarkup() {
+    return toMarkup.call(this, "span");
+  }
+
+}
+/**
+ * This node represents an anchor (<a>) element with a hyperlink.  See `span`
+ * for further details.
+ */
+
+class Anchor {
+  constructor(href, classes, children, options) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    initNode.call(this, classes, options);
+    this.children = children || [];
+    this.setAttribute('href', href);
+  }
+
+  setAttribute(attribute, value) {
+    this.attributes[attribute] = value;
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+
+  toNode() {
+    return toNode.call(this, "a");
+  }
+
+  toMarkup() {
+    return toMarkup.call(this, "a");
+  }
+
+}
+/**
+ * This node represents an image embed (<img>) element.
+ */
+
+class Img {
+  constructor(src, alt, style) {
+    this.src = void 0;
+    this.alt = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    this.alt = alt;
+    this.src = src;
+    this.classes = ["mord"];
+    this.style = style;
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+
+  toNode() {
+    const node = document.createElement("img");
+    node.src = this.src;
+    node.alt = this.alt;
+    node.className = "mord"; // Apply inline styles
+
+    for (const style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        // $FlowFixMe
+        node.style[style] = this.style[style];
+      }
+    }
+
+    return node;
+  }
+
+  toMarkup() {
+    let markup = `<img  src='${this.src} 'alt='${this.alt}' `; // Add the styles, after hyphenation
+
+    let styles = "";
+
+    for (const style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        styles += `${utils.hyphenate(style)}:${this.style[style]};`;
+      }
+    }
+
+    if (styles) {
+      markup += ` style="${utils.escape(styles)}"`;
+    }
+
+    markup += "'/>";
+    return markup;
+  }
+
+}
+const iCombinations = {
+  'î': '\u0131\u0302',
+  'ï': '\u0131\u0308',
+  'í': '\u0131\u0301',
+  // 'ī': '\u0131\u0304', // enable when we add Extended Latin
+  'ì': '\u0131\u0300'
+};
+/**
+ * A symbol node contains information about a single symbol. It either renders
+ * to a single text node, or a span with a single text node in it, depending on
+ * whether it has CSS classes, styles, or needs italic correction.
+ */
+
+class SymbolNode {
+  constructor(text, height, depth, italic, skew, width, classes, style) {
+    this.text = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.italic = void 0;
+    this.skew = void 0;
+    this.width = void 0;
+    this.maxFontSize = void 0;
+    this.classes = void 0;
+    this.style = void 0;
+    this.text = text;
+    this.height = height || 0;
+    this.depth = depth || 0;
+    this.italic = italic || 0;
+    this.skew = skew || 0;
+    this.width = width || 0;
+    this.classes = classes || [];
+    this.style = style || {};
+    this.maxFontSize = 0; // Mark text from non-Latin scripts with specific classes so that we
+    // can specify which fonts to use.  This allows us to render these
+    // characters with a serif font in situations where the browser would
+    // either default to a sans serif or render a placeholder character.
+    // We use CSS class names like cjk_fallback, hangul_fallback and
+    // brahmic_fallback. See ./unicodeScripts.js for the set of possible
+    // script names
+
+    const script = scriptFromCodepoint(this.text.charCodeAt(0));
+
+    if (script) {
+      this.classes.push(script + "_fallback");
+    }
+
+    if (/[îïíì]/.test(this.text)) {
+      // add ī when we add Extended Latin
+      this.text = iCombinations[this.text];
+    }
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+  /**
+   * Creates a text node or span from a symbol node. Note that a span is only
+   * created if it is needed.
+   */
+
+
+  toNode() {
+    const node = document.createTextNode(this.text);
+    let span = null;
+
+    if (this.italic > 0) {
+      span = document.createElement("span");
+      span.style.marginRight = this.italic + "em";
+    }
+
+    if (this.classes.length > 0) {
+      span = span || document.createElement("span");
+      span.className = createClass(this.classes);
+    }
+
+    for (const style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        span = span || document.createElement("span"); // $FlowFixMe Flow doesn't seem to understand span.style's type.
+
+        span.style[style] = this.style[style];
+      }
+    }
+
+    if (span) {
+      span.appendChild(node);
+      return span;
+    } else {
+      return node;
+    }
+  }
+  /**
+   * Creates markup for a symbol node.
+   */
+
+
+  toMarkup() {
+    // TODO(alpert): More duplication than I'd like from
+    // span.prototype.toMarkup and symbolNode.prototype.toNode...
+    let needsSpan = false;
+    let markup = "<span";
+
+    if (this.classes.length) {
+      needsSpan = true;
+      markup += " class=\"";
+      markup += utils.escape(createClass(this.classes));
+      markup += "\"";
+    }
+
+    let styles = "";
+
+    if (this.italic > 0) {
+      styles += "margin-right:" + this.italic + "em;";
+    }
+
+    for (const style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+      }
+    }
+
+    if (styles) {
+      needsSpan = true;
+      markup += " style=\"" + utils.escape(styles) + "\"";
+    }
+
+    const escaped = utils.escape(this.text);
+
+    if (needsSpan) {
+      markup += ">";
+      markup += escaped;
+      markup += "</span>";
+      return markup;
+    } else {
+      return escaped;
+    }
+  }
+
+}
+/**
+ * SVG nodes are used to render stretchy wide elements.
+ */
+
+class SvgNode {
+  constructor(children, attributes) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.children = children || [];
+    this.attributes = attributes || {};
+  }
+
+  toNode() {
+    const svgNS = "http://www.w3.org/2000/svg";
+    const node = document.createElementNS(svgNS, "svg"); // Apply attributes
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    for (let i = 0; i < this.children.length; i++) {
+      node.appendChild(this.children[i].toNode());
+    }
+
+    return node;
+  }
+
+  toMarkup() {
+    let markup = "<svg"; // Apply attributes
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += ` ${attr}='${this.attributes[attr]}'`;
+      }
+    }
+
+    markup += ">";
+
+    for (let i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    markup += "</svg>";
+    return markup;
+  }
+
+}
+class PathNode {
+  constructor(pathName, alternate) {
+    this.pathName = void 0;
+    this.alternate = void 0;
+    this.pathName = pathName;
+    this.alternate = alternate; // Used only for \sqrt
+  }
+
+  toNode() {
+    const svgNS = "http://www.w3.org/2000/svg";
+    const node = document.createElementNS(svgNS, "path");
+
+    if (this.alternate) {
+      node.setAttribute("d", this.alternate);
+    } else {
+      node.setAttribute("d", path[this.pathName]);
+    }
+
+    return node;
+  }
+
+  toMarkup() {
+    if (this.alternate) {
+      return `<path d='${this.alternate}'/>`;
+    } else {
+      return `<path d='${path[this.pathName]}'/>`;
+    }
+  }
+
+}
+class LineNode {
+  constructor(attributes) {
+    this.attributes = void 0;
+    this.attributes = attributes || {};
+  }
+
+  toNode() {
+    const svgNS = "http://www.w3.org/2000/svg";
+    const node = document.createElementNS(svgNS, "line"); // Apply attributes
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    return node;
+  }
+
+  toMarkup() {
+    let markup = "<line";
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += ` ${attr}='${this.attributes[attr]}'`;
+      }
+    }
+
+    markup += "/>";
+    return markup;
+  }
+
+}
+function assertSymbolDomNode(group) {
+  if (group instanceof SymbolNode) {
+    return group;
+  } else {
+    throw new Error(`Expected symbolNode but got ${String(group)}.`);
+  }
+}
+function assertSpan(group) {
+  if (group instanceof Span) {
+    return group;
+  } else {
+    throw new Error(`Expected span<HtmlDomNode> but got ${String(group)}.`);
+  }
+}
+
+// This file is GENERATED by buildMetrics.sh. DO NOT MODIFY.
+var metricMap = {
+  "AMS-Regular": {
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+    "8739": [0.08167, 0.58167, 0, 0, 0.22222],
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+    "73": [0, 0.61111, 0, 0, 0.525],
+    "74": [0, 0.61111, 0, 0, 0.525],
+    "75": [0, 0.61111, 0, 0, 0.525],
+    "76": [0, 0.61111, 0, 0, 0.525],
+    "77": [0, 0.61111, 0, 0, 0.525],
+    "78": [0, 0.61111, 0, 0, 0.525],
+    "79": [0, 0.61111, 0, 0, 0.525],
+    "80": [0, 0.61111, 0, 0, 0.525],
+    "81": [0.13889, 0.61111, 0, 0, 0.525],
+    "82": [0, 0.61111, 0, 0, 0.525],
+    "83": [0, 0.61111, 0, 0, 0.525],
+    "84": [0, 0.61111, 0, 0, 0.525],
+    "85": [0, 0.61111, 0, 0, 0.525],
+    "86": [0, 0.61111, 0, 0, 0.525],
+    "87": [0, 0.61111, 0, 0, 0.525],
+    "88": [0, 0.61111, 0, 0, 0.525],
+    "89": [0, 0.61111, 0, 0, 0.525],
+    "90": [0, 0.61111, 0, 0, 0.525],
+    "91": [0.08333, 0.69444, 0, 0, 0.525],
+    "92": [0.08333, 0.69444, 0, 0, 0.525],
+    "93": [0.08333, 0.69444, 0, 0, 0.525],
+    "94": [0, 0.61111, 0, 0, 0.525],
+    "95": [0.09514, 0, 0, 0, 0.525],
+    "96": [0, 0.61111, 0, 0, 0.525],
+    "97": [0, 0.43056, 0, 0, 0.525],
+    "98": [0, 0.61111, 0, 0, 0.525],
+    "99": [0, 0.43056, 0, 0, 0.525],
+    "100": [0, 0.61111, 0, 0, 0.525],
+    "101": [0, 0.43056, 0, 0, 0.525],
+    "102": [0, 0.61111, 0, 0, 0.525],
+    "103": [0.22222, 0.43056, 0, 0, 0.525],
+    "104": [0, 0.61111, 0, 0, 0.525],
+    "105": [0, 0.61111, 0, 0, 0.525],
+    "106": [0.22222, 0.61111, 0, 0, 0.525],
+    "107": [0, 0.61111, 0, 0, 0.525],
+    "108": [0, 0.61111, 0, 0, 0.525],
+    "109": [0, 0.43056, 0, 0, 0.525],
+    "110": [0, 0.43056, 0, 0, 0.525],
+    "111": [0, 0.43056, 0, 0, 0.525],
+    "112": [0.22222, 0.43056, 0, 0, 0.525],
+    "113": [0.22222, 0.43056, 0, 0, 0.525],
+    "114": [0, 0.43056, 0, 0, 0.525],
+    "115": [0, 0.43056, 0, 0, 0.525],
+    "116": [0, 0.55358, 0, 0, 0.525],
+    "117": [0, 0.43056, 0, 0, 0.525],
+    "118": [0, 0.43056, 0, 0, 0.525],
+    "119": [0, 0.43056, 0, 0, 0.525],
+    "120": [0, 0.43056, 0, 0, 0.525],
+    "121": [0.22222, 0.43056, 0, 0, 0.525],
+    "122": [0, 0.43056, 0, 0, 0.525],
+    "123": [0.08333, 0.69444, 0, 0, 0.525],
+    "124": [0.08333, 0.69444, 0, 0, 0.525],
+    "125": [0.08333, 0.69444, 0, 0, 0.525],
+    "126": [0, 0.61111, 0, 0, 0.525],
+    "127": [0, 0.61111, 0, 0, 0.525],
+    "160": [0, 0, 0, 0, 0.525],
+    "176": [0, 0.61111, 0, 0, 0.525],
+    "184": [0.19445, 0, 0, 0, 0.525],
+    "305": [0, 0.43056, 0, 0, 0.525],
+    "567": [0.22222, 0.43056, 0, 0, 0.525],
+    "711": [0, 0.56597, 0, 0, 0.525],
+    "713": [0, 0.56555, 0, 0, 0.525],
+    "714": [0, 0.61111, 0, 0, 0.525],
+    "715": [0, 0.61111, 0, 0, 0.525],
+    "728": [0, 0.61111, 0, 0, 0.525],
+    "730": [0, 0.61111, 0, 0, 0.525],
+    "770": [0, 0.61111, 0, 0, 0.525],
+    "771": [0, 0.61111, 0, 0, 0.525],
+    "776": [0, 0.61111, 0, 0, 0.525],
+    "915": [0, 0.61111, 0, 0, 0.525],
+    "916": [0, 0.61111, 0, 0, 0.525],
+    "920": [0, 0.61111, 0, 0, 0.525],
+    "923": [0, 0.61111, 0, 0, 0.525],
+    "926": [0, 0.61111, 0, 0, 0.525],
+    "928": [0, 0.61111, 0, 0, 0.525],
+    "931": [0, 0.61111, 0, 0, 0.525],
+    "933": [0, 0.61111, 0, 0, 0.525],
+    "934": [0, 0.61111, 0, 0, 0.525],
+    "936": [0, 0.61111, 0, 0, 0.525],
+    "937": [0, 0.61111, 0, 0, 0.525],
+    "8216": [0, 0.61111, 0, 0, 0.525],
+    "8217": [0, 0.61111, 0, 0, 0.525],
+    "8242": [0, 0.61111, 0, 0, 0.525],
+    "9251": [0.11111, 0.21944, 0, 0, 0.525]
+  }
+};
+
+/**
+ * This file contains metrics regarding fonts and individual symbols. The sigma
+ * and xi variables, as well as the metricMap map contain data extracted from
+ * TeX, TeX font metrics, and the TTF files. These data are then exposed via the
+ * `metrics` variable and the getCharacterMetrics function.
+ */
+// In TeX, there are actually three sets of dimensions, one for each of
+// textstyle (size index 5 and higher: >=9pt), scriptstyle (size index 3 and 4:
+// 7-8pt), and scriptscriptstyle (size index 1 and 2: 5-6pt).  These are
+// provided in the the arrays below, in that order.
+//
+// The font metrics are stored in fonts cmsy10, cmsy7, and cmsy5 respsectively.
+// This was determined by running the following script:
+//
+//     latex -interaction=nonstopmode \
+//     '\documentclass{article}\usepackage{amsmath}\begin{document}' \
+//     '$a$ \expandafter\show\the\textfont2' \
+//     '\expandafter\show\the\scriptfont2' \
+//     '\expandafter\show\the\scriptscriptfont2' \
+//     '\stop'
+//
+// The metrics themselves were retreived using the following commands:
+//
+//     tftopl cmsy10
+//     tftopl cmsy7
+//     tftopl cmsy5
+//
+// The output of each of these commands is quite lengthy.  The only part we
+// care about is the FONTDIMEN section. Each value is measured in EMs.
+const sigmasAndXis = {
+  slant: [0.250, 0.250, 0.250],
+  // sigma1
+  space: [0.000, 0.000, 0.000],
+  // sigma2
+  stretch: [0.000, 0.000, 0.000],
+  // sigma3
+  shrink: [0.000, 0.000, 0.000],
+  // sigma4
+  xHeight: [0.431, 0.431, 0.431],
+  // sigma5
+  quad: [1.000, 1.171, 1.472],
+  // sigma6
+  extraSpace: [0.000, 0.000, 0.000],
+  // sigma7
+  num1: [0.677, 0.732, 0.925],
+  // sigma8
+  num2: [0.394, 0.384, 0.387],
+  // sigma9
+  num3: [0.444, 0.471, 0.504],
+  // sigma10
+  denom1: [0.686, 0.752, 1.025],
+  // sigma11
+  denom2: [0.345, 0.344, 0.532],
+  // sigma12
+  sup1: [0.413, 0.503, 0.504],
+  // sigma13
+  sup2: [0.363, 0.431, 0.404],
+  // sigma14
+  sup3: [0.289, 0.286, 0.294],
+  // sigma15
+  sub1: [0.150, 0.143, 0.200],
+  // sigma16
+  sub2: [0.247, 0.286, 0.400],
+  // sigma17
+  supDrop: [0.386, 0.353, 0.494],
+  // sigma18
+  subDrop: [0.050, 0.071, 0.100],
+  // sigma19
+  delim1: [2.390, 1.700, 1.980],
+  // sigma20
+  delim2: [1.010, 1.157, 1.420],
+  // sigma21
+  axisHeight: [0.250, 0.250, 0.250],
+  // sigma22
+  // These font metrics are extracted from TeX by using tftopl on cmex10.tfm;
+  // they correspond to the font parameters of the extension fonts (family 3).
+  // See the TeXbook, page 441. In AMSTeX, the extension fonts scale; to
+  // match cmex7, we'd use cmex7.tfm values for script and scriptscript
+  // values.
+  defaultRuleThickness: [0.04, 0.049, 0.049],
+  // xi8; cmex7: 0.049
+  bigOpSpacing1: [0.111, 0.111, 0.111],
+  // xi9
+  bigOpSpacing2: [0.166, 0.166, 0.166],
+  // xi10
+  bigOpSpacing3: [0.2, 0.2, 0.2],
+  // xi11
+  bigOpSpacing4: [0.6, 0.611, 0.611],
+  // xi12; cmex7: 0.611
+  bigOpSpacing5: [0.1, 0.143, 0.143],
+  // xi13; cmex7: 0.143
+  // The \sqrt rule width is taken from the height of the surd character.
+  // Since we use the same font at all sizes, this thickness doesn't scale.
+  sqrtRuleThickness: [0.04, 0.04, 0.04],
+  // This value determines how large a pt is, for metrics which are defined
+  // in terms of pts.
+  // This value is also used in katex.less; if you change it make sure the
+  // values match.
+  ptPerEm: [10.0, 10.0, 10.0],
+  // The space between adjacent `|` columns in an array definition. From
+  // `\showthe\doublerulesep` in LaTeX. Equals 2.0 / ptPerEm.
+  doubleRuleSep: [0.2, 0.2, 0.2],
+  // The width of separator lines in {array} environments. From
+  // `\showthe\arrayrulewidth` in LaTeX. Equals 0.4 / ptPerEm.
+  arrayRuleWidth: [0.04, 0.04, 0.04],
+  // Two values from LaTeX source2e:
+  fboxsep: [0.3, 0.3, 0.3],
+  //        3 pt / ptPerEm
+  fboxrule: [0.04, 0.04, 0.04] // 0.4 pt / ptPerEm
+
+}; // This map contains a mapping from font name and character code to character
+// should have Latin-1 and Cyrillic characters, but may not depending on the
+// operating system.  The metrics do not account for extra height from the
+// accents.  In the case of Cyrillic characters which have both ascenders and
+// descenders we prefer approximations with ascenders, primarily to prevent
+// the fraction bar or root line from intersecting the glyph.
+// TODO(kevinb) allow union of multiple glyph metrics for better accuracy.
+
+const extraCharacterMap = {
+  // Latin-1
+  'Å': 'A',
+  'Ç': 'C',
+  'Ð': 'D',
+  'Þ': 'o',
+  'å': 'a',
+  'ç': 'c',
+  'ð': 'd',
+  'þ': 'o',
+  // Cyrillic
+  'А': 'A',
+  'Б': 'B',
+  'В': 'B',
+  'Г': 'F',
+  'Д': 'A',
+  'Е': 'E',
+  'Ж': 'K',
+  'З': '3',
+  'И': 'N',
+  'Й': 'N',
+  'К': 'K',
+  'Л': 'N',
+  'М': 'M',
+  'Н': 'H',
+  'О': 'O',
+  'П': 'N',
+  'Р': 'P',
+  'С': 'C',
+  'Т': 'T',
+  'У': 'y',
+  'Ф': 'O',
+  'Х': 'X',
+  'Ц': 'U',
+  'Ч': 'h',
+  'Ш': 'W',
+  'Щ': 'W',
+  'Ъ': 'B',
+  'Ы': 'X',
+  'Ь': 'B',
+  'Э': '3',
+  'Ю': 'X',
+  'Я': 'R',
+  'а': 'a',
+  'б': 'b',
+  'в': 'a',
+  'г': 'r',
+  'д': 'y',
+  'е': 'e',
+  'ж': 'm',
+  'з': 'e',
+  'и': 'n',
+  'й': 'n',
+  'к': 'n',
+  'л': 'n',
+  'м': 'm',
+  'н': 'n',
+  'о': 'o',
+  'п': 'n',
+  'р': 'p',
+  'с': 'c',
+  'т': 'o',
+  'у': 'y',
+  'ф': 'b',
+  'х': 'x',
+  'ц': 'n',
+  'ч': 'n',
+  'ш': 'w',
+  'щ': 'w',
+  'ъ': 'a',
+  'ы': 'm',
+  'ь': 'a',
+  'э': 'e',
+  'ю': 'm',
+  'я': 'r'
+};
+
+/**
+ * This function adds new font metrics to default metricMap
+ * It can also override existing metrics
+ */
+function setFontMetrics(fontName, metrics) {
+  metricMap[fontName] = metrics;
+}
+/**
+ * This function is a convenience function for looking up information in the
+ * metricMap table. It takes a character as a string, and a font.
+ *
+ * Note: the `width` property may be undefined if fontMetricsData.js wasn't
+ * built using `Make extended_metrics`.
+ */
+
+function getCharacterMetrics(character, font, mode) {
+  if (!metricMap[font]) {
+    throw new Error(`Font metrics not found for font: ${font}.`);
+  }
+
+  let ch = character.charCodeAt(0);
+  let metrics = metricMap[font][ch];
+
+  if (!metrics && character[0] in extraCharacterMap) {
+    ch = extraCharacterMap[character[0]].charCodeAt(0);
+    metrics = metricMap[font][ch];
+  }
+
+  if (!metrics && mode === 'text') {
+    // We don't typically have font metrics for Asian scripts.
+    // But since we support them in text mode, we need to return
+    // some sort of metrics.
+    // So if the character is in a script we support but we
+    // don't have metrics for it, just use the metrics for
+    // the Latin capital letter M. This is close enough because
+    // we (currently) only care about the height of the glpyh
+    // not its width.
+    if (supportedCodepoint(ch)) {
+      metrics = metricMap[font][77]; // 77 is the charcode for 'M'
+    }
+  }
+
+  if (metrics) {
+    return {
+      depth: metrics[0],
+      height: metrics[1],
+      italic: metrics[2],
+      skew: metrics[3],
+      width: metrics[4]
+    };
+  }
+}
+const fontMetricsBySizeIndex = {};
+/**
+ * Get the font metrics for a given size.
+ */
+
+function getGlobalMetrics(size) {
+  let sizeIndex;
+
+  if (size >= 5) {
+    sizeIndex = 0;
+  } else if (size >= 3) {
+    sizeIndex = 1;
+  } else {
+    sizeIndex = 2;
+  }
+
+  if (!fontMetricsBySizeIndex[sizeIndex]) {
+    const metrics = fontMetricsBySizeIndex[sizeIndex] = {
+      cssEmPerMu: sigmasAndXis.quad[sizeIndex] / 18
+    };
+
+    for (const key in sigmasAndXis) {
+      if (sigmasAndXis.hasOwnProperty(key)) {
+        metrics[key] = sigmasAndXis[key][sizeIndex];
+      }
+    }
+  }
+
+  return fontMetricsBySizeIndex[sizeIndex];
+}
+
+/**
+ * This file holds a list of all no-argument functions and single-character
+ * symbols (like 'a' or ';').
+ *
+ * For each of the symbols, there are three properties they can have:
+ * - font (required): the font to be used for this symbol. Either "main" (the
+     normal font), or "ams" (the ams fonts).
+ * - group (required): the ParseNode group type the symbol should have (i.e.
+     "textord", "mathord", etc).
+     See https://github.com/KaTeX/KaTeX/wiki/Examining-TeX#group-types
+ * - replace: the character that this symbol or function should be
+ *   replaced with (i.e. "\phi" has a replace value of "\u03d5", the phi
+ *   character in the main font).
+ *
+ * The outermost map in the table indicates what mode the symbols should be
+ * accepted in (e.g. "math" or "text").
+ */
+// Some of these have a "-token" suffix since these are also used as `ParseNode`
+// types for raw text tokens, and we want to avoid conflicts with higher-level
+// `ParseNode` types. These `ParseNode`s are constructed within `Parser` by
+// looking up the `symbols` map.
+const ATOMS = {
+  "bin": 1,
+  "close": 1,
+  "inner": 1,
+  "open": 1,
+  "punct": 1,
+  "rel": 1
+};
+const NON_ATOMS = {
+  "accent-token": 1,
+  "mathord": 1,
+  "op-token": 1,
+  "spacing": 1,
+  "textord": 1
+};
+const symbols = {
+  "math": {},
+  "text": {}
+};
+/** `acceptUnicodeChar = true` is only applicable if `replace` is set. */
+
+function defineSymbol(mode, font, group, replace, name, acceptUnicodeChar) {
+  symbols[mode][name] = {
+    font,
+    group,
+    replace
+  };
+
+  if (acceptUnicodeChar && replace) {
+    symbols[mode][replace] = symbols[mode][name];
+  }
+} // Some abbreviations for commonly used strings.
+// This helps minify the code, and also spotting typos using jshint.
+// modes:
+
+const math = "math";
+const text$1 = "text"; // fonts:
+
+const main = "main";
+const ams = "ams"; // groups:
+
+const accent = "accent-token";
+const bin = "bin";
+const close = "close";
+const inner = "inner";
+const mathord = "mathord";
+const op = "op-token";
+const open = "open";
+const punct = "punct";
+const rel = "rel";
+const spacing = "spacing";
+const textord = "textord"; // Now comes the symbol table
+// Relation Symbols
+
+defineSymbol(math, main, rel, "\u2261", "\\equiv", true);
+defineSymbol(math, main, rel, "\u227a", "\\prec", true);
+defineSymbol(math, main, rel, "\u227b", "\\succ", true);
+defineSymbol(math, main, rel, "\u223c", "\\sim", true);
+defineSymbol(math, main, rel, "\u22a5", "\\perp");
+defineSymbol(math, main, rel, "\u2aaf", "\\preceq", true);
+defineSymbol(math, main, rel, "\u2ab0", "\\succeq", true);
+defineSymbol(math, main, rel, "\u2243", "\\simeq", true);
+defineSymbol(math, main, rel, "\u2223", "\\mid", true);
+defineSymbol(math, main, rel, "\u226a", "\\ll", true);
+defineSymbol(math, main, rel, "\u226b", "\\gg", true);
+defineSymbol(math, main, rel, "\u224d", "\\asymp", true);
+defineSymbol(math, main, rel, "\u2225", "\\parallel");
+defineSymbol(math, main, rel, "\u22c8", "\\bowtie", true);
+defineSymbol(math, main, rel, "\u2323", "\\smile", true);
+defineSymbol(math, main, rel, "\u2291", "\\sqsubseteq", true);
+defineSymbol(math, main, rel, "\u2292", "\\sqsupseteq", true);
+defineSymbol(math, main, rel, "\u2250", "\\doteq", true);
+defineSymbol(math, main, rel, "\u2322", "\\frown", true);
+defineSymbol(math, main, rel, "\u220b", "\\ni", true);
+defineSymbol(math, main, rel, "\u221d", "\\propto", true);
+defineSymbol(math, main, rel, "\u22a2", "\\vdash", true);
+defineSymbol(math, main, rel, "\u22a3", "\\dashv", true);
+defineSymbol(math, main, rel, "\u220b", "\\owns"); // Punctuation
+
+defineSymbol(math, main, punct, "\u002e", "\\ldotp");
+defineSymbol(math, main, punct, "\u22c5", "\\cdotp"); // Misc Symbols
+
+defineSymbol(math, main, textord, "\u0023", "\\#");
+defineSymbol(text$1, main, textord, "\u0023", "\\#");
+defineSymbol(math, main, textord, "\u0026", "\\&");
+defineSymbol(text$1, main, textord, "\u0026", "\\&");
+defineSymbol(math, main, textord, "\u2135", "\\aleph", true);
+defineSymbol(math, main, textord, "\u2200", "\\forall", true);
+defineSymbol(math, main, textord, "\u210f", "\\hbar", true);
+defineSymbol(math, main, textord, "\u2203", "\\exists", true);
+defineSymbol(math, main, textord, "\u2207", "\\nabla", true);
+defineSymbol(math, main, textord, "\u266d", "\\flat", true);
+defineSymbol(math, main, textord, "\u2113", "\\ell", true);
+defineSymbol(math, main, textord, "\u266e", "\\natural", true);
+defineSymbol(math, main, textord, "\u2663", "\\clubsuit", true);
+defineSymbol(math, main, textord, "\u2118", "\\wp", true);
+defineSymbol(math, main, textord, "\u266f", "\\sharp", true);
+defineSymbol(math, main, textord, "\u2662", "\\diamondsuit", true);
+defineSymbol(math, main, textord, "\u211c", "\\Re", true);
+defineSymbol(math, main, textord, "\u2661", "\\heartsuit", true);
+defineSymbol(math, main, textord, "\u2111", "\\Im", true);
+defineSymbol(math, main, textord, "\u2660", "\\spadesuit", true);
+defineSymbol(text$1, main, textord, "\u00a7", "\\S", true);
+defineSymbol(text$1, main, textord, "\u00b6", "\\P", true); // Math and Text
+
+defineSymbol(math, main, textord, "\u2020", "\\dag");
+defineSymbol(text$1, main, textord, "\u2020", "\\dag");
+defineSymbol(text$1, main, textord, "\u2020", "\\textdagger");
+defineSymbol(math, main, textord, "\u2021", "\\ddag");
+defineSymbol(text$1, main, textord, "\u2021", "\\ddag");
+defineSymbol(text$1, main, textord, "\u2021", "\\textdaggerdbl"); // Large Delimiters
+
+defineSymbol(math, main, close, "\u23b1", "\\rmoustache", true);
+defineSymbol(math, main, open, "\u23b0", "\\lmoustache", true);
+defineSymbol(math, main, close, "\u27ef", "\\rgroup", true);
+defineSymbol(math, main, open, "\u27ee", "\\lgroup", true); // Binary Operators
+
+defineSymbol(math, main, bin, "\u2213", "\\mp", true);
+defineSymbol(math, main, bin, "\u2296", "\\ominus", true);
+defineSymbol(math, main, bin, "\u228e", "\\uplus", true);
+defineSymbol(math, main, bin, "\u2293", "\\sqcap", true);
+defineSymbol(math, main, bin, "\u2217", "\\ast");
+defineSymbol(math, main, bin, "\u2294", "\\sqcup", true);
+defineSymbol(math, main, bin, "\u25ef", "\\bigcirc");
+defineSymbol(math, main, bin, "\u2219", "\\bullet");
+defineSymbol(math, main, bin, "\u2021", "\\ddagger");
+defineSymbol(math, main, bin, "\u2240", "\\wr", true);
+defineSymbol(math, main, bin, "\u2a3f", "\\amalg");
+defineSymbol(math, main, bin, "\u0026", "\\And"); // from amsmath
+// Arrow Symbols
+
+defineSymbol(math, main, rel, "\u27f5", "\\longleftarrow", true);
+defineSymbol(math, main, rel, "\u21d0", "\\Leftarrow", true);
+defineSymbol(math, main, rel, "\u27f8", "\\Longleftarrow", true);
+defineSymbol(math, main, rel, "\u27f6", "\\longrightarrow", true);
+defineSymbol(math, main, rel, "\u21d2", "\\Rightarrow", true);
+defineSymbol(math, main, rel, "\u27f9", "\\Longrightarrow", true);
+defineSymbol(math, main, rel, "\u2194", "\\leftrightarrow", true);
+defineSymbol(math, main, rel, "\u27f7", "\\longleftrightarrow", true);
+defineSymbol(math, main, rel, "\u21d4", "\\Leftrightarrow", true);
+defineSymbol(math, main, rel, "\u27fa", "\\Longleftrightarrow", true);
+defineSymbol(math, main, rel, "\u21a6", "\\mapsto", true);
+defineSymbol(math, main, rel, "\u27fc", "\\longmapsto", true);
+defineSymbol(math, main, rel, "\u2197", "\\nearrow", true);
+defineSymbol(math, main, rel, "\u21a9", "\\hookleftarrow", true);
+defineSymbol(math, main, rel, "\u21aa", "\\hookrightarrow", true);
+defineSymbol(math, main, rel, "\u2198", "\\searrow", true);
+defineSymbol(math, main, rel, "\u21bc", "\\leftharpoonup", true);
+defineSymbol(math, main, rel, "\u21c0", "\\rightharpoonup", true);
+defineSymbol(math, main, rel, "\u2199", "\\swarrow", true);
+defineSymbol(math, main, rel, "\u21bd", "\\leftharpoondown", true);
+defineSymbol(math, main, rel, "\u21c1", "\\rightharpoondown", true);
+defineSymbol(math, main, rel, "\u2196", "\\nwarrow", true);
+defineSymbol(math, main, rel, "\u21cc", "\\rightleftharpoons", true); // AMS Negated Binary Relations
+
+defineSymbol(math, ams, rel, "\u226e", "\\nless", true); // Symbol names preceeded by "@" each have a corresponding macro.
+
+defineSymbol(math, ams, rel, "\ue010", "\\@nleqslant");
+defineSymbol(math, ams, rel, "\ue011", "\\@nleqq");
+defineSymbol(math, ams, rel, "\u2a87", "\\lneq", true);
+defineSymbol(math, ams, rel, "\u2268", "\\lneqq", true);
+defineSymbol(math, ams, rel, "\ue00c", "\\@lvertneqq");
+defineSymbol(math, ams, rel, "\u22e6", "\\lnsim", true);
+defineSymbol(math, ams, rel, "\u2a89", "\\lnapprox", true);
+defineSymbol(math, ams, rel, "\u2280", "\\nprec", true); // unicode-math maps \u22e0 to \npreccurlyeq. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u22e0", "\\npreceq", true);
+defineSymbol(math, ams, rel, "\u22e8", "\\precnsim", true);
+defineSymbol(math, ams, rel, "\u2ab9", "\\precnapprox", true);
+defineSymbol(math, ams, rel, "\u2241", "\\nsim", true);
+defineSymbol(math, ams, rel, "\ue006", "\\@nshortmid");
+defineSymbol(math, ams, rel, "\u2224", "\\nmid", true);
+defineSymbol(math, ams, rel, "\u22ac", "\\nvdash", true);
+defineSymbol(math, ams, rel, "\u22ad", "\\nvDash", true);
+defineSymbol(math, ams, rel, "\u22ea", "\\ntriangleleft");
+defineSymbol(math, ams, rel, "\u22ec", "\\ntrianglelefteq", true);
+defineSymbol(math, ams, rel, "\u228a", "\\subsetneq", true);
+defineSymbol(math, ams, rel, "\ue01a", "\\@varsubsetneq");
+defineSymbol(math, ams, rel, "\u2acb", "\\subsetneqq", true);
+defineSymbol(math, ams, rel, "\ue017", "\\@varsubsetneqq");
+defineSymbol(math, ams, rel, "\u226f", "\\ngtr", true);
+defineSymbol(math, ams, rel, "\ue00f", "\\@ngeqslant");
+defineSymbol(math, ams, rel, "\ue00e", "\\@ngeqq");
+defineSymbol(math, ams, rel, "\u2a88", "\\gneq", true);
+defineSymbol(math, ams, rel, "\u2269", "\\gneqq", true);
+defineSymbol(math, ams, rel, "\ue00d", "\\@gvertneqq");
+defineSymbol(math, ams, rel, "\u22e7", "\\gnsim", true);
+defineSymbol(math, ams, rel, "\u2a8a", "\\gnapprox", true);
+defineSymbol(math, ams, rel, "\u2281", "\\nsucc", true); // unicode-math maps \u22e1 to \nsucccurlyeq. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u22e1", "\\nsucceq", true);
+defineSymbol(math, ams, rel, "\u22e9", "\\succnsim", true);
+defineSymbol(math, ams, rel, "\u2aba", "\\succnapprox", true); // unicode-math maps \u2246 to \simneqq. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u2246", "\\ncong", true);
+defineSymbol(math, ams, rel, "\ue007", "\\@nshortparallel");
+defineSymbol(math, ams, rel, "\u2226", "\\nparallel", true);
+defineSymbol(math, ams, rel, "\u22af", "\\nVDash", true);
+defineSymbol(math, ams, rel, "\u22eb", "\\ntriangleright");
+defineSymbol(math, ams, rel, "\u22ed", "\\ntrianglerighteq", true);
+defineSymbol(math, ams, rel, "\ue018", "\\@nsupseteqq");
+defineSymbol(math, ams, rel, "\u228b", "\\supsetneq", true);
+defineSymbol(math, ams, rel, "\ue01b", "\\@varsupsetneq");
+defineSymbol(math, ams, rel, "\u2acc", "\\supsetneqq", true);
+defineSymbol(math, ams, rel, "\ue019", "\\@varsupsetneqq");
+defineSymbol(math, ams, rel, "\u22ae", "\\nVdash", true);
+defineSymbol(math, ams, rel, "\u2ab5", "\\precneqq", true);
+defineSymbol(math, ams, rel, "\u2ab6", "\\succneqq", true);
+defineSymbol(math, ams, rel, "\ue016", "\\@nsubseteqq");
+defineSymbol(math, ams, bin, "\u22b4", "\\unlhd");
+defineSymbol(math, ams, bin, "\u22b5", "\\unrhd"); // AMS Negated Arrows
+
+defineSymbol(math, ams, rel, "\u219a", "\\nleftarrow", true);
+defineSymbol(math, ams, rel, "\u219b", "\\nrightarrow", true);
+defineSymbol(math, ams, rel, "\u21cd", "\\nLeftarrow", true);
+defineSymbol(math, ams, rel, "\u21cf", "\\nRightarrow", true);
+defineSymbol(math, ams, rel, "\u21ae", "\\nleftrightarrow", true);
+defineSymbol(math, ams, rel, "\u21ce", "\\nLeftrightarrow", true); // AMS Misc
+
+defineSymbol(math, ams, rel, "\u25b3", "\\vartriangle");
+defineSymbol(math, ams, textord, "\u210f", "\\hslash");
+defineSymbol(math, ams, textord, "\u25bd", "\\triangledown");
+defineSymbol(math, ams, textord, "\u25ca", "\\lozenge");
+defineSymbol(math, ams, textord, "\u24c8", "\\circledS");
+defineSymbol(math, ams, textord, "\u00ae", "\\circledR");
+defineSymbol(text$1, ams, textord, "\u00ae", "\\circledR");
+defineSymbol(math, ams, textord, "\u2221", "\\measuredangle", true);
+defineSymbol(math, ams, textord, "\u2204", "\\nexists");
+defineSymbol(math, ams, textord, "\u2127", "\\mho");
+defineSymbol(math, ams, textord, "\u2132", "\\Finv", true);
+defineSymbol(math, ams, textord, "\u2141", "\\Game", true);
+defineSymbol(math, ams, textord, "\u2035", "\\backprime");
+defineSymbol(math, ams, textord, "\u25b2", "\\blacktriangle");
+defineSymbol(math, ams, textord, "\u25bc", "\\blacktriangledown");
+defineSymbol(math, ams, textord, "\u25a0", "\\blacksquare");
+defineSymbol(math, ams, textord, "\u29eb", "\\blacklozenge");
+defineSymbol(math, ams, textord, "\u2605", "\\bigstar");
+defineSymbol(math, ams, textord, "\u2222", "\\sphericalangle", true);
+defineSymbol(math, ams, textord, "\u2201", "\\complement", true); // unicode-math maps U+F0 to \matheth. We map to AMS function \eth
+
+defineSymbol(math, ams, textord, "\u00f0", "\\eth", true);
+defineSymbol(text$1, main, textord, "\u00f0", "\u00f0");
+defineSymbol(math, ams, textord, "\u2571", "\\diagup");
+defineSymbol(math, ams, textord, "\u2572", "\\diagdown");
+defineSymbol(math, ams, textord, "\u25a1", "\\square");
+defineSymbol(math, ams, textord, "\u25a1", "\\Box");
+defineSymbol(math, ams, textord, "\u25ca", "\\Diamond"); // unicode-math maps U+A5 to \mathyen. We map to AMS function \yen
+
+defineSymbol(math, ams, textord, "\u00a5", "\\yen", true);
+defineSymbol(text$1, ams, textord, "\u00a5", "\\yen", true);
+defineSymbol(math, ams, textord, "\u2713", "\\checkmark", true);
+defineSymbol(text$1, ams, textord, "\u2713", "\\checkmark"); // AMS Hebrew
+
+defineSymbol(math, ams, textord, "\u2136", "\\beth", true);
+defineSymbol(math, ams, textord, "\u2138", "\\daleth", true);
+defineSymbol(math, ams, textord, "\u2137", "\\gimel", true); // AMS Greek
+
+defineSymbol(math, ams, textord, "\u03dd", "\\digamma", true);
+defineSymbol(math, ams, textord, "\u03f0", "\\varkappa"); // AMS Delimiters
+
+defineSymbol(math, ams, open, "\u250c", "\\@ulcorner", true);
+defineSymbol(math, ams, close, "\u2510", "\\@urcorner", true);
+defineSymbol(math, ams, open, "\u2514", "\\@llcorner", true);
+defineSymbol(math, ams, close, "\u2518", "\\@lrcorner", true); // AMS Binary Relations
+
+defineSymbol(math, ams, rel, "\u2266", "\\leqq", true);
+defineSymbol(math, ams, rel, "\u2a7d", "\\leqslant", true);
+defineSymbol(math, ams, rel, "\u2a95", "\\eqslantless", true);
+defineSymbol(math, ams, rel, "\u2272", "\\lesssim", true);
+defineSymbol(math, ams, rel, "\u2a85", "\\lessapprox", true);
+defineSymbol(math, ams, rel, "\u224a", "\\approxeq", true);
+defineSymbol(math, ams, bin, "\u22d6", "\\lessdot");
+defineSymbol(math, ams, rel, "\u22d8", "\\lll", true);
+defineSymbol(math, ams, rel, "\u2276", "\\lessgtr", true);
+defineSymbol(math, ams, rel, "\u22da", "\\lesseqgtr", true);
+defineSymbol(math, ams, rel, "\u2a8b", "\\lesseqqgtr", true);
+defineSymbol(math, ams, rel, "\u2251", "\\doteqdot");
+defineSymbol(math, ams, rel, "\u2253", "\\risingdotseq", true);
+defineSymbol(math, ams, rel, "\u2252", "\\fallingdotseq", true);
+defineSymbol(math, ams, rel, "\u223d", "\\backsim", true);
+defineSymbol(math, ams, rel, "\u22cd", "\\backsimeq", true);
+defineSymbol(math, ams, rel, "\u2ac5", "\\subseteqq", true);
+defineSymbol(math, ams, rel, "\u22d0", "\\Subset", true);
+defineSymbol(math, ams, rel, "\u228f", "\\sqsubset", true);
+defineSymbol(math, ams, rel, "\u227c", "\\preccurlyeq", true);
+defineSymbol(math, ams, rel, "\u22de", "\\curlyeqprec", true);
+defineSymbol(math, ams, rel, "\u227e", "\\precsim", true);
+defineSymbol(math, ams, rel, "\u2ab7", "\\precapprox", true);
+defineSymbol(math, ams, rel, "\u22b2", "\\vartriangleleft");
+defineSymbol(math, ams, rel, "\u22b4", "\\trianglelefteq");
+defineSymbol(math, ams, rel, "\u22a8", "\\vDash", true);
+defineSymbol(math, ams, rel, "\u22aa", "\\Vvdash", true);
+defineSymbol(math, ams, rel, "\u2323", "\\smallsmile");
+defineSymbol(math, ams, rel, "\u2322", "\\smallfrown");
+defineSymbol(math, ams, rel, "\u224f", "\\bumpeq", true);
+defineSymbol(math, ams, rel, "\u224e", "\\Bumpeq", true);
+defineSymbol(math, ams, rel, "\u2267", "\\geqq", true);
+defineSymbol(math, ams, rel, "\u2a7e", "\\geqslant", true);
+defineSymbol(math, ams, rel, "\u2a96", "\\eqslantgtr", true);
+defineSymbol(math, ams, rel, "\u2273", "\\gtrsim", true);
+defineSymbol(math, ams, rel, "\u2a86", "\\gtrapprox", true);
+defineSymbol(math, ams, bin, "\u22d7", "\\gtrdot");
+defineSymbol(math, ams, rel, "\u22d9", "\\ggg", true);
+defineSymbol(math, ams, rel, "\u2277", "\\gtrless", true);
+defineSymbol(math, ams, rel, "\u22db", "\\gtreqless", true);
+defineSymbol(math, ams, rel, "\u2a8c", "\\gtreqqless", true);
+defineSymbol(math, ams, rel, "\u2256", "\\eqcirc", true);
+defineSymbol(math, ams, rel, "\u2257", "\\circeq", true);
+defineSymbol(math, ams, rel, "\u225c", "\\triangleq", true);
+defineSymbol(math, ams, rel, "\u223c", "\\thicksim");
+defineSymbol(math, ams, rel, "\u2248", "\\thickapprox");
+defineSymbol(math, ams, rel, "\u2ac6", "\\supseteqq", true);
+defineSymbol(math, ams, rel, "\u22d1", "\\Supset", true);
+defineSymbol(math, ams, rel, "\u2290", "\\sqsupset", true);
+defineSymbol(math, ams, rel, "\u227d", "\\succcurlyeq", true);
+defineSymbol(math, ams, rel, "\u22df", "\\curlyeqsucc", true);
+defineSymbol(math, ams, rel, "\u227f", "\\succsim", true);
+defineSymbol(math, ams, rel, "\u2ab8", "\\succapprox", true);
+defineSymbol(math, ams, rel, "\u22b3", "\\vartriangleright");
+defineSymbol(math, ams, rel, "\u22b5", "\\trianglerighteq");
+defineSymbol(math, ams, rel, "\u22a9", "\\Vdash", true);
+defineSymbol(math, ams, rel, "\u2223", "\\shortmid");
+defineSymbol(math, ams, rel, "\u2225", "\\shortparallel");
+defineSymbol(math, ams, rel, "\u226c", "\\between", true);
+defineSymbol(math, ams, rel, "\u22d4", "\\pitchfork", true);
+defineSymbol(math, ams, rel, "\u221d", "\\varpropto");
+defineSymbol(math, ams, rel, "\u25c0", "\\blacktriangleleft"); // unicode-math says that \therefore is a mathord atom.
+// We kept the amssymb atom type, which is rel.
+
+defineSymbol(math, ams, rel, "\u2234", "\\therefore", true);
+defineSymbol(math, ams, rel, "\u220d", "\\backepsilon");
+defineSymbol(math, ams, rel, "\u25b6", "\\blacktriangleright"); // unicode-math says that \because is a mathord atom.
+// We kept the amssymb atom type, which is rel.
+
+defineSymbol(math, ams, rel, "\u2235", "\\because", true);
+defineSymbol(math, ams, rel, "\u22d8", "\\llless");
+defineSymbol(math, ams, rel, "\u22d9", "\\gggtr");
+defineSymbol(math, ams, bin, "\u22b2", "\\lhd");
+defineSymbol(math, ams, bin, "\u22b3", "\\rhd");
+defineSymbol(math, ams, rel, "\u2242", "\\eqsim", true);
+defineSymbol(math, main, rel, "\u22c8", "\\Join");
+defineSymbol(math, ams, rel, "\u2251", "\\Doteq", true); // AMS Binary Operators
+
+defineSymbol(math, ams, bin, "\u2214", "\\dotplus", true);
+defineSymbol(math, ams, bin, "\u2216", "\\smallsetminus");
+defineSymbol(math, ams, bin, "\u22d2", "\\Cap", true);
+defineSymbol(math, ams, bin, "\u22d3", "\\Cup", true);
+defineSymbol(math, ams, bin, "\u2a5e", "\\doublebarwedge", true);
+defineSymbol(math, ams, bin, "\u229f", "\\boxminus", true);
+defineSymbol(math, ams, bin, "\u229e", "\\boxplus", true);
+defineSymbol(math, ams, bin, "\u22c7", "\\divideontimes", true);
+defineSymbol(math, ams, bin, "\u22c9", "\\ltimes", true);
+defineSymbol(math, ams, bin, "\u22ca", "\\rtimes", true);
+defineSymbol(math, ams, bin, "\u22cb", "\\leftthreetimes", true);
+defineSymbol(math, ams, bin, "\u22cc", "\\rightthreetimes", true);
+defineSymbol(math, ams, bin, "\u22cf", "\\curlywedge", true);
+defineSymbol(math, ams, bin, "\u22ce", "\\curlyvee", true);
+defineSymbol(math, ams, bin, "\u229d", "\\circleddash", true);
+defineSymbol(math, ams, bin, "\u229b", "\\circledast", true);
+defineSymbol(math, ams, bin, "\u22c5", "\\centerdot");
+defineSymbol(math, ams, bin, "\u22ba", "\\intercal", true);
+defineSymbol(math, ams, bin, "\u22d2", "\\doublecap");
+defineSymbol(math, ams, bin, "\u22d3", "\\doublecup");
+defineSymbol(math, ams, bin, "\u22a0", "\\boxtimes", true); // AMS Arrows
+// Note: unicode-math maps \u21e2 to their own function \rightdasharrow.
+// We'll map it to AMS function \dashrightarrow. It produces the same atom.
+
+defineSymbol(math, ams, rel, "\u21e2", "\\dashrightarrow", true); // unicode-math maps \u21e0 to \leftdasharrow. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u21e0", "\\dashleftarrow", true);
+defineSymbol(math, ams, rel, "\u21c7", "\\leftleftarrows", true);
+defineSymbol(math, ams, rel, "\u21c6", "\\leftrightarrows", true);
+defineSymbol(math, ams, rel, "\u21da", "\\Lleftarrow", true);
+defineSymbol(math, ams, rel, "\u219e", "\\twoheadleftarrow", true);
+defineSymbol(math, ams, rel, "\u21a2", "\\leftarrowtail", true);
+defineSymbol(math, ams, rel, "\u21ab", "\\looparrowleft", true);
+defineSymbol(math, ams, rel, "\u21cb", "\\leftrightharpoons", true);
+defineSymbol(math, ams, rel, "\u21b6", "\\curvearrowleft", true); // unicode-math maps \u21ba to \acwopencirclearrow. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u21ba", "\\circlearrowleft", true);
+defineSymbol(math, ams, rel, "\u21b0", "\\Lsh", true);
+defineSymbol(math, ams, rel, "\u21c8", "\\upuparrows", true);
+defineSymbol(math, ams, rel, "\u21bf", "\\upharpoonleft", true);
+defineSymbol(math, ams, rel, "\u21c3", "\\downharpoonleft", true);
+defineSymbol(math, ams, rel, "\u22b8", "\\multimap", true);
+defineSymbol(math, ams, rel, "\u21ad", "\\leftrightsquigarrow", true);
+defineSymbol(math, ams, rel, "\u21c9", "\\rightrightarrows", true);
+defineSymbol(math, ams, rel, "\u21c4", "\\rightleftarrows", true);
+defineSymbol(math, ams, rel, "\u21a0", "\\twoheadrightarrow", true);
+defineSymbol(math, ams, rel, "\u21a3", "\\rightarrowtail", true);
+defineSymbol(math, ams, rel, "\u21ac", "\\looparrowright", true);
+defineSymbol(math, ams, rel, "\u21b7", "\\curvearrowright", true); // unicode-math maps \u21bb to \cwopencirclearrow. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u21bb", "\\circlearrowright", true);
+defineSymbol(math, ams, rel, "\u21b1", "\\Rsh", true);
+defineSymbol(math, ams, rel, "\u21ca", "\\downdownarrows", true);
+defineSymbol(math, ams, rel, "\u21be", "\\upharpoonright", true);
+defineSymbol(math, ams, rel, "\u21c2", "\\downharpoonright", true);
+defineSymbol(math, ams, rel, "\u21dd", "\\rightsquigarrow", true);
+defineSymbol(math, ams, rel, "\u21dd", "\\leadsto");
+defineSymbol(math, ams, rel, "\u21db", "\\Rrightarrow", true);
+defineSymbol(math, ams, rel, "\u21be", "\\restriction");
+defineSymbol(math, main, textord, "\u2018", "`");
+defineSymbol(math, main, textord, "$", "\\$");
+defineSymbol(text$1, main, textord, "$", "\\$");
+defineSymbol(text$1, main, textord, "$", "\\textdollar");
+defineSymbol(math, main, textord, "%", "\\%");
+defineSymbol(text$1, main, textord, "%", "\\%");
+defineSymbol(math, main, textord, "_", "\\_");
+defineSymbol(text$1, main, textord, "_", "\\_");
+defineSymbol(text$1, main, textord, "_", "\\textunderscore");
+defineSymbol(math, main, textord, "\u2220", "\\angle", true);
+defineSymbol(math, main, textord, "\u221e", "\\infty", true);
+defineSymbol(math, main, textord, "\u2032", "\\prime");
+defineSymbol(math, main, textord, "\u25b3", "\\triangle");
+defineSymbol(math, main, textord, "\u0393", "\\Gamma", true);
+defineSymbol(math, main, textord, "\u0394", "\\Delta", true);
+defineSymbol(math, main, textord, "\u0398", "\\Theta", true);
+defineSymbol(math, main, textord, "\u039b", "\\Lambda", true);
+defineSymbol(math, main, textord, "\u039e", "\\Xi", true);
+defineSymbol(math, main, textord, "\u03a0", "\\Pi", true);
+defineSymbol(math, main, textord, "\u03a3", "\\Sigma", true);
+defineSymbol(math, main, textord, "\u03a5", "\\Upsilon", true);
+defineSymbol(math, main, textord, "\u03a6", "\\Phi", true);
+defineSymbol(math, main, textord, "\u03a8", "\\Psi", true);
+defineSymbol(math, main, textord, "\u03a9", "\\Omega", true);
+defineSymbol(math, main, textord, "A", "\u0391");
+defineSymbol(math, main, textord, "B", "\u0392");
+defineSymbol(math, main, textord, "E", "\u0395");
+defineSymbol(math, main, textord, "Z", "\u0396");
+defineSymbol(math, main, textord, "H", "\u0397");
+defineSymbol(math, main, textord, "I", "\u0399");
+defineSymbol(math, main, textord, "K", "\u039A");
+defineSymbol(math, main, textord, "M", "\u039C");
+defineSymbol(math, main, textord, "N", "\u039D");
+defineSymbol(math, main, textord, "O", "\u039F");
+defineSymbol(math, main, textord, "P", "\u03A1");
+defineSymbol(math, main, textord, "T", "\u03A4");
+defineSymbol(math, main, textord, "X", "\u03A7");
+defineSymbol(math, main, textord, "\u00ac", "\\neg", true);
+defineSymbol(math, main, textord, "\u00ac", "\\lnot");
+defineSymbol(math, main, textord, "\u22a4", "\\top");
+defineSymbol(math, main, textord, "\u22a5", "\\bot");
+defineSymbol(math, main, textord, "\u2205", "\\emptyset");
+defineSymbol(math, ams, textord, "\u2205", "\\varnothing");
+defineSymbol(math, main, mathord, "\u03b1", "\\alpha", true);
+defineSymbol(math, main, mathord, "\u03b2", "\\beta", true);
+defineSymbol(math, main, mathord, "\u03b3", "\\gamma", true);
+defineSymbol(math, main, mathord, "\u03b4", "\\delta", true);
+defineSymbol(math, main, mathord, "\u03f5", "\\epsilon", true);
+defineSymbol(math, main, mathord, "\u03b6", "\\zeta", true);
+defineSymbol(math, main, mathord, "\u03b7", "\\eta", true);
+defineSymbol(math, main, mathord, "\u03b8", "\\theta", true);
+defineSymbol(math, main, mathord, "\u03b9", "\\iota", true);
+defineSymbol(math, main, mathord, "\u03ba", "\\kappa", true);
+defineSymbol(math, main, mathord, "\u03bb", "\\lambda", true);
+defineSymbol(math, main, mathord, "\u03bc", "\\mu", true);
+defineSymbol(math, main, mathord, "\u03bd", "\\nu", true);
+defineSymbol(math, main, mathord, "\u03be", "\\xi", true);
+defineSymbol(math, main, mathord, "\u03bf", "\\omicron", true);
+defineSymbol(math, main, mathord, "\u03c0", "\\pi", true);
+defineSymbol(math, main, mathord, "\u03c1", "\\rho", true);
+defineSymbol(math, main, mathord, "\u03c3", "\\sigma", true);
+defineSymbol(math, main, mathord, "\u03c4", "\\tau", true);
+defineSymbol(math, main, mathord, "\u03c5", "\\upsilon", true);
+defineSymbol(math, main, mathord, "\u03d5", "\\phi", true);
+defineSymbol(math, main, mathord, "\u03c7", "\\chi", true);
+defineSymbol(math, main, mathord, "\u03c8", "\\psi", true);
+defineSymbol(math, main, mathord, "\u03c9", "\\omega", true);
+defineSymbol(math, main, mathord, "\u03b5", "\\varepsilon", true);
+defineSymbol(math, main, mathord, "\u03d1", "\\vartheta", true);
+defineSymbol(math, main, mathord, "\u03d6", "\\varpi", true);
+defineSymbol(math, main, mathord, "\u03f1", "\\varrho", true);
+defineSymbol(math, main, mathord, "\u03c2", "\\varsigma", true);
+defineSymbol(math, main, mathord, "\u03c6", "\\varphi", true);
+defineSymbol(math, main, bin, "\u2217", "*");
+defineSymbol(math, main, bin, "+", "+");
+defineSymbol(math, main, bin, "\u2212", "-");
+defineSymbol(math, main, bin, "\u22c5", "\\cdot", true);
+defineSymbol(math, main, bin, "\u2218", "\\circ");
+defineSymbol(math, main, bin, "\u00f7", "\\div", true);
+defineSymbol(math, main, bin, "\u00b1", "\\pm", true);
+defineSymbol(math, main, bin, "\u00d7", "\\times", true);
+defineSymbol(math, main, bin, "\u2229", "\\cap", true);
+defineSymbol(math, main, bin, "\u222a", "\\cup", true);
+defineSymbol(math, main, bin, "\u2216", "\\setminus");
+defineSymbol(math, main, bin, "\u2227", "\\land");
+defineSymbol(math, main, bin, "\u2228", "\\lor");
+defineSymbol(math, main, bin, "\u2227", "\\wedge", true);
+defineSymbol(math, main, bin, "\u2228", "\\vee", true);
+defineSymbol(math, main, textord, "\u221a", "\\surd");
+defineSymbol(math, main, open, "\u27e8", "\\langle", true);
+defineSymbol(math, main, open, "\u2223", "\\lvert");
+defineSymbol(math, main, open, "\u2225", "\\lVert");
+defineSymbol(math, main, close, "?", "?");
+defineSymbol(math, main, close, "!", "!");
+defineSymbol(math, main, close, "\u27e9", "\\rangle", true);
+defineSymbol(math, main, close, "\u2223", "\\rvert");
+defineSymbol(math, main, close, "\u2225", "\\rVert");
+defineSymbol(math, main, rel, "=", "=");
+defineSymbol(math, main, rel, ":", ":");
+defineSymbol(math, main, rel, "\u2248", "\\approx", true);
+defineSymbol(math, main, rel, "\u2245", "\\cong", true);
+defineSymbol(math, main, rel, "\u2265", "\\ge");
+defineSymbol(math, main, rel, "\u2265", "\\geq", true);
+defineSymbol(math, main, rel, "\u2190", "\\gets");
+defineSymbol(math, main, rel, ">", "\\gt", true);
+defineSymbol(math, main, rel, "\u2208", "\\in", true);
+defineSymbol(math, main, rel, "\ue020", "\\@not");
+defineSymbol(math, main, rel, "\u2282", "\\subset", true);
+defineSymbol(math, main, rel, "\u2283", "\\supset", true);
+defineSymbol(math, main, rel, "\u2286", "\\subseteq", true);
+defineSymbol(math, main, rel, "\u2287", "\\supseteq", true);
+defineSymbol(math, ams, rel, "\u2288", "\\nsubseteq", true);
+defineSymbol(math, ams, rel, "\u2289", "\\nsupseteq", true);
+defineSymbol(math, main, rel, "\u22a8", "\\models");
+defineSymbol(math, main, rel, "\u2190", "\\leftarrow", true);
+defineSymbol(math, main, rel, "\u2264", "\\le");
+defineSymbol(math, main, rel, "\u2264", "\\leq", true);
+defineSymbol(math, main, rel, "<", "\\lt", true);
+defineSymbol(math, main, rel, "\u2192", "\\rightarrow", true);
+defineSymbol(math, main, rel, "\u2192", "\\to");
+defineSymbol(math, ams, rel, "\u2271", "\\ngeq", true);
+defineSymbol(math, ams, rel, "\u2270", "\\nleq", true);
+defineSymbol(math, main, spacing, "\u00a0", "\\ ");
+defineSymbol(math, main, spacing, "\u00a0", "~");
+defineSymbol(math, main, spacing, "\u00a0", "\\space"); // Ref: LaTeX Source 2e: \DeclareRobustCommand{\nobreakspace}{%
+
+defineSymbol(math, main, spacing, "\u00a0", "\\nobreakspace");
+defineSymbol(text$1, main, spacing, "\u00a0", "\\ ");
+defineSymbol(text$1, main, spacing, "\u00a0", " ");
+defineSymbol(text$1, main, spacing, "\u00a0", "~");
+defineSymbol(text$1, main, spacing, "\u00a0", "\\space");
+defineSymbol(text$1, main, spacing, "\u00a0", "\\nobreakspace");
+defineSymbol(math, main, spacing, null, "\\nobreak");
+defineSymbol(math, main, spacing, null, "\\allowbreak");
+defineSymbol(math, main, punct, ",", ",");
+defineSymbol(math, main, punct, ";", ";");
+defineSymbol(math, ams, bin, "\u22bc", "\\barwedge", true);
+defineSymbol(math, ams, bin, "\u22bb", "\\veebar", true);
+defineSymbol(math, main, bin, "\u2299", "\\odot", true);
+defineSymbol(math, main, bin, "\u2295", "\\oplus", true);
+defineSymbol(math, main, bin, "\u2297", "\\otimes", true);
+defineSymbol(math, main, textord, "\u2202", "\\partial", true);
+defineSymbol(math, main, bin, "\u2298", "\\oslash", true);
+defineSymbol(math, ams, bin, "\u229a", "\\circledcirc", true);
+defineSymbol(math, ams, bin, "\u22a1", "\\boxdot", true);
+defineSymbol(math, main, bin, "\u25b3", "\\bigtriangleup");
+defineSymbol(math, main, bin, "\u25bd", "\\bigtriangledown");
+defineSymbol(math, main, bin, "\u2020", "\\dagger");
+defineSymbol(math, main, bin, "\u22c4", "\\diamond");
+defineSymbol(math, main, bin, "\u22c6", "\\star");
+defineSymbol(math, main, bin, "\u25c3", "\\triangleleft");
+defineSymbol(math, main, bin, "\u25b9", "\\triangleright");
+defineSymbol(math, main, open, "{", "\\{");
+defineSymbol(text$1, main, textord, "{", "\\{");
+defineSymbol(text$1, main, textord, "{", "\\textbraceleft");
+defineSymbol(math, main, close, "}", "\\}");
+defineSymbol(text$1, main, textord, "}", "\\}");
+defineSymbol(text$1, main, textord, "}", "\\textbraceright");
+defineSymbol(math, main, open, "{", "\\lbrace");
+defineSymbol(math, main, close, "}", "\\rbrace");
+defineSymbol(math, main, open, "[", "\\lbrack", true);
+defineSymbol(text$1, main, textord, "[", "\\lbrack", true);
+defineSymbol(math, main, close, "]", "\\rbrack", true);
+defineSymbol(text$1, main, textord, "]", "\\rbrack", true);
+defineSymbol(math, main, open, "(", "\\lparen", true);
+defineSymbol(math, main, close, ")", "\\rparen", true);
+defineSymbol(text$1, main, textord, "<", "\\textless", true); // in T1 fontenc
+
+defineSymbol(text$1, main, textord, ">", "\\textgreater", true); // in T1 fontenc
+
+defineSymbol(math, main, open, "\u230a", "\\lfloor", true);
+defineSymbol(math, main, close, "\u230b", "\\rfloor", true);
+defineSymbol(math, main, open, "\u2308", "\\lceil", true);
+defineSymbol(math, main, close, "\u2309", "\\rceil", true);
+defineSymbol(math, main, textord, "\\", "\\backslash");
+defineSymbol(math, main, textord, "\u2223", "|");
+defineSymbol(math, main, textord, "\u2223", "\\vert");
+defineSymbol(text$1, main, textord, "|", "\\textbar", true); // in T1 fontenc
+
+defineSymbol(math, main, textord, "\u2225", "\\|");
+defineSymbol(math, main, textord, "\u2225", "\\Vert");
+defineSymbol(text$1, main, textord, "\u2225", "\\textbardbl");
+defineSymbol(text$1, main, textord, "~", "\\textasciitilde");
+defineSymbol(text$1, main, textord, "\\", "\\textbackslash");
+defineSymbol(text$1, main, textord, "^", "\\textasciicircum");
+defineSymbol(math, main, rel, "\u2191", "\\uparrow", true);
+defineSymbol(math, main, rel, "\u21d1", "\\Uparrow", true);
+defineSymbol(math, main, rel, "\u2193", "\\downarrow", true);
+defineSymbol(math, main, rel, "\u21d3", "\\Downarrow", true);
+defineSymbol(math, main, rel, "\u2195", "\\updownarrow", true);
+defineSymbol(math, main, rel, "\u21d5", "\\Updownarrow", true);
+defineSymbol(math, main, op, "\u2210", "\\coprod");
+defineSymbol(math, main, op, "\u22c1", "\\bigvee");
+defineSymbol(math, main, op, "\u22c0", "\\bigwedge");
+defineSymbol(math, main, op, "\u2a04", "\\biguplus");
+defineSymbol(math, main, op, "\u22c2", "\\bigcap");
+defineSymbol(math, main, op, "\u22c3", "\\bigcup");
+defineSymbol(math, main, op, "\u222b", "\\int");
+defineSymbol(math, main, op, "\u222b", "\\intop");
+defineSymbol(math, main, op, "\u222c", "\\iint");
+defineSymbol(math, main, op, "\u222d", "\\iiint");
+defineSymbol(math, main, op, "\u220f", "\\prod");
+defineSymbol(math, main, op, "\u2211", "\\sum");
+defineSymbol(math, main, op, "\u2a02", "\\bigotimes");
+defineSymbol(math, main, op, "\u2a01", "\\bigoplus");
+defineSymbol(math, main, op, "\u2a00", "\\bigodot");
+defineSymbol(math, main, op, "\u222e", "\\oint");
+defineSymbol(math, main, op, "\u2a06", "\\bigsqcup");
+defineSymbol(math, main, op, "\u222b", "\\smallint");
+defineSymbol(text$1, main, inner, "\u2026", "\\textellipsis");
+defineSymbol(math, main, inner, "\u2026", "\\mathellipsis");
+defineSymbol(text$1, main, inner, "\u2026", "\\ldots", true);
+defineSymbol(math, main, inner, "\u2026", "\\ldots", true);
+defineSymbol(math, main, inner, "\u22ef", "\\@cdots", true);
+defineSymbol(math, main, inner, "\u22f1", "\\ddots", true);
+defineSymbol(math, main, textord, "\u22ee", "\\varvdots"); // \vdots is a macro
+
+defineSymbol(math, main, accent, "\u02ca", "\\acute");
+defineSymbol(math, main, accent, "\u02cb", "\\grave");
+defineSymbol(math, main, accent, "\u00a8", "\\ddot");
+defineSymbol(math, main, accent, "\u007e", "\\tilde");
+defineSymbol(math, main, accent, "\u02c9", "\\bar");
+defineSymbol(math, main, accent, "\u02d8", "\\breve");
+defineSymbol(math, main, accent, "\u02c7", "\\check");
+defineSymbol(math, main, accent, "\u005e", "\\hat");
+defineSymbol(math, main, accent, "\u20d7", "\\vec");
+defineSymbol(math, main, accent, "\u02d9", "\\dot");
+defineSymbol(math, main, accent, "\u02da", "\\mathring"); // \imath and \jmath should be invariant to \mathrm, \mathbf, etc., so use PUA
+
+defineSymbol(math, main, mathord, "\ue131", "\\@imath");
+defineSymbol(math, main, mathord, "\ue237", "\\@jmath");
+defineSymbol(math, main, textord, "\u0131", "\u0131");
+defineSymbol(math, main, textord, "\u0237", "\u0237");
+defineSymbol(text$1, main, textord, "\u0131", "\\i", true);
+defineSymbol(text$1, main, textord, "\u0237", "\\j", true);
+defineSymbol(text$1, main, textord, "\u00df", "\\ss", true);
+defineSymbol(text$1, main, textord, "\u00e6", "\\ae", true);
+defineSymbol(text$1, main, textord, "\u0153", "\\oe", true);
+defineSymbol(text$1, main, textord, "\u00f8", "\\o", true);
+defineSymbol(text$1, main, textord, "\u00c6", "\\AE", true);
+defineSymbol(text$1, main, textord, "\u0152", "\\OE", true);
+defineSymbol(text$1, main, textord, "\u00d8", "\\O", true);
+defineSymbol(text$1, main, accent, "\u02ca", "\\'"); // acute
+
+defineSymbol(text$1, main, accent, "\u02cb", "\\`"); // grave
+
+defineSymbol(text$1, main, accent, "\u02c6", "\\^"); // circumflex
+
+defineSymbol(text$1, main, accent, "\u02dc", "\\~"); // tilde
+
+defineSymbol(text$1, main, accent, "\u02c9", "\\="); // macron
+
+defineSymbol(text$1, main, accent, "\u02d8", "\\u"); // breve
+
+defineSymbol(text$1, main, accent, "\u02d9", "\\."); // dot above
+
+defineSymbol(text$1, main, accent, "\u02da", "\\r"); // ring above
+
+defineSymbol(text$1, main, accent, "\u02c7", "\\v"); // caron
+
+defineSymbol(text$1, main, accent, "\u00a8", '\\"'); // diaresis
+
+defineSymbol(text$1, main, accent, "\u02dd", "\\H"); // double acute
+
+defineSymbol(text$1, main, accent, "\u25ef", "\\textcircled"); // \bigcirc glyph
+// These ligatures are detected and created in Parser.js's `formLigatures`.
+
+const ligatures = {
+  "--": true,
+  "---": true,
+  "``": true,
+  "''": true
+};
+defineSymbol(text$1, main, textord, "\u2013", "--", true);
+defineSymbol(text$1, main, textord, "\u2013", "\\textendash");
+defineSymbol(text$1, main, textord, "\u2014", "---", true);
+defineSymbol(text$1, main, textord, "\u2014", "\\textemdash");
+defineSymbol(text$1, main, textord, "\u2018", "`", true);
+defineSymbol(text$1, main, textord, "\u2018", "\\textquoteleft");
+defineSymbol(text$1, main, textord, "\u2019", "'", true);
+defineSymbol(text$1, main, textord, "\u2019", "\\textquoteright");
+defineSymbol(text$1, main, textord, "\u201c", "``", true);
+defineSymbol(text$1, main, textord, "\u201c", "\\textquotedblleft");
+defineSymbol(text$1, main, textord, "\u201d", "''", true);
+defineSymbol(text$1, main, textord, "\u201d", "\\textquotedblright"); //  \degree from gensymb package
+
+defineSymbol(math, main, textord, "\u00b0", "\\degree", true);
+defineSymbol(text$1, main, textord, "\u00b0", "\\degree"); // \textdegree from inputenc package
+
+defineSymbol(text$1, main, textord, "\u00b0", "\\textdegree", true); // TODO: In LaTeX, \pounds can generate a different character in text and math
+// mode, but among our fonts, only Main-Regular defines this character "163".
+
+defineSymbol(math, main, textord, "\u00a3", "\\pounds");
+defineSymbol(math, main, textord, "\u00a3", "\\mathsterling", true);
+defineSymbol(text$1, main, textord, "\u00a3", "\\pounds");
+defineSymbol(text$1, main, textord, "\u00a3", "\\textsterling", true);
+defineSymbol(math, ams, textord, "\u2720", "\\maltese");
+defineSymbol(text$1, ams, textord, "\u2720", "\\maltese"); // There are lots of symbols which are the same, so we add them in afterwards.
+// All of these are textords in math mode
+
+const mathTextSymbols = "0123456789/@.\"";
+
+for (let i = 0; i < mathTextSymbols.length; i++) {
+  const ch = mathTextSymbols.charAt(i);
+  defineSymbol(math, main, textord, ch, ch);
+} // All of these are textords in text mode
+
+
+const textSymbols = "0123456789!@*()-=+\";:?/.,";
+
+for (let i = 0; i < textSymbols.length; i++) {
+  const ch = textSymbols.charAt(i);
+  defineSymbol(text$1, main, textord, ch, ch);
+} // All of these are textords in text mode, and mathords in math mode
+
+
+const letters = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";
+
+for (let i = 0; i < letters.length; i++) {
+  const ch = letters.charAt(i);
+  defineSymbol(math, main, mathord, ch, ch);
+  defineSymbol(text$1, main, textord, ch, ch);
+} // Blackboard bold and script letters in Unicode range
+
+
+defineSymbol(math, ams, textord, "C", "\u2102"); // blackboard bold
+
+defineSymbol(text$1, ams, textord, "C", "\u2102");
+defineSymbol(math, ams, textord, "H", "\u210D");
+defineSymbol(text$1, ams, textord, "H", "\u210D");
+defineSymbol(math, ams, textord, "N", "\u2115");
+defineSymbol(text$1, ams, textord, "N", "\u2115");
+defineSymbol(math, ams, textord, "P", "\u2119");
+defineSymbol(text$1, ams, textord, "P", "\u2119");
+defineSymbol(math, ams, textord, "Q", "\u211A");
+defineSymbol(text$1, ams, textord, "Q", "\u211A");
+defineSymbol(math, ams, textord, "R", "\u211D");
+defineSymbol(text$1, ams, textord, "R", "\u211D");
+defineSymbol(math, ams, textord, "Z", "\u2124");
+defineSymbol(text$1, ams, textord, "Z", "\u2124");
+defineSymbol(math, main, mathord, "h", "\u210E"); // italic h, Planck constant
+
+defineSymbol(text$1, main, mathord, "h", "\u210E"); // The next loop loads wide (surrogate pair) characters.
+// We support some letters in the Unicode range U+1D400 to U+1D7FF,
+// Mathematical Alphanumeric Symbols.
+// Some editors do not deal well with wide characters. So don't write the
+// string into this file. Instead, create the string from the surrogate pair.
+
+let wideChar = "";
+
+for (let i = 0; i < letters.length; i++) {
+  const ch = letters.charAt(i); // The hex numbers in the next line are a surrogate pair.
+  // 0xD835 is the high surrogate for all letters in the range we support.
+  // 0xDC00 is the low surrogate for bold A.
+
+  wideChar = String.fromCharCode(0xD835, 0xDC00 + i); // A-Z a-z bold
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDC34 + i); // A-Z a-z italic
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDC68 + i); // A-Z a-z bold italic
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDD04 + i); // A-Z a-z Fractur
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDDA0 + i); // A-Z a-z sans-serif
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDDD4 + i); // A-Z a-z sans bold
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDE08 + i); // A-Z a-z sans italic
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDE70 + i); // A-Z a-z monospace
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+
+  if (i < 26) {
+    // KaTeX fonts have only capital letters for blackboard bold and script.
+    // See exception for k below.
+    wideChar = String.fromCharCode(0xD835, 0xDD38 + i); // A-Z double struck
+
+    defineSymbol(math, main, mathord, ch, wideChar);
+    defineSymbol(text$1, main, textord, ch, wideChar);
+    wideChar = String.fromCharCode(0xD835, 0xDC9C + i); // A-Z script
+
+    defineSymbol(math, main, mathord, ch, wideChar);
+    defineSymbol(text$1, main, textord, ch, wideChar);
+  } // TODO: Add bold script when it is supported by a KaTeX font.
+
+} // "k" is the only double struck lower case letter in the KaTeX fonts.
+
+
+wideChar = String.fromCharCode(0xD835, 0xDD5C); // k double struck
+
+defineSymbol(math, main, mathord, "k", wideChar);
+defineSymbol(text$1, main, textord, "k", wideChar); // Next, some wide character numerals
+
+for (let i = 0; i < 10; i++) {
+  const ch = i.toString();
+  wideChar = String.fromCharCode(0xD835, 0xDFCE + i); // 0-9 bold
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDFE2 + i); // 0-9 sans serif
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDFEC + i); // 0-9 bold sans
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDFF6 + i); // 0-9 monospace
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+} // We add these Latin-1 letters as symbols for backwards-compatibility,
+// but they are not actually in the font, nor are they supported by the
+// Unicode accent mechanism, so they fall back to Times font and look ugly.
+// TODO(edemaine): Fix this.
+
+
+const extraLatin = "\u00c7\u00d0\u00de\u00e7\u00fe";
+
+for (let i = 0; i < extraLatin.length; i++) {
+  const ch = extraLatin.charAt(i);
+  defineSymbol(math, main, mathord, ch, ch);
+  defineSymbol(text$1, main, textord, ch, ch);
+}
+
+/**
+ * This file provides support for Unicode range U+1D400 to U+1D7FF,
+ * Mathematical Alphanumeric Symbols.
+ *
+ * Function wideCharacterFont takes a wide character as input and returns
+ * the font information necessary to render it properly.
+ */
+/**
+ * Data below is from https://www.unicode.org/charts/PDF/U1D400.pdf
+ * That document sorts characters into groups by font type, say bold or italic.
+ *
+ * In the arrays below, each subarray consists three elements:
+ *      * The CSS class of that group when in math mode.
+ *      * The CSS class of that group when in text mode.
+ *      * The font name, so that KaTeX can get font metrics.
+ */
+
+const wideLatinLetterData = [["mathbf", "textbf", "Main-Bold"], // A-Z bold upright
+["mathbf", "textbf", "Main-Bold"], // a-z bold upright
+["mathnormal", "textit", "Math-Italic"], // A-Z italic
+["mathnormal", "textit", "Math-Italic"], // a-z italic
+["boldsymbol", "boldsymbol", "Main-BoldItalic"], // A-Z bold italic
+["boldsymbol", "boldsymbol", "Main-BoldItalic"], // a-z bold italic
+// Map fancy A-Z letters to script, not calligraphic.
+// This aligns with unicode-math and math fonts (except Cambria Math).
+["mathscr", "textscr", "Script-Regular"], // A-Z script
+["", "", ""], // a-z script.  No font
+["", "", ""], // A-Z bold script. No font
+["", "", ""], // a-z bold script. No font
+["mathfrak", "textfrak", "Fraktur-Regular"], // A-Z Fraktur
+["mathfrak", "textfrak", "Fraktur-Regular"], // a-z Fraktur
+["mathbb", "textbb", "AMS-Regular"], // A-Z double-struck
+["mathbb", "textbb", "AMS-Regular"], // k double-struck
+["", "", ""], // A-Z bold Fraktur No font metrics
+["", "", ""], // a-z bold Fraktur.   No font.
+["mathsf", "textsf", "SansSerif-Regular"], // A-Z sans-serif
+["mathsf", "textsf", "SansSerif-Regular"], // a-z sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // A-Z bold sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // a-z bold sans-serif
+["mathitsf", "textitsf", "SansSerif-Italic"], // A-Z italic sans-serif
+["mathitsf", "textitsf", "SansSerif-Italic"], // a-z italic sans-serif
+["", "", ""], // A-Z bold italic sans. No font
+["", "", ""], // a-z bold italic sans. No font
+["mathtt", "texttt", "Typewriter-Regular"], // A-Z monospace
+["mathtt", "texttt", "Typewriter-Regular"]];
+const wideNumeralData = [["mathbf", "textbf", "Main-Bold"], // 0-9 bold
+["", "", ""], // 0-9 double-struck. No KaTeX font.
+["mathsf", "textsf", "SansSerif-Regular"], // 0-9 sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // 0-9 bold sans-serif
+["mathtt", "texttt", "Typewriter-Regular"]];
+const wideCharacterFont = function wideCharacterFont(wideChar, mode) {
+  // IE doesn't support codePointAt(). So work with the surrogate pair.
+  const H = wideChar.charCodeAt(0); // high surrogate
+
+  const L = wideChar.charCodeAt(1); // low surrogate
+
+  const codePoint = (H - 0xD800) * 0x400 + (L - 0xDC00) + 0x10000;
+  const j = mode === "math" ? 0 : 1; // column index for CSS class.
+
+  if (0x1D400 <= codePoint && codePoint < 0x1D6A4) {
+    // wideLatinLetterData contains exactly 26 chars on each row.
+    // So we can calculate the relevant row. No traverse necessary.
+    const i = Math.floor((codePoint - 0x1D400) / 26);
+    return [wideLatinLetterData[i][2], wideLatinLetterData[i][j]];
+  } else if (0x1D7CE <= codePoint && codePoint <= 0x1D7FF) {
+    // Numerals, ten per row.
+    const i = Math.floor((codePoint - 0x1D7CE) / 10);
+    return [wideNumeralData[i][2], wideNumeralData[i][j]];
+  } else if (codePoint === 0x1D6A5 || codePoint === 0x1D6A6) {
+    // dotless i or j
+    return [wideLatinLetterData[0][2], wideLatinLetterData[0][j]];
+  } else if (0x1D6A6 < codePoint && codePoint < 0x1D7CE) {
+    // Greek letters. Not supported, yet.
+    return ["", ""];
+  } else {
+    // We don't support any wide characters outside 1D400–1D7FF.
+    throw new ParseError("Unsupported character: " + wideChar);
+  }
+};
+
+/**
+ * This file contains information about the options that the Parser carries
+ * around with it while parsing. Data is held in an `Options` object, and when
+ * recursing, a new `Options` object can be created with the `.with*` and
+ * `.reset` functions.
+ */
+const sizeStyleMap = [// Each element contains [textsize, scriptsize, scriptscriptsize].
+// The size mappings are taken from TeX with \normalsize=10pt.
+[1, 1, 1], // size1: [5, 5, 5]              \tiny
+[2, 1, 1], // size2: [6, 5, 5]
+[3, 1, 1], // size3: [7, 5, 5]              \scriptsize
+[4, 2, 1], // size4: [8, 6, 5]              \footnotesize
+[5, 2, 1], // size5: [9, 6, 5]              \small
+[6, 3, 1], // size6: [10, 7, 5]             \normalsize
+[7, 4, 2], // size7: [12, 8, 6]             \large
+[8, 6, 3], // size8: [14.4, 10, 7]          \Large
+[9, 7, 6], // size9: [17.28, 12, 10]        \LARGE
+[10, 8, 7], // size10: [20.74, 14.4, 12]     \huge
+[11, 10, 9]];
+const sizeMultipliers = [// fontMetrics.js:getGlobalMetrics also uses size indexes, so if
+// you change size indexes, change that function.
+0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.44, 1.728, 2.074, 2.488];
+
+const sizeAtStyle = function sizeAtStyle(size, style) {
+  return style.size < 2 ? size : sizeStyleMap[size - 1][style.size - 1];
+}; // In these types, "" (empty string) means "no change".
+
+
+/**
+ * This is the main options class. It contains the current style, size, color,
+ * and font.
+ *
+ * Options objects should not be modified. To create a new Options with
+ * different properties, call a `.having*` method.
+ */
+class Options {
+  // A font family applies to a group of fonts (i.e. SansSerif), while a font
+  // represents a specific font (i.e. SansSerif Bold).
+  // See: https://tex.stackexchange.com/questions/22350/difference-between-textrm-and-mathrm
+
+  /**
+   * The base size index.
+   */
+  constructor(data) {
+    this.style = void 0;
+    this.color = void 0;
+    this.size = void 0;
+    this.textSize = void 0;
+    this.phantom = void 0;
+    this.font = void 0;
+    this.fontFamily = void 0;
+    this.fontWeight = void 0;
+    this.fontShape = void 0;
+    this.sizeMultiplier = void 0;
+    this.maxSize = void 0;
+    this.minRuleThickness = void 0;
+    this._fontMetrics = void 0;
+    this.style = data.style;
+    this.color = data.color;
+    this.size = data.size || Options.BASESIZE;
+    this.textSize = data.textSize || this.size;
+    this.phantom = !!data.phantom;
+    this.font = data.font || "";
+    this.fontFamily = data.fontFamily || "";
+    this.fontWeight = data.fontWeight || '';
+    this.fontShape = data.fontShape || '';
+    this.sizeMultiplier = sizeMultipliers[this.size - 1];
+    this.maxSize = data.maxSize;
+    this.minRuleThickness = data.minRuleThickness;
+    this._fontMetrics = undefined;
+  }
+  /**
+   * Returns a new options object with the same properties as "this".  Properties
+   * from "extension" will be copied to the new options object.
+   */
+
+
+  extend(extension) {
+    const data = {
+      style: this.style,
+      size: this.size,
+      textSize: this.textSize,
+      color: this.color,
+      phantom: this.phantom,
+      font: this.font,
+      fontFamily: this.fontFamily,
+      fontWeight: this.fontWeight,
+      fontShape: this.fontShape,
+      maxSize: this.maxSize,
+      minRuleThickness: this.minRuleThickness
+    };
+
+    for (const key in extension) {
+      if (extension.hasOwnProperty(key)) {
+        data[key] = extension[key];
+      }
+    }
+
+    return new Options(data);
+  }
+  /**
+   * Return an options object with the given style. If `this.style === style`,
+   * returns `this`.
+   */
+
+
+  havingStyle(style) {
+    if (this.style === style) {
+      return this;
+    } else {
+      return this.extend({
+        style: style,
+        size: sizeAtStyle(this.textSize, style)
+      });
+    }
+  }
+  /**
+   * Return an options object with a cramped version of the current style. If
+   * the current style is cramped, returns `this`.
+   */
+
+
+  havingCrampedStyle() {
+    return this.havingStyle(this.style.cramp());
+  }
+  /**
+   * Return an options object with the given size and in at least `\textstyle`.
+   * Returns `this` if appropriate.
+   */
+
+
+  havingSize(size) {
+    if (this.size === size && this.textSize === size) {
+      return this;
+    } else {
+      return this.extend({
+        style: this.style.text(),
+        size: size,
+        textSize: size,
+        sizeMultiplier: sizeMultipliers[size - 1]
+      });
+    }
+  }
+  /**
+   * Like `this.havingSize(BASESIZE).havingStyle(style)`. If `style` is omitted,
+   * changes to at least `\textstyle`.
+   */
+
+
+  havingBaseStyle(style) {
+    style = style || this.style.text();
+    const wantSize = sizeAtStyle(Options.BASESIZE, style);
+
+    if (this.size === wantSize && this.textSize === Options.BASESIZE && this.style === style) {
+      return this;
+    } else {
+      return this.extend({
+        style: style,
+        size: wantSize
+      });
+    }
+  }
+  /**
+   * Remove the effect of sizing changes such as \Huge.
+   * Keep the effect of the current style, such as \scriptstyle.
+   */
+
+
+  havingBaseSizing() {
+    let size;
+
+    switch (this.style.id) {
+      case 4:
+      case 5:
+        size = 3; // normalsize in scriptstyle
+
+        break;
+
+      case 6:
+      case 7:
+        size = 1; // normalsize in scriptscriptstyle
+
+        break;
+
+      default:
+        size = 6;
+      // normalsize in textstyle or displaystyle
+    }
+
+    return this.extend({
+      style: this.style.text(),
+      size: size
+    });
+  }
+  /**
+   * Create a new options object with the given color.
+   */
+
+
+  withColor(color) {
+    return this.extend({
+      color: color
+    });
+  }
+  /**
+   * Create a new options object with "phantom" set to true.
+   */
+
+
+  withPhantom() {
+    return this.extend({
+      phantom: true
+    });
+  }
+  /**
+   * Creates a new options object with the given math font or old text font.
+   * @type {[type]}
+   */
+
+
+  withFont(font) {
+    return this.extend({
+      font
+    });
+  }
+  /**
+   * Create a new options objects with the given fontFamily.
+   */
+
+
+  withTextFontFamily(fontFamily) {
+    return this.extend({
+      fontFamily,
+      font: ""
+    });
+  }
+  /**
+   * Creates a new options object with the given font weight
+   */
+
+
+  withTextFontWeight(fontWeight) {
+    return this.extend({
+      fontWeight,
+      font: ""
+    });
+  }
+  /**
+   * Creates a new options object with the given font weight
+   */
+
+
+  withTextFontShape(fontShape) {
+    return this.extend({
+      fontShape,
+      font: ""
+    });
+  }
+  /**
+   * Return the CSS sizing classes required to switch from enclosing options
+   * `oldOptions` to `this`. Returns an array of classes.
+   */
+
+
+  sizingClasses(oldOptions) {
+    if (oldOptions.size !== this.size) {
+      return ["sizing", "reset-size" + oldOptions.size, "size" + this.size];
+    } else {
+      return [];
+    }
+  }
+  /**
+   * Return the CSS sizing classes required to switch to the base size. Like
+   * `this.havingSize(BASESIZE).sizingClasses(this)`.
+   */
+
+
+  baseSizingClasses() {
+    if (this.size !== Options.BASESIZE) {
+      return ["sizing", "reset-size" + this.size, "size" + Options.BASESIZE];
+    } else {
+      return [];
+    }
+  }
+  /**
+   * Return the font metrics for this size.
+   */
+
+
+  fontMetrics() {
+    if (!this._fontMetrics) {
+      this._fontMetrics = getGlobalMetrics(this.size);
+    }
+
+    return this._fontMetrics;
+  }
+  /**
+   * Gets the CSS color of the current options object
+   */
+
+
+  getColor() {
+    if (this.phantom) {
+      return "transparent";
+    } else {
+      return this.color;
+    }
+  }
+
+}
+
+Options.BASESIZE = 6;
+
+/**
+ * This file does conversion between units.  In particular, it provides
+ * calculateSize to convert other units into ems.
+ */
+// Thus, multiplying a length by this number converts the length from units
+// into pts.  Dividing the result by ptPerEm gives the number of ems
+// *assuming* a font size of ptPerEm (normal size, normal style).
+
+const ptPerUnit = {
+  // https://en.wikibooks.org/wiki/LaTeX/Lengths and
+  // https://tex.stackexchange.com/a/8263
+  "pt": 1,
+  // TeX point
+  "mm": 7227 / 2540,
+  // millimeter
+  "cm": 7227 / 254,
+  // centimeter
+  "in": 72.27,
+  // inch
+  "bp": 803 / 800,
+  // big (PostScript) points
+  "pc": 12,
+  // pica
+  "dd": 1238 / 1157,
+  // didot
+  "cc": 14856 / 1157,
+  // cicero (12 didot)
+  "nd": 685 / 642,
+  // new didot
+  "nc": 1370 / 107,
+  // new cicero (12 new didot)
+  "sp": 1 / 65536,
+  // scaled point (TeX's internal smallest unit)
+  // https://tex.stackexchange.com/a/41371
+  "px": 803 / 800 // \pdfpxdimen defaults to 1 bp in pdfTeX and LuaTeX
+
+}; // Dictionary of relative units, for fast validity testing.
+
+const relativeUnit = {
+  "ex": true,
+  "em": true,
+  "mu": true
+};
+
+/**
+ * Determine whether the specified unit (either a string defining the unit
+ * or a "size" parse node containing a unit field) is valid.
+ */
+const validUnit = function validUnit(unit) {
+  if (typeof unit !== "string") {
+    unit = unit.unit;
+  }
+
+  return unit in ptPerUnit || unit in relativeUnit || unit === "ex";
+};
+/*
+ * Convert a "size" parse node (with numeric "number" and string "unit" fields,
+ * as parsed by functions.js argType "size") into a CSS em value for the
+ * current style/scale.  `options` gives the current options.
+ */
+
+const calculateSize = function calculateSize(sizeValue, options) {
+  let scale;
+
+  if (sizeValue.unit in ptPerUnit) {
+    // Absolute units
+    scale = ptPerUnit[sizeValue.unit] // Convert unit to pt
+    / options.fontMetrics().ptPerEm // Convert pt to CSS em
+    / options.sizeMultiplier; // Unscale to make absolute units
+  } else if (sizeValue.unit === "mu") {
+    // `mu` units scale with scriptstyle/scriptscriptstyle.
+    scale = options.fontMetrics().cssEmPerMu;
+  } else {
+    // Other relative units always refer to the *textstyle* font
+    // in the current size.
+    let unitOptions;
+
+    if (options.style.isTight()) {
+      // isTight() means current style is script/scriptscript.
+      unitOptions = options.havingStyle(options.style.text());
+    } else {
+      unitOptions = options;
+    } // TODO: In TeX these units are relative to the quad of the current
+    // *text* font, e.g. cmr10. KaTeX instead uses values from the
+    // comparably-sized *Computer Modern symbol* font. At 10pt, these
+    // match. At 7pt and 5pt, they differ: cmr7=1.138894, cmsy7=1.170641;
+    // cmr5=1.361133, cmsy5=1.472241. Consider $\scriptsize a\kern1emb$.
+    // TeX \showlists shows a kern of 1.13889 * fontsize;
+    // KaTeX shows a kern of 1.171 * fontsize.
+
+
+    if (sizeValue.unit === "ex") {
+      scale = unitOptions.fontMetrics().xHeight;
+    } else if (sizeValue.unit === "em") {
+      scale = unitOptions.fontMetrics().quad;
+    } else {
+      throw new ParseError("Invalid unit: '" + sizeValue.unit + "'");
+    }
+
+    if (unitOptions !== options) {
+      scale *= unitOptions.sizeMultiplier / options.sizeMultiplier;
+    }
+  }
+
+  return Math.min(sizeValue.number * scale, options.maxSize);
+};
+
+/* eslint no-console:0 */
+
+/**
+ * Looks up the given symbol in fontMetrics, after applying any symbol
+ * replacements defined in symbol.js
+ */
+const lookupSymbol = function lookupSymbol(value, // TODO(#963): Use a union type for this.
+fontName, mode) {
+  // Replace the value with its replaced value from symbol.js
+  if (symbols[mode][value] && symbols[mode][value].replace) {
+    value = symbols[mode][value].replace;
+  }
+
+  return {
+    value: value,
+    metrics: getCharacterMetrics(value, fontName, mode)
+  };
+};
+/**
+ * Makes a symbolNode after translation via the list of symbols in symbols.js.
+ * Correctly pulls out metrics for the character, and optionally takes a list of
+ * classes to be attached to the node.
+ *
+ * TODO: make argument order closer to makeSpan
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ * TODO(#953): Make `options` mandatory and always pass it in.
+ */
+
+
+const makeSymbol = function makeSymbol(value, fontName, mode, options, classes) {
+  const lookup = lookupSymbol(value, fontName, mode);
+  const metrics = lookup.metrics;
+  value = lookup.value;
+  let symbolNode;
+
+  if (metrics) {
+    let italic = metrics.italic;
+
+    if (mode === "text" || options && options.font === "mathit") {
+      italic = 0;
+    }
+
+    symbolNode = new SymbolNode(value, metrics.height, metrics.depth, italic, metrics.skew, metrics.width, classes);
+  } else {
+    // TODO(emily): Figure out a good way to only print this in development
+    typeof console !== "undefined" && console.warn("No character metrics " + `for '${value}' in style '${fontName}' and mode '${mode}'`);
+    symbolNode = new SymbolNode(value, 0, 0, 0, 0, 0, classes);
+  }
+
+  if (options) {
+    symbolNode.maxFontSize = options.sizeMultiplier;
+
+    if (options.style.isTight()) {
+      symbolNode.classes.push("mtight");
+    }
+
+    const color = options.getColor();
+
+    if (color) {
+      symbolNode.style.color = color;
+    }
+  }
+
+  return symbolNode;
+};
+/**
+ * Makes a symbol in Main-Regular or AMS-Regular.
+ * Used for rel, bin, open, close, inner, and punct.
+ */
+
+
+const mathsym = function mathsym(value, mode, options, classes) {
+  if (classes === void 0) {
+    classes = [];
+  }
+
+  // Decide what font to render the symbol in by its entry in the symbols
+  // table.
+  // Have a special case for when the value = \ because the \ is used as a
+  // textord in unsupported command errors but cannot be parsed as a regular
+  // text ordinal and is therefore not present as a symbol in the symbols
+  // table for text, as well as a special case for boldsymbol because it
+  // can be used for bold + and -
+  if (options.font === "boldsymbol" && lookupSymbol(value, "Main-Bold", mode).metrics) {
+    return makeSymbol(value, "Main-Bold", mode, options, classes.concat(["mathbf"]));
+  } else if (value === "\\" || symbols[mode][value].font === "main") {
+    return makeSymbol(value, "Main-Regular", mode, options, classes);
+  } else {
+    return makeSymbol(value, "AMS-Regular", mode, options, classes.concat(["amsrm"]));
+  }
+};
+/**
+ * Determines which of the two font names (Main-Bold and Math-BoldItalic) and
+ * corresponding style tags (mathbf or boldsymbol) to use for font "boldsymbol",
+ * depending on the symbol.  Use this function instead of fontMap for font
+ * "boldsymbol".
+ */
+
+
+const boldsymbol = function boldsymbol(value, mode, options, classes, type) {
+  if (type !== "textord" && lookupSymbol(value, "Math-BoldItalic", mode).metrics) {
+    return {
+      fontName: "Math-BoldItalic",
+      fontClass: "boldsymbol"
+    };
+  } else {
+    // Some glyphs do not exist in Math-BoldItalic so we need to use
+    // Main-Bold instead.
+    return {
+      fontName: "Main-Bold",
+      fontClass: "mathbf"
+    };
+  }
+};
+/**
+ * Makes either a mathord or textord in the correct font and color.
+ */
+
+
+const makeOrd = function makeOrd(group, options, type) {
+  const mode = group.mode;
+  const text = group.text;
+  const classes = ["mord"]; // Math mode or Old font (i.e. \rm)
+
+  const isFont = mode === "math" || mode === "text" && options.font;
+  const fontOrFamily = isFont ? options.font : options.fontFamily;
+
+  if (text.charCodeAt(0) === 0xD835) {
+    // surrogate pairs get special treatment
+    const _wideCharacterFont = wideCharacterFont(text, mode),
+          wideFontName = _wideCharacterFont[0],
+          wideFontClass = _wideCharacterFont[1];
+
+    return makeSymbol(text, wideFontName, mode, options, classes.concat(wideFontClass));
+  } else if (fontOrFamily) {
+    let fontName;
+    let fontClasses;
+
+    if (fontOrFamily === "boldsymbol") {
+      const fontData = boldsymbol(text, mode, options, classes, type);
+      fontName = fontData.fontName;
+      fontClasses = [fontData.fontClass];
+    } else if (isFont) {
+      fontName = fontMap[fontOrFamily].fontName;
+      fontClasses = [fontOrFamily];
+    } else {
+      fontName = retrieveTextFontName(fontOrFamily, options.fontWeight, options.fontShape);
+      fontClasses = [fontOrFamily, options.fontWeight, options.fontShape];
+    }
+
+    if (lookupSymbol(text, fontName, mode).metrics) {
+      return makeSymbol(text, fontName, mode, options, classes.concat(fontClasses));
+    } else if (ligatures.hasOwnProperty(text) && fontName.substr(0, 10) === "Typewriter") {
+      // Deconstruct ligatures in monospace fonts (\texttt, \tt).
+      const parts = [];
+
+      for (let i = 0; i < text.length; i++) {
+        parts.push(makeSymbol(text[i], fontName, mode, options, classes.concat(fontClasses)));
+      }
+
+      return makeFragment(parts);
+    }
+  } // Makes a symbol in the default font for mathords and textords.
+
+
+  if (type === "mathord") {
+    return makeSymbol(text, "Math-Italic", mode, options, classes.concat(["mathnormal"]));
+  } else if (type === "textord") {
+    const font = symbols[mode][text] && symbols[mode][text].font;
+
+    if (font === "ams") {
+      const fontName = retrieveTextFontName("amsrm", options.fontWeight, options.fontShape);
+      return makeSymbol(text, fontName, mode, options, classes.concat("amsrm", options.fontWeight, options.fontShape));
+    } else if (font === "main" || !font) {
+      const fontName = retrieveTextFontName("textrm", options.fontWeight, options.fontShape);
+      return makeSymbol(text, fontName, mode, options, classes.concat(options.fontWeight, options.fontShape));
+    } else {
+      // fonts added by plugins
+      const fontName = retrieveTextFontName(font, options.fontWeight, options.fontShape); // We add font name as a css class
+
+      return makeSymbol(text, fontName, mode, options, classes.concat(fontName, options.fontWeight, options.fontShape));
+    }
+  } else {
+    throw new Error("unexpected type: " + type + " in makeOrd");
+  }
+};
+/**
+ * Returns true if subsequent symbolNodes have the same classes, skew, maxFont,
+ * and styles.
+ */
+
+
+const canCombine = (prev, next) => {
+  if (createClass(prev.classes) !== createClass(next.classes) || prev.skew !== next.skew || prev.maxFontSize !== next.maxFontSize) {
+    return false;
+  }
+
+  for (const style in prev.style) {
+    if (prev.style.hasOwnProperty(style) && prev.style[style] !== next.style[style]) {
+      return false;
+    }
+  }
+
+  for (const style in next.style) {
+    if (next.style.hasOwnProperty(style) && prev.style[style] !== next.style[style]) {
+      return false;
+    }
+  }
+
+  return true;
+};
+/**
+ * Combine consequetive domTree.symbolNodes into a single symbolNode.
+ * Note: this function mutates the argument.
+ */
+
+
+const tryCombineChars = chars => {
+  for (let i = 0; i < chars.length - 1; i++) {
+    const prev = chars[i];
+    const next = chars[i + 1];
+
+    if (prev instanceof SymbolNode && next instanceof SymbolNode && canCombine(prev, next)) {
+      prev.text += next.text;
+      prev.height = Math.max(prev.height, next.height);
+      prev.depth = Math.max(prev.depth, next.depth); // Use the last character's italic correction since we use
+      // it to add padding to the right of the span created from
+      // the combined characters.
+
+      prev.italic = next.italic;
+      chars.splice(i + 1, 1);
+      i--;
+    }
+  }
+
+  return chars;
+};
+/**
+ * Calculate the height, depth, and maxFontSize of an element based on its
+ * children.
+ */
+
+
+const sizeElementFromChildren = function sizeElementFromChildren(elem) {
+  let height = 0;
+  let depth = 0;
+  let maxFontSize = 0;
+
+  for (let i = 0; i < elem.children.length; i++) {
+    const child = elem.children[i];
+
+    if (child.height > height) {
+      height = child.height;
+    }
+
+    if (child.depth > depth) {
+      depth = child.depth;
+    }
+
+    if (child.maxFontSize > maxFontSize) {
+      maxFontSize = child.maxFontSize;
+    }
+  }
+
+  elem.height = height;
+  elem.depth = depth;
+  elem.maxFontSize = maxFontSize;
+};
+/**
+ * Makes a span with the given list of classes, list of children, and options.
+ *
+ * TODO(#953): Ensure that `options` is always provided (currently some call
+ * sites don't pass it) and make the type below mandatory.
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ */
+
+
+const makeSpan = function makeSpan(classes, children, options, style) {
+  const span = new Span(classes, children, options, style);
+  sizeElementFromChildren(span);
+  return span;
+}; // SVG one is simpler -- doesn't require height, depth, max-font setting.
+// This is also a separate method for typesafety.
+
+
+const makeSvgSpan = (classes, children, options, style) => new Span(classes, children, options, style);
+
+const makeLineSpan = function makeLineSpan(className, options, thickness) {
+  const line = makeSpan([className], [], options);
+  line.height = Math.max(thickness || options.fontMetrics().defaultRuleThickness, options.minRuleThickness);
+  line.style.borderBottomWidth = line.height + "em";
+  line.maxFontSize = 1.0;
+  return line;
+};
+/**
+ * Makes an anchor with the given href, list of classes, list of children,
+ * and options.
+ */
+
+
+const makeAnchor = function makeAnchor(href, classes, children, options) {
+  const anchor = new Anchor(href, classes, children, options);
+  sizeElementFromChildren(anchor);
+  return anchor;
+};
+/**
+ * Makes a document fragment with the given list of children.
+ */
+
+
+const makeFragment = function makeFragment(children) {
+  const fragment = new DocumentFragment(children);
+  sizeElementFromChildren(fragment);
+  return fragment;
+};
+/**
+ * Wraps group in a span if it's a document fragment, allowing to apply classes
+ * and styles
+ */
+
+
+const wrapFragment = function wrapFragment(group, options) {
+  if (group instanceof DocumentFragment) {
+    return makeSpan([], [group], options);
+  }
+
+  return group;
+}; // These are exact object types to catch typos in the names of the optional fields.
+
+
+// Computes the updated `children` list and the overall depth.
+//
+// This helper function for makeVList makes it easier to enforce type safety by
+// allowing early exits (returns) in the logic.
+const getVListChildrenAndDepth = function getVListChildrenAndDepth(params) {
+  if (params.positionType === "individualShift") {
+    const oldChildren = params.children;
+    const children = [oldChildren[0]]; // Add in kerns to the list of params.children to get each element to be
+    // shifted to the correct specified shift
+
+    const depth = -oldChildren[0].shift - oldChildren[0].elem.depth;
+    let currPos = depth;
+
+    for (let i = 1; i < oldChildren.length; i++) {
+      const diff = -oldChildren[i].shift - currPos - oldChildren[i].elem.depth;
+      const size = diff - (oldChildren[i - 1].elem.height + oldChildren[i - 1].elem.depth);
+      currPos = currPos + diff;
+      children.push({
+        type: "kern",
+        size
+      });
+      children.push(oldChildren[i]);
+    }
+
+    return {
+      children,
+      depth
+    };
+  }
+
+  let depth;
+
+  if (params.positionType === "top") {
+    // We always start at the bottom, so calculate the bottom by adding up
+    // all the sizes
+    let bottom = params.positionData;
+
+    for (let i = 0; i < params.children.length; i++) {
+      const child = params.children[i];
+      bottom -= child.type === "kern" ? child.size : child.elem.height + child.elem.depth;
+    }
+
+    depth = bottom;
+  } else if (params.positionType === "bottom") {
+    depth = -params.positionData;
+  } else {
+    const firstChild = params.children[0];
+
+    if (firstChild.type !== "elem") {
+      throw new Error('First child must have type "elem".');
+    }
+
+    if (params.positionType === "shift") {
+      depth = -firstChild.elem.depth - params.positionData;
+    } else if (params.positionType === "firstBaseline") {
+      depth = -firstChild.elem.depth;
+    } else {
+      throw new Error(`Invalid positionType ${params.positionType}.`);
+    }
+  }
+
+  return {
+    children: params.children,
+    depth
+  };
+};
+/**
+ * Makes a vertical list by stacking elements and kerns on top of each other.
+ * Allows for many different ways of specifying the positioning method.
+ *
+ * See VListParam documentation above.
+ */
+
+
+const makeVList = function makeVList(params, options) {
+  const _getVListChildrenAndD = getVListChildrenAndDepth(params),
+        children = _getVListChildrenAndD.children,
+        depth = _getVListChildrenAndD.depth; // Create a strut that is taller than any list item. The strut is added to
+  // each item, where it will determine the item's baseline. Since it has
+  // `overflow:hidden`, the strut's top edge will sit on the item's line box's
+  // top edge and the strut's bottom edge will sit on the item's baseline,
+  // with no additional line-height spacing. This allows the item baseline to
+  // be positioned precisely without worrying about font ascent and
+  // line-height.
+
+
+  let pstrutSize = 0;
+
+  for (let i = 0; i < children.length; i++) {
+    const child = children[i];
+
+    if (child.type === "elem") {
+      const elem = child.elem;
+      pstrutSize = Math.max(pstrutSize, elem.maxFontSize, elem.height);
+    }
+  }
+
+  pstrutSize += 2;
+  const pstrut = makeSpan(["pstrut"], []);
+  pstrut.style.height = pstrutSize + "em"; // Create a new list of actual children at the correct offsets
+
+  const realChildren = [];
+  let minPos = depth;
+  let maxPos = depth;
+  let currPos = depth;
+
+  for (let i = 0; i < children.length; i++) {
+    const child = children[i];
+
+    if (child.type === "kern") {
+      currPos += child.size;
+    } else {
+      const elem = child.elem;
+      const classes = child.wrapperClasses || [];
+      const style = child.wrapperStyle || {};
+      const childWrap = makeSpan(classes, [pstrut, elem], undefined, style);
+      childWrap.style.top = -pstrutSize - currPos - elem.depth + "em";
+
+      if (child.marginLeft) {
+        childWrap.style.marginLeft = child.marginLeft;
+      }
+
+      if (child.marginRight) {
+        childWrap.style.marginRight = child.marginRight;
+      }
+
+      realChildren.push(childWrap);
+      currPos += elem.height + elem.depth;
+    }
+
+    minPos = Math.min(minPos, currPos);
+    maxPos = Math.max(maxPos, currPos);
+  } // The vlist contents go in a table-cell with `vertical-align:bottom`.
+  // This cell's bottom edge will determine the containing table's baseline
+  // without overly expanding the containing line-box.
+
+
+  const vlist = makeSpan(["vlist"], realChildren);
+  vlist.style.height = maxPos + "em"; // A second row is used if necessary to represent the vlist's depth.
+
+  let rows;
+
+  if (minPos < 0) {
+    // We will define depth in an empty span with display: table-cell.
+    // It should render with the height that we define. But Chrome, in
+    // contenteditable mode only, treats that span as if it contains some
+    // text content. And that min-height over-rides our desired height.
+    // So we put another empty span inside the depth strut span.
+    const emptySpan = makeSpan([], []);
+    const depthStrut = makeSpan(["vlist"], [emptySpan]);
+    depthStrut.style.height = -minPos + "em"; // Safari wants the first row to have inline content; otherwise it
+    // puts the bottom of the *second* row on the baseline.
+
+    const topStrut = makeSpan(["vlist-s"], [new SymbolNode("\u200b")]);
+    rows = [makeSpan(["vlist-r"], [vlist, topStrut]), makeSpan(["vlist-r"], [depthStrut])];
+  } else {
+    rows = [makeSpan(["vlist-r"], [vlist])];
+  }
+
+  const vtable = makeSpan(["vlist-t"], rows);
+
+  if (rows.length === 2) {
+    vtable.classes.push("vlist-t2");
+  }
+
+  vtable.height = maxPos;
+  vtable.depth = -minPos;
+  return vtable;
+}; // Glue is a concept from TeX which is a flexible space between elements in
+// either a vertical or horizontal list. In KaTeX, at least for now, it's
+// static space between elements in a horizontal layout.
+
+
+const makeGlue = (measurement, options) => {
+  // Make an empty span for the space
+  const rule = makeSpan(["mspace"], [], options);
+  const size = calculateSize(measurement, options);
+  rule.style.marginRight = `${size}em`;
+  return rule;
+}; // Takes font options, and returns the appropriate fontLookup name
+
+
+const retrieveTextFontName = function retrieveTextFontName(fontFamily, fontWeight, fontShape) {
+  let baseFontName = "";
+
+  switch (fontFamily) {
+    case "amsrm":
+      baseFontName = "AMS";
+      break;
+
+    case "textrm":
+      baseFontName = "Main";
+      break;
+
+    case "textsf":
+      baseFontName = "SansSerif";
+      break;
+
+    case "texttt":
+      baseFontName = "Typewriter";
+      break;
+
+    default:
+      baseFontName = fontFamily;
+    // use fonts added by a plugin
+  }
+
+  let fontStylesName;
+
+  if (fontWeight === "textbf" && fontShape === "textit") {
+    fontStylesName = "BoldItalic";
+  } else if (fontWeight === "textbf") {
+    fontStylesName = "Bold";
+  } else if (fontWeight === "textit") {
+    fontStylesName = "Italic";
+  } else {
+    fontStylesName = "Regular";
+  }
+
+  return `${baseFontName}-${fontStylesName}`;
+};
+/**
+ * Maps TeX font commands to objects containing:
+ * - variant: string used for "mathvariant" attribute in buildMathML.js
+ * - fontName: the "style" parameter to fontMetrics.getCharacterMetrics
+ */
+// A map between tex font commands an MathML mathvariant attribute values
+
+
+const fontMap = {
+  // styles
+  "mathbf": {
+    variant: "bold",
+    fontName: "Main-Bold"
+  },
+  "mathrm": {
+    variant: "normal",
+    fontName: "Main-Regular"
+  },
+  "textit": {
+    variant: "italic",
+    fontName: "Main-Italic"
+  },
+  "mathit": {
+    variant: "italic",
+    fontName: "Main-Italic"
+  },
+  "mathnormal": {
+    variant: "italic",
+    fontName: "Math-Italic"
+  },
+  // "boldsymbol" is missing because they require the use of multiple fonts:
+  // Math-BoldItalic and Main-Bold.  This is handled by a special case in
+  // makeOrd which ends up calling boldsymbol.
+  // families
+  "mathbb": {
+    variant: "double-struck",
+    fontName: "AMS-Regular"
+  },
+  "mathcal": {
+    variant: "script",
+    fontName: "Caligraphic-Regular"
+  },
+  "mathfrak": {
+    variant: "fraktur",
+    fontName: "Fraktur-Regular"
+  },
+  "mathscr": {
+    variant: "script",
+    fontName: "Script-Regular"
+  },
+  "mathsf": {
+    variant: "sans-serif",
+    fontName: "SansSerif-Regular"
+  },
+  "mathtt": {
+    variant: "monospace",
+    fontName: "Typewriter-Regular"
+  }
+};
+const svgData = {
+  //   path, width, height
+  vec: ["vec", 0.471, 0.714],
+  // values from the font glyph
+  oiintSize1: ["oiintSize1", 0.957, 0.499],
+  // oval to overlay the integrand
+  oiintSize2: ["oiintSize2", 1.472, 0.659],
+  oiiintSize1: ["oiiintSize1", 1.304, 0.499],
+  oiiintSize2: ["oiiintSize2", 1.98, 0.659],
+  leftParenInner: ["leftParenInner", 0.875, 0.3],
+  rightParenInner: ["rightParenInner", 0.875, 0.3]
+};
+
+const staticSvg = function staticSvg(value, options) {
+  // Create a span with inline SVG for the element.
+  const _svgData$value = svgData[value],
+        pathName = _svgData$value[0],
+        width = _svgData$value[1],
+        height = _svgData$value[2];
+  const path = new PathNode(pathName);
+  const svgNode = new SvgNode([path], {
+    "width": width + "em",
+    "height": height + "em",
+    // Override CSS rule `.katex svg { width: 100% }`
+    "style": "width:" + width + "em",
+    "viewBox": "0 0 " + 1000 * width + " " + 1000 * height,
+    "preserveAspectRatio": "xMinYMin"
+  });
+  const span = makeSvgSpan(["overlay"], [svgNode], options);
+  span.height = height;
+  span.style.height = height + "em";
+  span.style.width = width + "em";
+  return span;
+};
+
+var buildCommon = {
+  fontMap,
+  makeSymbol,
+  mathsym,
+  makeSpan,
+  makeSvgSpan,
+  makeLineSpan,
+  makeAnchor,
+  makeFragment,
+  wrapFragment,
+  makeVList,
+  makeOrd,
+  makeGlue,
+  staticSvg,
+  svgData,
+  tryCombineChars
+};
+
+/**
+ * Describes spaces between different classes of atoms.
+ */
+const thinspace = {
+  number: 3,
+  unit: "mu"
+};
+const mediumspace = {
+  number: 4,
+  unit: "mu"
+};
+const thickspace = {
+  number: 5,
+  unit: "mu"
+}; // Making the type below exact with all optional fields doesn't work due to
+// - https://github.com/facebook/flow/issues/4582
+// - https://github.com/facebook/flow/issues/5688
+// However, since *all* fields are optional, $Shape<> works as suggested in 5688
+// above.
+
+// Spacing relationships for display and text styles
+const spacings = {
+  mord: {
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mop: {
+    mord: thinspace,
+    mop: thinspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mbin: {
+    mord: mediumspace,
+    mop: mediumspace,
+    mopen: mediumspace,
+    minner: mediumspace
+  },
+  mrel: {
+    mord: thickspace,
+    mop: thickspace,
+    mopen: thickspace,
+    minner: thickspace
+  },
+  mopen: {},
+  mclose: {
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mpunct: {
+    mord: thinspace,
+    mop: thinspace,
+    mrel: thickspace,
+    mopen: thinspace,
+    mclose: thinspace,
+    mpunct: thinspace,
+    minner: thinspace
+  },
+  minner: {
+    mord: thinspace,
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    mopen: thinspace,
+    mpunct: thinspace,
+    minner: thinspace
+  }
+}; // Spacing relationships for script and scriptscript styles
+
+const tightSpacings = {
+  mord: {
+    mop: thinspace
+  },
+  mop: {
+    mord: thinspace,
+    mop: thinspace
+  },
+  mbin: {},
+  mrel: {},
+  mopen: {},
+  mclose: {
+    mop: thinspace
+  },
+  mpunct: {},
+  minner: {
+    mop: thinspace
+  }
+};
+
+/** Context provided to function handlers for error messages. */
+// Note: reverse the order of the return type union will cause a flow error.
+// See https://github.com/facebook/flow/issues/3663.
+// More general version of `HtmlBuilder` for nodes (e.g. \sum, accent types)
+// whose presence impacts super/subscripting. In this case, ParseNode<"supsub">
+// delegates its HTML building to the HtmlBuilder corresponding to these nodes.
+
+/**
+ * Final function spec for use at parse time.
+ * This is almost identical to `FunctionPropSpec`, except it
+ * 1. includes the function handler, and
+ * 2. requires all arguments except argTypes.
+ * It is generated by `defineFunction()` below.
+ */
+
+/**
+ * All registered functions.
+ * `functions.js` just exports this same dictionary again and makes it public.
+ * `Parser.js` requires this dictionary.
+ */
+const _functions = {};
+/**
+ * All HTML builders. Should be only used in the `define*` and the `build*ML`
+ * functions.
+ */
+
+const _htmlGroupBuilders = {};
+/**
+ * All MathML builders. Should be only used in the `define*` and the `build*ML`
+ * functions.
+ */
+
+const _mathmlGroupBuilders = {};
+function defineFunction(_ref) {
+  let type = _ref.type,
+      names = _ref.names,
+      props = _ref.props,
+      handler = _ref.handler,
+      htmlBuilder = _ref.htmlBuilder,
+      mathmlBuilder = _ref.mathmlBuilder;
+  // Set default values of functions
+  const data = {
+    type,
+    numArgs: props.numArgs,
+    argTypes: props.argTypes,
+    greediness: props.greediness === undefined ? 1 : props.greediness,
+    allowedInText: !!props.allowedInText,
+    allowedInMath: props.allowedInMath === undefined ? true : props.allowedInMath,
+    numOptionalArgs: props.numOptionalArgs || 0,
+    infix: !!props.infix,
+    handler: handler
+  };
+
+  for (let i = 0; i < names.length; ++i) {
+    _functions[names[i]] = data;
+  }
+
+  if (type) {
+    if (htmlBuilder) {
+      _htmlGroupBuilders[type] = htmlBuilder;
+    }
+
+    if (mathmlBuilder) {
+      _mathmlGroupBuilders[type] = mathmlBuilder;
+    }
+  }
+}
+/**
+ * Use this to register only the HTML and MathML builders for a function (e.g.
+ * if the function's ParseNode is generated in Parser.js rather than via a
+ * stand-alone handler provided to `defineFunction`).
+ */
+
+function defineFunctionBuilders(_ref2) {
+  let type = _ref2.type,
+      htmlBuilder = _ref2.htmlBuilder,
+      mathmlBuilder = _ref2.mathmlBuilder;
+  defineFunction({
+    type,
+    names: [],
+    props: {
+      numArgs: 0
+    },
+
+    handler() {
+      throw new Error('Should never be called.');
+    },
+
+    htmlBuilder,
+    mathmlBuilder
+  });
+} // Since the corresponding buildHTML/buildMathML function expects a
+// list of elements, we normalize for different kinds of arguments
+
+const ordargument = function ordargument(arg) {
+  return arg.type === "ordgroup" ? arg.body : [arg];
+};
+
+/**
+ * This file does the main work of building a domTree structure from a parse
+ * tree. The entry point is the `buildHTML` function, which takes a parse tree.
+ * Then, the buildExpression, buildGroup, and various groupBuilders functions
+ * are called, to produce a final HTML tree.
+ */
+const makeSpan$1 = buildCommon.makeSpan; // Binary atoms (first class `mbin`) change into ordinary atoms (`mord`)
+// depending on their surroundings. See TeXbook pg. 442-446, Rules 5 and 6,
+// and the text before Rule 19.
+
+const binLeftCanceller = ["leftmost", "mbin", "mopen", "mrel", "mop", "mpunct"];
+const binRightCanceller = ["rightmost", "mrel", "mclose", "mpunct"];
+const styleMap = {
+  "display": Style$1.DISPLAY,
+  "text": Style$1.TEXT,
+  "script": Style$1.SCRIPT,
+  "scriptscript": Style$1.SCRIPTSCRIPT
+};
+const DomEnum = {
+  mord: "mord",
+  mop: "mop",
+  mbin: "mbin",
+  mrel: "mrel",
+  mopen: "mopen",
+  mclose: "mclose",
+  mpunct: "mpunct",
+  minner: "minner"
+};
+
+/**
+ * Take a list of nodes, build them in order, and return a list of the built
+ * nodes. documentFragments are flattened into their contents, so the
+ * returned list contains no fragments. `isRealGroup` is true if `expression`
+ * is a real group (no atoms will be added on either side), as opposed to
+ * a partial group (e.g. one created by \color). `surrounding` is an array
+ * consisting type of nodes that will be added to the left and right.
+ */
+const buildExpression = function buildExpression(expression, options, isRealGroup, surrounding) {
+  if (surrounding === void 0) {
+    surrounding = [null, null];
+  }
+
+  // Parse expressions into `groups`.
+  const groups = [];
+
+  for (let i = 0; i < expression.length; i++) {
+    const output = buildGroup(expression[i], options);
+
+    if (output instanceof DocumentFragment) {
+      const children = output.children;
+      groups.push(...children);
+    } else {
+      groups.push(output);
+    }
+  } // If `expression` is a partial group, let the parent handle spacings
+  // to avoid processing groups multiple times.
+
+
+  if (!isRealGroup) {
+    return groups;
+  }
+
+  let glueOptions = options;
+
+  if (expression.length === 1) {
+    const node = expression[0];
+
+    if (node.type === "sizing") {
+      glueOptions = options.havingSize(node.size);
+    } else if (node.type === "styling") {
+      glueOptions = options.havingStyle(styleMap[node.style]);
+    }
+  } // Dummy spans for determining spacings between surrounding atoms.
+  // If `expression` has no atoms on the left or right, class "leftmost"
+  // or "rightmost", respectively, is used to indicate it.
+
+
+  const dummyPrev = makeSpan$1([surrounding[0] || "leftmost"], [], options);
+  const dummyNext = makeSpan$1([surrounding[1] || "rightmost"], [], options); // TODO: These code assumes that a node's math class is the first element
+  // of its `classes` array. A later cleanup should ensure this, for
+  // instance by changing the signature of `makeSpan`.
+  // Before determining what spaces to insert, perform bin cancellation.
+  // Binary operators change to ordinary symbols in some contexts.
+
+  const isRoot = isRealGroup === "root";
+  traverseNonSpaceNodes(groups, (node, prev) => {
+    const prevType = prev.classes[0];
+    const type = node.classes[0];
+
+    if (prevType === "mbin" && utils.contains(binRightCanceller, type)) {
+      prev.classes[0] = "mord";
+    } else if (type === "mbin" && utils.contains(binLeftCanceller, prevType)) {
+      node.classes[0] = "mord";
+    }
+  }, {
+    node: dummyPrev
+  }, dummyNext, isRoot);
+  traverseNonSpaceNodes(groups, (node, prev) => {
+    const prevType = getTypeOfDomTree(prev);
+    const type = getTypeOfDomTree(node); // 'mtight' indicates that the node is script or scriptscript style.
+
+    const space = prevType && type ? node.hasClass("mtight") ? tightSpacings[prevType][type] : spacings[prevType][type] : null;
+
+    if (space) {
+      // Insert glue (spacing) after the `prev`.
+      return buildCommon.makeGlue(space, glueOptions);
+    }
+  }, {
+    node: dummyPrev
+  }, dummyNext, isRoot);
+  return groups;
+}; // Depth-first traverse non-space `nodes`, calling `callback` with the current and
+// previous node as arguments, optionally returning a node to insert after the
+// previous node. `prev` is an object with the previous node and `insertAfter`
+// function to insert after it. `next` is a node that will be added to the right.
+// Used for bin cancellation and inserting spacings.
+
+const traverseNonSpaceNodes = function traverseNonSpaceNodes(nodes, callback, prev, next, isRoot) {
+  if (next) {
+    // temporarily append the right node, if exists
+    nodes.push(next);
+  }
+
+  let i = 0;
+
+  for (; i < nodes.length; i++) {
+    const node = nodes[i];
+    const partialGroup = checkPartialGroup(node);
+
+    if (partialGroup) {
+      // Recursive DFS
+      // $FlowFixMe: make nodes a $ReadOnlyArray by returning a new array
+      traverseNonSpaceNodes(partialGroup.children, callback, prev, null, isRoot);
+      continue;
+    } // Ignore explicit spaces (e.g., \;, \,) when determining what implicit
+    // spacing should go between atoms of different classes
+
+
+    const nonspace = !node.hasClass("mspace");
+
+    if (nonspace) {
+      const result = callback(node, prev.node);
+
+      if (result) {
+        if (prev.insertAfter) {
+          prev.insertAfter(result);
+        } else {
+          // insert at front
+          nodes.unshift(result);
+          i++;
+        }
+      }
+    }
+
+    if (nonspace) {
+      prev.node = node;
+    } else if (isRoot && node.hasClass("newline")) {
+      prev.node = makeSpan$1(["leftmost"]); // treat like beginning of line
+    }
+
+    prev.insertAfter = (index => n => {
+      nodes.splice(index + 1, 0, n);
+      i++;
+    })(i);
+  }
+
+  if (next) {
+    nodes.pop();
+  }
+}; // Check if given node is a partial group, i.e., does not affect spacing around.
+
+
+const checkPartialGroup = function checkPartialGroup(node) {
+  if (node instanceof DocumentFragment || node instanceof Anchor || node instanceof Span && node.hasClass("enclosing")) {
+    return node;
+  }
+
+  return null;
+}; // Return the outermost node of a domTree.
+
+
+const getOutermostNode = function getOutermostNode(node, side) {
+  const partialGroup = checkPartialGroup(node);
+
+  if (partialGroup) {
+    const children = partialGroup.children;
+
+    if (children.length) {
+      if (side === "right") {
+        return getOutermostNode(children[children.length - 1], "right");
+      } else if (side === "left") {
+        return getOutermostNode(children[0], "left");
+      }
+    }
+  }
+
+  return node;
+}; // Return math atom class (mclass) of a domTree.
+// If `side` is given, it will get the type of the outermost node at given side.
+
+
+const getTypeOfDomTree = function getTypeOfDomTree(node, side) {
+  if (!node) {
+    return null;
+  }
+
+  if (side) {
+    node = getOutermostNode(node, side);
+  } // This makes a lot of assumptions as to where the type of atom
+  // appears.  We should do a better job of enforcing this.
+
+
+  return DomEnum[node.classes[0]] || null;
+};
+const makeNullDelimiter = function makeNullDelimiter(options, classes) {
+  const moreClasses = ["nulldelimiter"].concat(options.baseSizingClasses());
+  return makeSpan$1(classes.concat(moreClasses));
+};
+/**
+ * buildGroup is the function that takes a group and calls the correct groupType
+ * function for it. It also handles the interaction of size and style changes
+ * between parents and children.
+ */
+
+const buildGroup = function buildGroup(group, options, baseOptions) {
+  if (!group) {
+    return makeSpan$1();
+  }
+
+  if (_htmlGroupBuilders[group.type]) {
+    // Call the groupBuilders function
+    let groupNode = _htmlGroupBuilders[group.type](group, options); // If the size changed between the parent and the current group, account
+    // for that size difference.
+
+    if (baseOptions && options.size !== baseOptions.size) {
+      groupNode = makeSpan$1(options.sizingClasses(baseOptions), [groupNode], options);
+      const multiplier = options.sizeMultiplier / baseOptions.sizeMultiplier;
+      groupNode.height *= multiplier;
+      groupNode.depth *= multiplier;
+    }
+
+    return groupNode;
+  } else {
+    throw new ParseError("Got group of unknown type: '" + group.type + "'");
+  }
+};
+/**
+ * Combine an array of HTML DOM nodes (e.g., the output of `buildExpression`)
+ * into an unbreakable HTML node of class .base, with proper struts to
+ * guarantee correct vertical extent.  `buildHTML` calls this repeatedly to
+ * make up the entire expression as a sequence of unbreakable units.
+ */
+
+function buildHTMLUnbreakable(children, options) {
+  // Compute height and depth of this chunk.
+  const body = makeSpan$1(["base"], children, options); // Add strut, which ensures that the top of the HTML element falls at
+  // the height of the expression, and the bottom of the HTML element
+  // falls at the depth of the expression.
+
+  const strut = makeSpan$1(["strut"]);
+  strut.style.height = body.height + body.depth + "em";
+  strut.style.verticalAlign = -body.depth + "em";
+  body.children.unshift(strut);
+  return body;
+}
+/**
+ * Take an entire parse tree, and build it into an appropriate set of HTML
+ * nodes.
+ */
+
+
+function buildHTML(tree, options) {
+  // Strip off outer tag wrapper for processing below.
+  let tag = null;
+
+  if (tree.length === 1 && tree[0].type === "tag") {
+    tag = tree[0].tag;
+    tree = tree[0].body;
+  } // Build the expression contained in the tree
+
+
+  const expression = buildExpression(tree, options, "root");
+  const children = []; // Create one base node for each chunk between potential line breaks.
+  // The TeXBook [p.173] says "A formula will be broken only after a
+  // relation symbol like $=$ or $<$ or $\rightarrow$, or after a binary
+  // operation symbol like $+$ or $-$ or $\times$, where the relation or
+  // binary operation is on the ``outer level'' of the formula (i.e., not
+  // enclosed in {...} and not part of an \over construction)."
+
+  let parts = [];
+
+  for (let i = 0; i < expression.length; i++) {
+    parts.push(expression[i]);
+
+    if (expression[i].hasClass("mbin") || expression[i].hasClass("mrel") || expression[i].hasClass("allowbreak")) {
+      // Put any post-operator glue on same line as operator.
+      // Watch for \nobreak along the way, and stop at \newline.
+      let nobreak = false;
+
+      while (i < expression.length - 1 && expression[i + 1].hasClass("mspace") && !expression[i + 1].hasClass("newline")) {
+        i++;
+        parts.push(expression[i]);
+
+        if (expression[i].hasClass("nobreak")) {
+          nobreak = true;
+        }
+      } // Don't allow break if \nobreak among the post-operator glue.
+
+
+      if (!nobreak) {
+        children.push(buildHTMLUnbreakable(parts, options));
+        parts = [];
+      }
+    } else if (expression[i].hasClass("newline")) {
+      // Write the line except the newline
+      parts.pop();
+
+      if (parts.length > 0) {
+        children.push(buildHTMLUnbreakable(parts, options));
+        parts = [];
+      } // Put the newline at the top level
+
+
+      children.push(expression[i]);
+    }
+  }
+
+  if (parts.length > 0) {
+    children.push(buildHTMLUnbreakable(parts, options));
+  } // Now, if there was a tag, build it too and append it as a final child.
+
+
+  let tagChild;
+
+  if (tag) {
+    tagChild = buildHTMLUnbreakable(buildExpression(tag, options, true));
+    tagChild.classes = ["tag"];
+    children.push(tagChild);
+  }
+
+  const htmlNode = makeSpan$1(["katex-html"], children);
+  htmlNode.setAttribute("aria-hidden", "true"); // Adjust the strut of the tag to be the maximum height of all children
+  // (the height of the enclosing htmlNode) for proper vertical alignment.
+
+  if (tagChild) {
+    const strut = tagChild.children[0];
+    strut.style.height = htmlNode.height + htmlNode.depth + "em";
+    strut.style.verticalAlign = -htmlNode.depth + "em";
+  }
+
+  return htmlNode;
+}
+
+/**
+ * These objects store data about MathML nodes. This is the MathML equivalent
+ * of the types in domTree.js. Since MathML handles its own rendering, and
+ * since we're mainly using MathML to improve accessibility, we don't manage
+ * any of the styling state that the plain DOM nodes do.
+ *
+ * The `toNode` and `toMarkup` functions work simlarly to how they do in
+ * domTree.js, creating namespaced DOM nodes and HTML text markup respectively.
+ */
+function newDocumentFragment(children) {
+  return new DocumentFragment(children);
+}
+/**
+ * This node represents a general purpose MathML node of any type. The
+ * constructor requires the type of node to create (for example, `"mo"` or
+ * `"mspace"`, corresponding to `<mo>` and `<mspace>` tags).
+ */
+
+class MathNode {
+  constructor(type, children) {
+    this.type = void 0;
+    this.attributes = void 0;
+    this.children = void 0;
+    this.type = type;
+    this.attributes = {};
+    this.children = children || [];
+  }
+  /**
+   * Sets an attribute on a MathML node. MathML depends on attributes to convey a
+   * semantic content, so this is used heavily.
+   */
+
+
+  setAttribute(name, value) {
+    this.attributes[name] = value;
+  }
+  /**
+   * Gets an attribute on a MathML node.
+   */
+
+
+  getAttribute(name) {
+    return this.attributes[name];
+  }
+  /**
+   * Converts the math node into a MathML-namespaced DOM element.
+   */
+
+
+  toNode() {
+    const node = document.createElementNS("http://www.w3.org/1998/Math/MathML", this.type);
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    for (let i = 0; i < this.children.length; i++) {
+      node.appendChild(this.children[i].toNode());
+    }
+
+    return node;
+  }
+  /**
+   * Converts the math node into an HTML markup string.
+   */
+
+
+  toMarkup() {
+    let markup = "<" + this.type; // Add the attributes
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += " " + attr + "=\"";
+        markup += utils.escape(this.attributes[attr]);
+        markup += "\"";
+      }
+    }
+
+    markup += ">";
+
+    for (let i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    markup += "</" + this.type + ">";
+    return markup;
+  }
+  /**
+   * Converts the math node into a string, similar to innerText, but escaped.
+   */
+
+
+  toText() {
+    return this.children.map(child => child.toText()).join("");
+  }
+
+}
+/**
+ * This node represents a piece of text.
+ */
+
+class TextNode {
+  constructor(text) {
+    this.text = void 0;
+    this.text = text;
+  }
+  /**
+   * Converts the text node into a DOM text node.
+   */
+
+
+  toNode() {
+    return document.createTextNode(this.text);
+  }
+  /**
+   * Converts the text node into escaped HTML markup
+   * (representing the text itself).
+   */
+
+
+  toMarkup() {
+    return utils.escape(this.toText());
+  }
+  /**
+   * Converts the text node into a string
+   * (representing the text iteself).
+   */
+
+
+  toText() {
+    return this.text;
+  }
+
+}
+/**
+ * This node represents a space, but may render as <mspace.../> or as text,
+ * depending on the width.
+ */
+
+class SpaceNode {
+  /**
+   * Create a Space node with width given in CSS ems.
+   */
+  constructor(width) {
+    this.width = void 0;
+    this.character = void 0;
+    this.width = width; // See https://www.w3.org/TR/2000/WD-MathML2-20000328/chapter6.html
+    // for a table of space-like characters.  We use Unicode
+    // representations instead of &LongNames; as it's not clear how to
+    // make the latter via document.createTextNode.
+
+    if (width >= 0.05555 && width <= 0.05556) {
+      this.character = "\u200a"; // &VeryThinSpace;
+    } else if (width >= 0.1666 && width <= 0.1667) {
+      this.character = "\u2009"; // &ThinSpace;
+    } else if (width >= 0.2222 && width <= 0.2223) {
+      this.character = "\u2005"; // &MediumSpace;
+    } else if (width >= 0.2777 && width <= 0.2778) {
+      this.character = "\u2005\u200a"; // &ThickSpace;
+    } else if (width >= -0.05556 && width <= -0.05555) {
+      this.character = "\u200a\u2063"; // &NegativeVeryThinSpace;
+    } else if (width >= -0.1667 && width <= -0.1666) {
+      this.character = "\u2009\u2063"; // &NegativeThinSpace;
+    } else if (width >= -0.2223 && width <= -0.2222) {
+      this.character = "\u205f\u2063"; // &NegativeMediumSpace;
+    } else if (width >= -0.2778 && width <= -0.2777) {
+      this.character = "\u2005\u2063"; // &NegativeThickSpace;
+    } else {
+      this.character = null;
+    }
+  }
+  /**
+   * Converts the math node into a MathML-namespaced DOM element.
+   */
+
+
+  toNode() {
+    if (this.character) {
+      return document.createTextNode(this.character);
+    } else {
+      const node = document.createElementNS("http://www.w3.org/1998/Math/MathML", "mspace");
+      node.setAttribute("width", this.width + "em");
+      return node;
+    }
+  }
+  /**
+   * Converts the math node into an HTML markup string.
+   */
+
+
+  toMarkup() {
+    if (this.character) {
+      return `<mtext>${this.character}</mtext>`;
+    } else {
+      return `<mspace width="${this.width}em"/>`;
+    }
+  }
+  /**
+   * Converts the math node into a string, similar to innerText.
+   */
+
+
+  toText() {
+    if (this.character) {
+      return this.character;
+    } else {
+      return " ";
+    }
+  }
+
+}
+
+var mathMLTree = {
+  MathNode,
+  TextNode,
+  SpaceNode,
+  newDocumentFragment
+};
+
+/**
+ * This file converts a parse tree into a cooresponding MathML tree. The main
+ * entry point is the `buildMathML` function, which takes a parse tree from the
+ * parser.
+ */
+
+/**
+ * Takes a symbol and converts it into a MathML text node after performing
+ * optional replacement from symbols.js.
+ */
+const makeText = function makeText(text, mode, options) {
+  if (symbols[mode][text] && symbols[mode][text].replace && text.charCodeAt(0) !== 0xD835 && !(ligatures.hasOwnProperty(text) && options && (options.fontFamily && options.fontFamily.substr(4, 2) === "tt" || options.font && options.font.substr(4, 2) === "tt"))) {
+    text = symbols[mode][text].replace;
+  }
+
+  return new mathMLTree.TextNode(text);
+};
+/**
+ * Wrap the given array of nodes in an <mrow> node if needed, i.e.,
+ * unless the array has length 1.  Always returns a single node.
+ */
+
+const makeRow = function makeRow(body) {
+  if (body.length === 1) {
+    return body[0];
+  } else {
+    return new mathMLTree.MathNode("mrow", body);
+  }
+};
+/**
+ * Returns the math variant as a string or null if none is required.
+ */
+
+const getVariant = function getVariant(group, options) {
+  // Handle \text... font specifiers as best we can.
+  // MathML has a limited list of allowable mathvariant specifiers; see
+  // https://www.w3.org/TR/MathML3/chapter3.html#presm.commatt
+  if (options.fontFamily === "texttt") {
+    return "monospace";
+  } else if (options.fontFamily === "textsf") {
+    if (options.fontShape === "textit" && options.fontWeight === "textbf") {
+      return "sans-serif-bold-italic";
+    } else if (options.fontShape === "textit") {
+      return "sans-serif-italic";
+    } else if (options.fontWeight === "textbf") {
+      return "bold-sans-serif";
+    } else {
+      return "sans-serif";
+    }
+  } else if (options.fontShape === "textit" && options.fontWeight === "textbf") {
+    return "bold-italic";
+  } else if (options.fontShape === "textit") {
+    return "italic";
+  } else if (options.fontWeight === "textbf") {
+    return "bold";
+  }
+
+  const font = options.font;
+
+  if (!font || font === "mathnormal") {
+    return null;
+  }
+
+  const mode = group.mode;
+
+  if (font === "mathit") {
+    return "italic";
+  } else if (font === "boldsymbol") {
+    return group.type === "textord" ? "bold" : "bold-italic";
+  } else if (font === "mathbf") {
+    return "bold";
+  } else if (font === "mathbb") {
+    return "double-struck";
+  } else if (font === "mathfrak") {
+    return "fraktur";
+  } else if (font === "mathscr" || font === "mathcal") {
+    // MathML makes no distinction between script and caligrahpic
+    return "script";
+  } else if (font === "mathsf") {
+    return "sans-serif";
+  } else if (font === "mathtt") {
+    return "monospace";
+  }
+
+  let text = group.text;
+
+  if (utils.contains(["\\imath", "\\jmath"], text)) {
+    return null;
+  }
+
+  if (symbols[mode][text] && symbols[mode][text].replace) {
+    text = symbols[mode][text].replace;
+  }
+
+  const fontName = buildCommon.fontMap[font].fontName;
+
+  if (getCharacterMetrics(text, fontName, mode)) {
+    return buildCommon.fontMap[font].variant;
+  }
+
+  return null;
+};
+/**
+ * Takes a list of nodes, builds them, and returns a list of the generated
+ * MathML nodes.  Also combine consecutive <mtext> outputs into a single
+ * <mtext> tag.
+ */
+
+const buildExpression$1 = function buildExpression(expression, options, isOrdgroup) {
+  if (expression.length === 1) {
+    const group = buildGroup$1(expression[0], options);
+
+    if (isOrdgroup && group instanceof MathNode && group.type === "mo") {
+      // When TeX writers want to suppress spacing on an operator,
+      // they often put the operator by itself inside braces.
+      group.setAttribute("lspace", "0em");
+      group.setAttribute("rspace", "0em");
+    }
+
+    return [group];
+  }
+
+  const groups = [];
+  let lastGroup;
+
+  for (let i = 0; i < expression.length; i++) {
+    const group = buildGroup$1(expression[i], options);
+
+    if (group instanceof MathNode && lastGroup instanceof MathNode) {
+      // Concatenate adjacent <mtext>s
+      if (group.type === 'mtext' && lastGroup.type === 'mtext' && group.getAttribute('mathvariant') === lastGroup.getAttribute('mathvariant')) {
+        lastGroup.children.push(...group.children);
+        continue; // Concatenate adjacent <mn>s
+      } else if (group.type === 'mn' && lastGroup.type === 'mn') {
+        lastGroup.children.push(...group.children);
+        continue; // Concatenate <mn>...</mn> followed by <mi>.</mi>
+      } else if (group.type === 'mi' && group.children.length === 1 && lastGroup.type === 'mn') {
+        const child = group.children[0];
+
+        if (child instanceof TextNode && child.text === '.') {
+          lastGroup.children.push(...group.children);
+          continue;
+        }
+      } else if (lastGroup.type === 'mi' && lastGroup.children.length === 1) {
+        const lastChild = lastGroup.children[0];
+
+        if (lastChild instanceof TextNode && lastChild.text === '\u0338' && (group.type === 'mo' || group.type === 'mi' || group.type === 'mn')) {
+          const child = group.children[0];
+
+          if (child instanceof TextNode && child.text.length > 0) {
+            // Overlay with combining character long solidus
+            child.text = child.text.slice(0, 1) + "\u0338" + child.text.slice(1);
+            groups.pop();
+          }
+        }
+      }
+    }
+
+    groups.push(group);
+    lastGroup = group;
+  }
+
+  return groups;
+};
+/**
+ * Equivalent to buildExpression, but wraps the elements in an <mrow>
+ * if there's more than one.  Returns a single node instead of an array.
+ */
+
+const buildExpressionRow = function buildExpressionRow(expression, options, isOrdgroup) {
+  return makeRow(buildExpression$1(expression, options, isOrdgroup));
+};
+/**
+ * Takes a group from the parser and calls the appropriate groupBuilders function
+ * on it to produce a MathML node.
+ */
+
+const buildGroup$1 = function buildGroup(group, options) {
+  if (!group) {
+    return new mathMLTree.MathNode("mrow");
+  }
+
+  if (_mathmlGroupBuilders[group.type]) {
+    // Call the groupBuilders function
+    const result = _mathmlGroupBuilders[group.type](group, options);
+    return result;
+  } else {
+    throw new ParseError("Got group of unknown type: '" + group.type + "'");
+  }
+};
+/**
+ * Takes a full parse tree and settings and builds a MathML representation of
+ * it. In particular, we put the elements from building the parse tree into a
+ * <semantics> tag so we can also include that TeX source as an annotation.
+ *
+ * Note that we actually return a domTree element with a `<math>` inside it so
+ * we can do appropriate styling.
+ */
+
+function buildMathML(tree, texExpression, options, isDisplayMode, forMathmlOnly) {
+  const expression = buildExpression$1(tree, options); // Wrap up the expression in an mrow so it is presented in the semantics
+  // tag correctly, unless it's a single <mrow> or <mtable>.
+
+  let wrapper;
+
+  if (expression.length === 1 && expression[0] instanceof MathNode && utils.contains(["mrow", "mtable"], expression[0].type)) {
+    wrapper = expression[0];
+  } else {
+    wrapper = new mathMLTree.MathNode("mrow", expression);
+  } // Build a TeX annotation of the source
+
+
+  const annotation = new mathMLTree.MathNode("annotation", [new mathMLTree.TextNode(texExpression)]);
+  annotation.setAttribute("encoding", "application/x-tex");
+  const semantics = new mathMLTree.MathNode("semantics", [wrapper, annotation]);
+  const math = new mathMLTree.MathNode("math", [semantics]);
+  math.setAttribute("xmlns", "http://www.w3.org/1998/Math/MathML");
+
+  if (isDisplayMode) {
+    math.setAttribute("display", "block");
+  } // You can't style <math> nodes, so we wrap the node in a span.
+  // NOTE: The span class is not typed to have <math> nodes as children, and
+  // we don't want to make the children type more generic since the children
+  // of span are expected to have more fields in `buildHtml` contexts.
+
+
+  const wrapperClass = forMathmlOnly ? "katex" : "katex-mathml"; // $FlowFixMe
+
+  return buildCommon.makeSpan([wrapperClass], [math]);
+}
+
+const optionsFromSettings = function optionsFromSettings(settings) {
+  return new Options({
+    style: settings.displayMode ? Style$1.DISPLAY : Style$1.TEXT,
+    maxSize: settings.maxSize,
+    minRuleThickness: settings.minRuleThickness
+  });
+};
+
+const displayWrap = function displayWrap(node, settings) {
+  if (settings.displayMode) {
+    const classes = ["katex-display"];
+
+    if (settings.leqno) {
+      classes.push("leqno");
+    }
+
+    if (settings.fleqn) {
+      classes.push("fleqn");
+    }
+
+    node = buildCommon.makeSpan(classes, [node]);
+  }
+
+  return node;
+};
+
+const buildTree = function buildTree(tree, expression, settings) {
+  const options = optionsFromSettings(settings);
+  let katexNode;
+
+  if (settings.output === "mathml") {
+    return buildMathML(tree, expression, options, settings.displayMode, true);
+  } else if (settings.output === "html") {
+    const htmlNode = buildHTML(tree, options);
+    katexNode = buildCommon.makeSpan(["katex"], [htmlNode]);
+  } else {
+    const mathMLNode = buildMathML(tree, expression, options, settings.displayMode, false);
+    const htmlNode = buildHTML(tree, options);
+    katexNode = buildCommon.makeSpan(["katex"], [mathMLNode, htmlNode]);
+  }
+
+  return displayWrap(katexNode, settings);
+};
+const buildHTMLTree = function buildHTMLTree(tree, expression, settings) {
+  const options = optionsFromSettings(settings);
+  const htmlNode = buildHTML(tree, options);
+  const katexNode = buildCommon.makeSpan(["katex"], [htmlNode]);
+  return displayWrap(katexNode, settings);
+};
+
+/**
+ * This file provides support to buildMathML.js and buildHTML.js
+ * for stretchy wide elements rendered from SVG files
+ * and other CSS trickery.
+ */
+const stretchyCodePoint = {
+  widehat: "^",
+  widecheck: "ˇ",
+  widetilde: "~",
+  utilde: "~",
+  overleftarrow: "\u2190",
+  underleftarrow: "\u2190",
+  xleftarrow: "\u2190",
+  overrightarrow: "\u2192",
+  underrightarrow: "\u2192",
+  xrightarrow: "\u2192",
+  underbrace: "\u23df",
+  overbrace: "\u23de",
+  overgroup: "\u23e0",
+  undergroup: "\u23e1",
+  overleftrightarrow: "\u2194",
+  underleftrightarrow: "\u2194",
+  xleftrightarrow: "\u2194",
+  Overrightarrow: "\u21d2",
+  xRightarrow: "\u21d2",
+  overleftharpoon: "\u21bc",
+  xleftharpoonup: "\u21bc",
+  overrightharpoon: "\u21c0",
+  xrightharpoonup: "\u21c0",
+  xLeftarrow: "\u21d0",
+  xLeftrightarrow: "\u21d4",
+  xhookleftarrow: "\u21a9",
+  xhookrightarrow: "\u21aa",
+  xmapsto: "\u21a6",
+  xrightharpoondown: "\u21c1",
+  xleftharpoondown: "\u21bd",
+  xrightleftharpoons: "\u21cc",
+  xleftrightharpoons: "\u21cb",
+  xtwoheadleftarrow: "\u219e",
+  xtwoheadrightarrow: "\u21a0",
+  xlongequal: "=",
+  xtofrom: "\u21c4",
+  xrightleftarrows: "\u21c4",
+  xrightequilibrium: "\u21cc",
+  // Not a perfect match.
+  xleftequilibrium: "\u21cb" // None better available.
+
+};
+
+const mathMLnode = function mathMLnode(label) {
+  const node = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(stretchyCodePoint[label.substr(1)])]);
+  node.setAttribute("stretchy", "true");
+  return node;
+}; // Many of the KaTeX SVG images have been adapted from glyphs in KaTeX fonts.
+// Copyright (c) 2009-2010, Design Science, Inc. (<www.mathjax.org>)
+// Copyright (c) 2014-2017 Khan Academy (<www.khanacademy.org>)
+// Licensed under the SIL Open Font License, Version 1.1.
+// See \nhttp://scripts.sil.org/OFL
+// Very Long SVGs
+//    Many of the KaTeX stretchy wide elements use a long SVG image and an
+//    overflow: hidden tactic to achieve a stretchy image while avoiding
+//    distortion of arrowheads or brace corners.
+//    The SVG typically contains a very long (400 em) arrow.
+//    The SVG is in a container span that has overflow: hidden, so the span
+//    acts like a window that exposes only part of the  SVG.
+//    The SVG always has a longer, thinner aspect ratio than the container span.
+//    After the SVG fills 100% of the height of the container span,
+//    there is a long arrow shaft left over. That left-over shaft is not shown.
+//    Instead, it is sliced off because the span's CSS has overflow: hidden.
+//    Thus, the reader sees an arrow that matches the subject matter width
+//    without distortion.
+//    Some functions, such as \cancel, need to vary their aspect ratio. These
+//    functions do not get the overflow SVG treatment.
+// Second Brush Stroke
+//    Low resolution monitors struggle to display images in fine detail.
+//    So browsers apply anti-aliasing. A long straight arrow shaft therefore
+//    will sometimes appear as if it has a blurred edge.
+//    To mitigate this, these SVG files contain a second "brush-stroke" on the
+//    arrow shafts. That is, a second long thin rectangular SVG path has been
+//    written directly on top of each arrow shaft. This reinforcement causes
+//    some of the screen pixels to display as black instead of the anti-aliased
+//    gray pixel that a  single path would generate. So we get arrow shafts
+//    whose edges appear to be sharper.
+// In the katexImagesData object just below, the dimensions all
+// correspond to path geometry inside the relevant SVG.
+// For example, \overrightarrow uses the same arrowhead as glyph U+2192
+// from the KaTeX Main font. The scaling factor is 1000.
+// That is, inside the font, that arrowhead is 522 units tall, which
+// corresponds to 0.522 em inside the document.
+
+
+const katexImagesData = {
+  //   path(s), minWidth, height, align
+  overrightarrow: [["rightarrow"], 0.888, 522, "xMaxYMin"],
+  overleftarrow: [["leftarrow"], 0.888, 522, "xMinYMin"],
+  underrightarrow: [["rightarrow"], 0.888, 522, "xMaxYMin"],
+  underleftarrow: [["leftarrow"], 0.888, 522, "xMinYMin"],
+  xrightarrow: [["rightarrow"], 1.469, 522, "xMaxYMin"],
+  xleftarrow: [["leftarrow"], 1.469, 522, "xMinYMin"],
+  Overrightarrow: [["doublerightarrow"], 0.888, 560, "xMaxYMin"],
+  xRightarrow: [["doublerightarrow"], 1.526, 560, "xMaxYMin"],
+  xLeftarrow: [["doubleleftarrow"], 1.526, 560, "xMinYMin"],
+  overleftharpoon: [["leftharpoon"], 0.888, 522, "xMinYMin"],
+  xleftharpoonup: [["leftharpoon"], 0.888, 522, "xMinYMin"],
+  xleftharpoondown: [["leftharpoondown"], 0.888, 522, "xMinYMin"],
+  overrightharpoon: [["rightharpoon"], 0.888, 522, "xMaxYMin"],
+  xrightharpoonup: [["rightharpoon"], 0.888, 522, "xMaxYMin"],
+  xrightharpoondown: [["rightharpoondown"], 0.888, 522, "xMaxYMin"],
+  xlongequal: [["longequal"], 0.888, 334, "xMinYMin"],
+  xtwoheadleftarrow: [["twoheadleftarrow"], 0.888, 334, "xMinYMin"],
+  xtwoheadrightarrow: [["twoheadrightarrow"], 0.888, 334, "xMaxYMin"],
+  overleftrightarrow: [["leftarrow", "rightarrow"], 0.888, 522],
+  overbrace: [["leftbrace", "midbrace", "rightbrace"], 1.6, 548],
+  underbrace: [["leftbraceunder", "midbraceunder", "rightbraceunder"], 1.6, 548],
+  underleftrightarrow: [["leftarrow", "rightarrow"], 0.888, 522],
+  xleftrightarrow: [["leftarrow", "rightarrow"], 1.75, 522],
+  xLeftrightarrow: [["doubleleftarrow", "doublerightarrow"], 1.75, 560],
+  xrightleftharpoons: [["leftharpoondownplus", "rightharpoonplus"], 1.75, 716],
+  xleftrightharpoons: [["leftharpoonplus", "rightharpoondownplus"], 1.75, 716],
+  xhookleftarrow: [["leftarrow", "righthook"], 1.08, 522],
+  xhookrightarrow: [["lefthook", "rightarrow"], 1.08, 522],
+  overlinesegment: [["leftlinesegment", "rightlinesegment"], 0.888, 522],
+  underlinesegment: [["leftlinesegment", "rightlinesegment"], 0.888, 522],
+  overgroup: [["leftgroup", "rightgroup"], 0.888, 342],
+  undergroup: [["leftgroupunder", "rightgroupunder"], 0.888, 342],
+  xmapsto: [["leftmapsto", "rightarrow"], 1.5, 522],
+  xtofrom: [["leftToFrom", "rightToFrom"], 1.75, 528],
+  // The next three arrows are from the mhchem package.
+  // In mhchem.sty, min-length is 2.0em. But these arrows might appear in the
+  // document as \xrightarrow or \xrightleftharpoons. Those have
+  // min-length = 1.75em, so we set min-length on these next three to match.
+  xrightleftarrows: [["baraboveleftarrow", "rightarrowabovebar"], 1.75, 901],
+  xrightequilibrium: [["baraboveshortleftharpoon", "rightharpoonaboveshortbar"], 1.75, 716],
+  xleftequilibrium: [["shortbaraboveleftharpoon", "shortrightharpoonabovebar"], 1.75, 716]
+};
+
+const groupLength = function groupLength(arg) {
+  if (arg.type === "ordgroup") {
+    return arg.body.length;
+  } else {
+    return 1;
+  }
+};
+
+const svgSpan = function svgSpan(group, options) {
+  // Create a span with inline SVG for the element.
+  function buildSvgSpan_() {
+    let viewBoxWidth = 400000; // default
+
+    const label = group.label.substr(1);
+
+    if (utils.contains(["widehat", "widecheck", "widetilde", "utilde"], label)) {
+      // Each type in the `if` statement corresponds to one of the ParseNode
+      // types below. This narrowing is required to access `grp.base`.
+      const grp = group; // There are four SVG images available for each function.
+      // Choose a taller image when there are more characters.
+
+      const numChars = groupLength(grp.base);
+      let viewBoxHeight;
+      let pathName;
+      let height;
+
+      if (numChars > 5) {
+        if (label === "widehat" || label === "widecheck") {
+          viewBoxHeight = 420;
+          viewBoxWidth = 2364;
+          height = 0.42;
+          pathName = label + "4";
+        } else {
+          viewBoxHeight = 312;
+          viewBoxWidth = 2340;
+          height = 0.34;
+          pathName = "tilde4";
+        }
+      } else {
+        const imgIndex = [1, 1, 2, 2, 3, 3][numChars];
+
+        if (label === "widehat" || label === "widecheck") {
+          viewBoxWidth = [0, 1062, 2364, 2364, 2364][imgIndex];
+          viewBoxHeight = [0, 239, 300, 360, 420][imgIndex];
+          height = [0, 0.24, 0.3, 0.3, 0.36, 0.42][imgIndex];
+          pathName = label + imgIndex;
+        } else {
+          viewBoxWidth = [0, 600, 1033, 2339, 2340][imgIndex];
+          viewBoxHeight = [0, 260, 286, 306, 312][imgIndex];
+          height = [0, 0.26, 0.286, 0.3, 0.306, 0.34][imgIndex];
+          pathName = "tilde" + imgIndex;
+        }
+      }
+
+      const path = new PathNode(pathName);
+      const svgNode = new SvgNode([path], {
+        "width": "100%",
+        "height": height + "em",
+        "viewBox": `0 0 ${viewBoxWidth} ${viewBoxHeight}`,
+        "preserveAspectRatio": "none"
+      });
+      return {
+        span: buildCommon.makeSvgSpan([], [svgNode], options),
+        minWidth: 0,
+        height
+      };
+    } else {
+      const spans = [];
+      const data = katexImagesData[label];
+      const paths = data[0],
+            minWidth = data[1],
+            viewBoxHeight = data[2];
+      const height = viewBoxHeight / 1000;
+      const numSvgChildren = paths.length;
+      let widthClasses;
+      let aligns;
+
+      if (numSvgChildren === 1) {
+        // $FlowFixMe: All these cases must be of the 4-tuple type.
+        const align1 = data[3];
+        widthClasses = ["hide-tail"];
+        aligns = [align1];
+      } else if (numSvgChildren === 2) {
+        widthClasses = ["halfarrow-left", "halfarrow-right"];
+        aligns = ["xMinYMin", "xMaxYMin"];
+      } else if (numSvgChildren === 3) {
+        widthClasses = ["brace-left", "brace-center", "brace-right"];
+        aligns = ["xMinYMin", "xMidYMin", "xMaxYMin"];
+      } else {
+        throw new Error(`Correct katexImagesData or update code here to support
+                    ${numSvgChildren} children.`);
+      }
+
+      for (let i = 0; i < numSvgChildren; i++) {
+        const path = new PathNode(paths[i]);
+        const svgNode = new SvgNode([path], {
+          "width": "400em",
+          "height": height + "em",
+          "viewBox": `0 0 ${viewBoxWidth} ${viewBoxHeight}`,
+          "preserveAspectRatio": aligns[i] + " slice"
+        });
+        const span = buildCommon.makeSvgSpan([widthClasses[i]], [svgNode], options);
+
+        if (numSvgChildren === 1) {
+          return {
+            span,
+            minWidth,
+            height
+          };
+        } else {
+          span.style.height = height + "em";
+          spans.push(span);
+        }
+      }
+
+      return {
+        span: buildCommon.makeSpan(["stretchy"], spans, options),
+        minWidth,
+        height
+      };
+    }
+  } // buildSvgSpan_()
+
+
+  const _buildSvgSpan_ = buildSvgSpan_(),
+        span = _buildSvgSpan_.span,
+        minWidth = _buildSvgSpan_.minWidth,
+        height = _buildSvgSpan_.height; // Note that we are returning span.depth = 0.
+  // Any adjustments relative to the baseline must be done in buildHTML.
+
+
+  span.height = height;
+  span.style.height = height + "em";
+
+  if (minWidth > 0) {
+    span.style.minWidth = minWidth + "em";
+  }
+
+  return span;
+};
+
+const encloseSpan = function encloseSpan(inner, label, pad, options) {
+  // Return an image span for \cancel, \bcancel, \xcancel, or \fbox
+  let img;
+  const totalHeight = inner.height + inner.depth + 2 * pad;
+
+  if (/fbox|color/.test(label)) {
+    img = buildCommon.makeSpan(["stretchy", label], [], options);
+
+    if (label === "fbox") {
+      const color = options.color && options.getColor();
+
+      if (color) {
+        img.style.borderColor = color;
+      }
+    }
+  } else {
+    // \cancel, \bcancel, or \xcancel
+    // Since \cancel's SVG is inline and it omits the viewBox attribute,
+    // its stroke-width will not vary with span area.
+    const lines = [];
+
+    if (/^[bx]cancel$/.test(label)) {
+      lines.push(new LineNode({
+        "x1": "0",
+        "y1": "0",
+        "x2": "100%",
+        "y2": "100%",
+        "stroke-width": "0.046em"
+      }));
+    }
+
+    if (/^x?cancel$/.test(label)) {
+      lines.push(new LineNode({
+        "x1": "0",
+        "y1": "100%",
+        "x2": "100%",
+        "y2": "0",
+        "stroke-width": "0.046em"
+      }));
+    }
+
+    const svgNode = new SvgNode(lines, {
+      "width": "100%",
+      "height": totalHeight + "em"
+    });
+    img = buildCommon.makeSvgSpan([], [svgNode], options);
+  }
+
+  img.height = totalHeight;
+  img.style.height = totalHeight + "em";
+  return img;
+};
+
+var stretchy = {
+  encloseSpan,
+  mathMLnode,
+  svgSpan
+};
+
+/**
+ * Asserts that the node is of the given type and returns it with stricter
+ * typing. Throws if the node's type does not match.
+ */
+function assertNodeType(node, type) {
+  if (!node || node.type !== type) {
+    throw new Error(`Expected node of type ${type}, but got ` + (node ? `node of type ${node.type}` : String(node)));
+  }
+
+  return node;
+}
+/**
+ * Returns the node more strictly typed iff it is of the given type. Otherwise,
+ * returns null.
+ */
+
+function assertSymbolNodeType(node) {
+  const typedNode = checkSymbolNodeType(node);
+
+  if (!typedNode) {
+    throw new Error(`Expected node of symbol group type, but got ` + (node ? `node of type ${node.type}` : String(node)));
+  }
+
+  return typedNode;
+}
+/**
+ * Returns the node more strictly typed iff it is of the given type. Otherwise,
+ * returns null.
+ */
+
+function checkSymbolNodeType(node) {
+  if (node && (node.type === "atom" || NON_ATOMS.hasOwnProperty(node.type))) {
+    // $FlowFixMe
+    return node;
+  }
+
+  return null;
+}
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only "accent", but
+const htmlBuilder = (grp, options) => {
+  // Accents are handled in the TeXbook pg. 443, rule 12.
+  let base;
+  let group;
+  let supSubGroup;
+
+  if (grp && grp.type === "supsub") {
+    // If our base is a character box, and we have superscripts and
+    // subscripts, the supsub will defer to us. In particular, we want
+    // to attach the superscripts and subscripts to the inner body (so
+    // that the position of the superscripts and subscripts won't be
+    // affected by the height of the accent). We accomplish this by
+    // sticking the base of the accent into the base of the supsub, and
+    // rendering that, while keeping track of where the accent is.
+    // The real accent group is the base of the supsub group
+    group = assertNodeType(grp.base, "accent"); // The character box is the base of the accent group
+
+    base = group.base; // Stick the character box into the base of the supsub group
+
+    grp.base = base; // Rerender the supsub group with its new base, and store that
+    // result.
+
+    supSubGroup = assertSpan(buildGroup(grp, options)); // reset original base
+
+    grp.base = group;
+  } else {
+    group = assertNodeType(grp, "accent");
+    base = group.base;
+  } // Build the base group
+
+
+  const body = buildGroup(base, options.havingCrampedStyle()); // Does the accent need to shift for the skew of a character?
+
+  const mustShift = group.isShifty && utils.isCharacterBox(base); // Calculate the skew of the accent. This is based on the line "If the
+  // nucleus is not a single character, let s = 0; otherwise set s to the
+  // kern amount for the nucleus followed by the \skewchar of its font."
+  // Note that our skew metrics are just the kern between each character
+  // and the skewchar.
+
+  let skew = 0;
+
+  if (mustShift) {
+    // If the base is a character box, then we want the skew of the
+    // innermost character. To do that, we find the innermost character:
+    const baseChar = utils.getBaseElem(base); // Then, we render its group to get the symbol inside it
+
+    const baseGroup = buildGroup(baseChar, options.havingCrampedStyle()); // Finally, we pull the skew off of the symbol.
+
+    skew = assertSymbolDomNode(baseGroup).skew; // Note that we now throw away baseGroup, because the layers we
+    // removed with getBaseElem might contain things like \color which
+    // we can't get rid of.
+    // TODO(emily): Find a better way to get the skew
+  } // calculate the amount of space between the body and the accent
+
+
+  let clearance = Math.min(body.height, options.fontMetrics().xHeight); // Build the accent
+
+  let accentBody;
+
+  if (!group.isStretchy) {
+    let accent;
+    let width;
+
+    if (group.label === "\\vec") {
+      // Before version 0.9, \vec used the combining font glyph U+20D7.
+      // But browsers, especially Safari, are not consistent in how they
+      // render combining characters when not preceded by a character.
+      // So now we use an SVG.
+      // If Safari reforms, we should consider reverting to the glyph.
+      accent = buildCommon.staticSvg("vec", options);
+      width = buildCommon.svgData.vec[1];
+    } else {
+      accent = buildCommon.makeOrd({
+        mode: group.mode,
+        text: group.label
+      }, options, "textord");
+      accent = assertSymbolDomNode(accent); // Remove the italic correction of the accent, because it only serves to
+      // shift the accent over to a place we don't want.
+
+      accent.italic = 0;
+      width = accent.width;
+    }
+
+    accentBody = buildCommon.makeSpan(["accent-body"], [accent]); // "Full" accents expand the width of the resulting symbol to be
+    // at least the width of the accent, and overlap directly onto the
+    // character without any vertical offset.
+
+    const accentFull = group.label === "\\textcircled";
+
+    if (accentFull) {
+      accentBody.classes.push('accent-full');
+      clearance = body.height;
+    } // Shift the accent over by the skew.
+
+
+    let left = skew; // CSS defines `.katex .accent .accent-body:not(.accent-full) { width: 0 }`
+    // so that the accent doesn't contribute to the bounding box.
+    // We need to shift the character by its width (effectively half
+    // its width) to compensate.
+
+    if (!accentFull) {
+      left -= width / 2;
+    }
+
+    accentBody.style.left = left + "em"; // \textcircled uses the \bigcirc glyph, so it needs some
+    // vertical adjustment to match LaTeX.
+
+    if (group.label === "\\textcircled") {
+      accentBody.style.top = ".2em";
+    }
+
+    accentBody = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "kern",
+        size: -clearance
+      }, {
+        type: "elem",
+        elem: accentBody
+      }]
+    }, options);
+  } else {
+    accentBody = stretchy.svgSpan(group, options);
+    accentBody = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "elem",
+        elem: accentBody,
+        wrapperClasses: ["svg-align"],
+        wrapperStyle: skew > 0 ? {
+          width: `calc(100% - ${2 * skew}em)`,
+          marginLeft: `${2 * skew}em`
+        } : undefined
+      }]
+    }, options);
+  }
+
+  const accentWrap = buildCommon.makeSpan(["mord", "accent"], [accentBody], options);
+
+  if (supSubGroup) {
+    // Here, we replace the "base" child of the supsub with our newly
+    // generated accent.
+    supSubGroup.children[0] = accentWrap; // Since we don't rerun the height calculation after replacing the
+    // accent, we manually recalculate height.
+
+    supSubGroup.height = Math.max(accentWrap.height, supSubGroup.height); // Accents should always be ords, even when their innards are not.
+
+    supSubGroup.classes[0] = "mord";
+    return supSubGroup;
+  } else {
+    return accentWrap;
+  }
+};
+
+const mathmlBuilder = (group, options) => {
+  const accentNode = group.isStretchy ? stretchy.mathMLnode(group.label) : new mathMLTree.MathNode("mo", [makeText(group.label, group.mode)]);
+  const node = new mathMLTree.MathNode("mover", [buildGroup$1(group.base, options), accentNode]);
+  node.setAttribute("accent", "true");
+  return node;
+};
+
+const NON_STRETCHY_ACCENT_REGEX = new RegExp(["\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve", "\\check", "\\hat", "\\vec", "\\dot", "\\mathring"].map(accent => `\\${accent}`).join("|")); // Accents
+
+defineFunction({
+  type: "accent",
+  names: ["\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve", "\\check", "\\hat", "\\vec", "\\dot", "\\mathring", "\\widecheck", "\\widehat", "\\widetilde", "\\overrightarrow", "\\overleftarrow", "\\Overrightarrow", "\\overleftrightarrow", "\\overgroup", "\\overlinesegment", "\\overleftharpoon", "\\overrightharpoon"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    const base = args[0];
+    const isStretchy = !NON_STRETCHY_ACCENT_REGEX.test(context.funcName);
+    const isShifty = !isStretchy || context.funcName === "\\widehat" || context.funcName === "\\widetilde" || context.funcName === "\\widecheck";
+    return {
+      type: "accent",
+      mode: context.parser.mode,
+      label: context.funcName,
+      isStretchy: isStretchy,
+      isShifty: isShifty,
+      base: base
+    };
+  },
+  htmlBuilder,
+  mathmlBuilder
+}); // Text-mode accents
+
+defineFunction({
+  type: "accent",
+  names: ["\\'", "\\`", "\\^", "\\~", "\\=", "\\u", "\\.", '\\"', "\\r", "\\H", "\\v", "\\textcircled"],
+  props: {
+    numArgs: 1,
+    allowedInText: true,
+    allowedInMath: false
+  },
+  handler: (context, args) => {
+    const base = args[0];
+    return {
+      type: "accent",
+      mode: context.parser.mode,
+      label: context.funcName,
+      isStretchy: false,
+      isShifty: true,
+      base: base
+    };
+  },
+  htmlBuilder,
+  mathmlBuilder
+});
+
+// Horizontal overlap functions
+defineFunction({
+  type: "accentUnder",
+  names: ["\\underleftarrow", "\\underrightarrow", "\\underleftrightarrow", "\\undergroup", "\\underlinesegment", "\\utilde"],
+  props: {
+    numArgs: 1
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const base = args[0];
+    return {
+      type: "accentUnder",
+      mode: parser.mode,
+      label: funcName,
+      base: base
+    };
+  },
+  htmlBuilder: (group, options) => {
+    // Treat under accents much like underlines.
+    const innerGroup = buildGroup(group.base, options);
+    const accentBody = stretchy.svgSpan(group, options);
+    const kern = group.label === "\\utilde" ? 0.12 : 0; // Generate the vlist, with the appropriate kerns
+
+    const vlist = buildCommon.makeVList({
+      positionType: "top",
+      positionData: innerGroup.height,
+      children: [{
+        type: "elem",
+        elem: accentBody,
+        wrapperClasses: ["svg-align"]
+      }, {
+        type: "kern",
+        size: kern
+      }, {
+        type: "elem",
+        elem: innerGroup
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "accentunder"], [vlist], options);
+  },
+  mathmlBuilder: (group, options) => {
+    const accentNode = stretchy.mathMLnode(group.label);
+    const node = new mathMLTree.MathNode("munder", [buildGroup$1(group.base, options), accentNode]);
+    node.setAttribute("accentunder", "true");
+    return node;
+  }
+});
+
+// Helper function
+const paddedNode = group => {
+  const node = new mathMLTree.MathNode("mpadded", group ? [group] : []);
+  node.setAttribute("width", "+0.6em");
+  node.setAttribute("lspace", "0.3em");
+  return node;
+}; // Stretchy arrows with an optional argument
+
+
+defineFunction({
+  type: "xArrow",
+  names: ["\\xleftarrow", "\\xrightarrow", "\\xLeftarrow", "\\xRightarrow", "\\xleftrightarrow", "\\xLeftrightarrow", "\\xhookleftarrow", "\\xhookrightarrow", "\\xmapsto", "\\xrightharpoondown", "\\xrightharpoonup", "\\xleftharpoondown", "\\xleftharpoonup", "\\xrightleftharpoons", "\\xleftrightharpoons", "\\xlongequal", "\\xtwoheadrightarrow", "\\xtwoheadleftarrow", "\\xtofrom", // The next 3 functions are here to support the mhchem extension.
+  // Direct use of these functions is discouraged and may break someday.
+  "\\xrightleftarrows", "\\xrightequilibrium", "\\xleftequilibrium"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    return {
+      type: "xArrow",
+      mode: parser.mode,
+      label: funcName,
+      body: args[0],
+      below: optArgs[0]
+    };
+  },
+
+  // Flow is unable to correctly infer the type of `group`, even though it's
+  // unamibiguously determined from the passed-in `type` above.
+  htmlBuilder(group, options) {
+    const style = options.style; // Build the argument groups in the appropriate style.
+    // Ref: amsmath.dtx:   \hbox{$\scriptstyle\mkern#3mu{#6}\mkern#4mu$}%
+    // Some groups can return document fragments.  Handle those by wrapping
+    // them in a span.
+
+    let newOptions = options.havingStyle(style.sup());
+    const upperGroup = buildCommon.wrapFragment(buildGroup(group.body, newOptions, options), options);
+    upperGroup.classes.push("x-arrow-pad");
+    let lowerGroup;
+
+    if (group.below) {
+      // Build the lower group
+      newOptions = options.havingStyle(style.sub());
+      lowerGroup = buildCommon.wrapFragment(buildGroup(group.below, newOptions, options), options);
+      lowerGroup.classes.push("x-arrow-pad");
+    }
+
+    const arrowBody = stretchy.svgSpan(group, options); // Re shift: Note that stretchy.svgSpan returned arrowBody.depth = 0.
+    // The point we want on the math axis is at 0.5 * arrowBody.height.
+
+    const arrowShift = -options.fontMetrics().axisHeight + 0.5 * arrowBody.height; // 2 mu kern. Ref: amsmath.dtx: #7\if0#2\else\mkern#2mu\fi
+
+    let upperShift = -options.fontMetrics().axisHeight - 0.5 * arrowBody.height - 0.111; // 0.111 em = 2 mu
+
+    if (upperGroup.depth > 0.25 || group.label === "\\xleftequilibrium") {
+      upperShift -= upperGroup.depth; // shift up if depth encroaches
+    } // Generate the vlist
+
+
+    let vlist;
+
+    if (lowerGroup) {
+      const lowerShift = -options.fontMetrics().axisHeight + lowerGroup.height + 0.5 * arrowBody.height + 0.111;
+      vlist = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: upperGroup,
+          shift: upperShift
+        }, {
+          type: "elem",
+          elem: arrowBody,
+          shift: arrowShift
+        }, {
+          type: "elem",
+          elem: lowerGroup,
+          shift: lowerShift
+        }]
+      }, options);
+    } else {
+      vlist = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: upperGroup,
+          shift: upperShift
+        }, {
+          type: "elem",
+          elem: arrowBody,
+          shift: arrowShift
+        }]
+      }, options);
+    } // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+
+    vlist.children[0].children[0].children[1].classes.push("svg-align");
+    return buildCommon.makeSpan(["mrel", "x-arrow"], [vlist], options);
+  },
+
+  mathmlBuilder(group, options) {
+    const arrowNode = stretchy.mathMLnode(group.label);
+    let node;
+
+    if (group.body) {
+      const upperNode = paddedNode(buildGroup$1(group.body, options));
+
+      if (group.below) {
+        const lowerNode = paddedNode(buildGroup$1(group.below, options));
+        node = new mathMLTree.MathNode("munderover", [arrowNode, lowerNode, upperNode]);
+      } else {
+        node = new mathMLTree.MathNode("mover", [arrowNode, upperNode]);
+      }
+    } else if (group.below) {
+      const lowerNode = paddedNode(buildGroup$1(group.below, options));
+      node = new mathMLTree.MathNode("munder", [arrowNode, lowerNode]);
+    } else {
+      // This should never happen.
+      // Parser.js throws an error if there is no argument.
+      node = paddedNode();
+      node = new mathMLTree.MathNode("mover", [arrowNode, node]);
+    }
+
+    return node;
+  }
+
+});
+
+// {123} and converts into symbol with code 123.  It is used by the *macro*
+// \char defined in macros.js.
+
+defineFunction({
+  type: "textord",
+  names: ["\\@char"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    const arg = assertNodeType(args[0], "ordgroup");
+    const group = arg.body;
+    let number = "";
+
+    for (let i = 0; i < group.length; i++) {
+      const node = assertNodeType(group[i], "textord");
+      number += node.text;
+    }
+
+    const code = parseInt(number);
+
+    if (isNaN(code)) {
+      throw new ParseError(`\\@char has non-numeric argument ${number}`);
+    }
+
+    return {
+      type: "textord",
+      mode: parser.mode,
+      text: String.fromCharCode(code)
+    };
+  }
+
+});
+
+const htmlBuilder$1 = (group, options) => {
+  const elements = buildExpression(group.body, options.withColor(group.color), false); // \color isn't supposed to affect the type of the elements it contains.
+  // To accomplish this, we wrap the results in a fragment, so the inner
+  // elements will be able to directly interact with their neighbors. For
+  // example, `\color{red}{2 +} 3` has the same spacing as `2 + 3`
+
+  return buildCommon.makeFragment(elements);
+};
+
+const mathmlBuilder$1 = (group, options) => {
+  const inner = buildExpression$1(group.body, options.withColor(group.color));
+  const node = new mathMLTree.MathNode("mstyle", inner);
+  node.setAttribute("mathcolor", group.color);
+  return node;
+};
+
+defineFunction({
+  type: "color",
+  names: ["\\textcolor"],
+  props: {
+    numArgs: 2,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "original"]
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    const color = assertNodeType(args[0], "color-token").color;
+    const body = args[1];
+    return {
+      type: "color",
+      mode: parser.mode,
+      color,
+      body: ordargument(body)
+    };
+  },
+
+  htmlBuilder: htmlBuilder$1,
+  mathmlBuilder: mathmlBuilder$1
+});
+defineFunction({
+  type: "color",
+  names: ["\\color"],
+  props: {
+    numArgs: 1,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color"]
+  },
+
+  handler(_ref2, args) {
+    let parser = _ref2.parser,
+        breakOnTokenText = _ref2.breakOnTokenText;
+    const color = assertNodeType(args[0], "color-token").color; // Set macro \current@color in current namespace to store the current
+    // color, mimicking the behavior of color.sty.
+    // This is currently used just to correctly color a \right
+    // that follows a \color command.
+
+    parser.gullet.macros.set("\\current@color", color); // Parse out the implicit body that should be colored.
+
+    const body = parser.parseExpression(true, breakOnTokenText);
+    return {
+      type: "color",
+      mode: parser.mode,
+      color,
+      body
+    };
+  },
+
+  htmlBuilder: htmlBuilder$1,
+  mathmlBuilder: mathmlBuilder$1
+});
+
+// Row breaks within tabular environments, and line breaks at top level
+// same signature, we implement them as one megafunction, with newRow
+// indicating whether we're in the \cr case, and newLine indicating whether
+// to break the line in the \newline case.
+
+defineFunction({
+  type: "cr",
+  names: ["\\cr", "\\newline"],
+  props: {
+    numArgs: 0,
+    numOptionalArgs: 1,
+    argTypes: ["size"],
+    allowedInText: true
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const size = optArgs[0];
+    const newRow = funcName === "\\cr";
+    let newLine = false;
+
+    if (!newRow) {
+      if (parser.settings.displayMode && parser.settings.useStrictBehavior("newLineInDisplayMode", "In LaTeX, \\\\ or \\newline " + "does nothing in display mode")) {
+        newLine = false;
+      } else {
+        newLine = true;
+      }
+    }
+
+    return {
+      type: "cr",
+      mode: parser.mode,
+      newLine,
+      newRow,
+      size: size && assertNodeType(size, "size").value
+    };
+  },
+
+  // The following builders are called only at the top level,
+  // not within tabular/array environments.
+  htmlBuilder(group, options) {
+    if (group.newRow) {
+      throw new ParseError("\\cr valid only within a tabular/array environment");
+    }
+
+    const span = buildCommon.makeSpan(["mspace"], [], options);
+
+    if (group.newLine) {
+      span.classes.push("newline");
+
+      if (group.size) {
+        span.style.marginTop = calculateSize(group.size, options) + "em";
+      }
+    }
+
+    return span;
+  },
+
+  mathmlBuilder(group, options) {
+    const node = new mathMLTree.MathNode("mspace");
+
+    if (group.newLine) {
+      node.setAttribute("linebreak", "newline");
+
+      if (group.size) {
+        node.setAttribute("height", calculateSize(group.size, options) + "em");
+      }
+    }
+
+    return node;
+  }
+
+});
+
+const globalMap = {
+  "\\global": "\\global",
+  "\\long": "\\\\globallong",
+  "\\\\globallong": "\\\\globallong",
+  "\\def": "\\gdef",
+  "\\gdef": "\\gdef",
+  "\\edef": "\\xdef",
+  "\\xdef": "\\xdef",
+  "\\let": "\\\\globallet",
+  "\\futurelet": "\\\\globalfuture"
+};
+
+const checkControlSequence = tok => {
+  const name = tok.text;
+
+  if (/^(?:[\\{}$&#^_]|EOF)$/.test(name)) {
+    throw new ParseError("Expected a control sequence", tok);
+  }
+
+  return name;
+};
+
+const getRHS = parser => {
+  let tok = parser.gullet.popToken();
+
+  if (tok.text === "=") {
+    // consume optional equals
+    tok = parser.gullet.popToken();
+
+    if (tok.text === " ") {
+      // consume one optional space
+      tok = parser.gullet.popToken();
+    }
+  }
+
+  return tok;
+};
+
+const letCommand = (parser, name, tok, global) => {
+  let macro = parser.gullet.macros.get(tok.text);
+
+  if (macro == null) {
+    // don't expand it later even if a macro with the same name is defined
+    // e.g., \let\foo=\frac \def\frac{\relax} \frac12
+    tok.noexpand = true;
+    macro = {
+      tokens: [tok],
+      numArgs: 0,
+      // reproduce the same behavior in expansion
+      unexpandable: !parser.gullet.isExpandable(tok.text)
+    };
+  }
+
+  parser.gullet.macros.set(name, macro, global);
+}; // <assignment> -> <non-macro assignment>|<macro assignment>
+// <non-macro assignment> -> <simple assignment>|\global<non-macro assignment>
+// <macro assignment> -> <definition>|<prefix><macro assignment>
+// <prefix> -> \global|\long|\outer
+
+
+defineFunction({
+  type: "internal",
+  names: ["\\global", "\\long", "\\\\globallong"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    parser.consumeSpaces();
+    const token = parser.fetch();
+
+    if (globalMap[token.text]) {
+      // KaTeX doesn't have \par, so ignore \long
+      if (funcName === "\\global" || funcName === "\\\\globallong") {
+        token.text = globalMap[token.text];
+      }
+
+      return assertNodeType(parser.parseFunction(), "internal");
+    }
+
+    throw new ParseError(`Invalid token after macro prefix`, token);
+  }
+
+}); // Basic support for macro definitions: \def, \gdef, \edef, \xdef
+// <definition> -> <def><control sequence><definition text>
+// <def> -> \def|\gdef|\edef|\xdef
+// <definition text> -> <parameter text><left brace><balanced text><right brace>
+
+defineFunction({
+  type: "internal",
+  names: ["\\def", "\\gdef", "\\edef", "\\xdef"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref2) {
+    let parser = _ref2.parser,
+        funcName = _ref2.funcName;
+    let arg = parser.gullet.consumeArgs(1)[0];
+
+    if (arg.length !== 1) {
+      throw new ParseError("\\gdef's first argument must be a macro name");
+    }
+
+    const name = arg[0].text; // Count argument specifiers, and check they are in the order #1 #2 ...
+
+    let numArgs = 0;
+    arg = parser.gullet.consumeArgs(1)[0];
+
+    while (arg.length === 1 && arg[0].text === "#") {
+      arg = parser.gullet.consumeArgs(1)[0];
+
+      if (arg.length !== 1) {
+        throw new ParseError(`Invalid argument number length "${arg.length}"`);
+      }
+
+      if (!/^[1-9]$/.test(arg[0].text)) {
+        throw new ParseError(`Invalid argument number "${arg[0].text}"`);
+      }
+
+      numArgs++;
+
+      if (parseInt(arg[0].text) !== numArgs) {
+        throw new ParseError(`Argument number "${arg[0].text}" out of order`);
+      }
+
+      arg = parser.gullet.consumeArgs(1)[0];
+    }
+
+    if (funcName === "\\edef" || funcName === "\\xdef") {
+      arg = parser.gullet.expandTokens(arg);
+      arg.reverse(); // to fit in with stack order
+    } // Final arg is the expansion of the macro
+
+
+    parser.gullet.macros.set(name, {
+      tokens: arg,
+      numArgs
+    }, funcName === globalMap[funcName]);
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+
+}); // <simple assignment> -> <let assignment>
+// <let assignment> -> \futurelet<control sequence><token><token>
+//     | \let<control sequence><equals><one optional space><token>
+// <equals> -> <optional spaces>|<optional spaces>=
+
+defineFunction({
+  type: "internal",
+  names: ["\\let", "\\\\globallet"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref3) {
+    let parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    const name = checkControlSequence(parser.gullet.popToken());
+    parser.gullet.consumeSpaces();
+    const tok = getRHS(parser);
+    letCommand(parser, name, tok, funcName === "\\\\globallet");
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+
+}); // ref: https://www.tug.org/TUGboat/tb09-3/tb22bechtolsheim.pdf
+
+defineFunction({
+  type: "internal",
+  names: ["\\futurelet", "\\\\globalfuture"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref4) {
+    let parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    const name = checkControlSequence(parser.gullet.popToken());
+    const middle = parser.gullet.popToken();
+    const tok = parser.gullet.popToken();
+    letCommand(parser, name, tok, funcName === "\\\\globalfuture");
+    parser.gullet.pushToken(tok);
+    parser.gullet.pushToken(middle);
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+
+});
+
+/**
+ * This file deals with creating delimiters of various sizes. The TeXbook
+ * discusses these routines on page 441-442, in the "Another subroutine sets box
+ * x to a specified variable delimiter" paragraph.
+ *
+ * There are three main routines here. `makeSmallDelim` makes a delimiter in the
+ * normal font, but in either text, script, or scriptscript style.
+ * `makeLargeDelim` makes a delimiter in textstyle, but in one of the Size1,
+ * Size2, Size3, or Size4 fonts. `makeStackedDelim` makes a delimiter out of
+ * smaller pieces that are stacked on top of one another.
+ *
+ * The functions take a parameter `center`, which determines if the delimiter
+ * should be centered around the axis.
+ *
+ * Then, there are three exposed functions. `sizedDelim` makes a delimiter in
+ * one of the given sizes. This is used for things like `\bigl`.
+ * `customSizedDelim` makes a delimiter with a given total height+depth. It is
+ * called in places like `\sqrt`. `leftRightDelim` makes an appropriate
+ * delimiter which surrounds an expression of a given height an depth. It is
+ * used in `\left` and `\right`.
+ */
+
+/**
+ * Get the metrics for a given symbol and font, after transformation (i.e.
+ * after following replacement from symbols.js)
+ */
+const getMetrics = function getMetrics(symbol, font, mode) {
+  const replace = symbols.math[symbol] && symbols.math[symbol].replace;
+  const metrics = getCharacterMetrics(replace || symbol, font, mode);
+
+  if (!metrics) {
+    throw new Error(`Unsupported symbol ${symbol} and font size ${font}.`);
+  }
+
+  return metrics;
+};
+/**
+ * Puts a delimiter span in a given style, and adds appropriate height, depth,
+ * and maxFontSizes.
+ */
+
+
+const styleWrap = function styleWrap(delim, toStyle, options, classes) {
+  const newOptions = options.havingBaseStyle(toStyle);
+  const span = buildCommon.makeSpan(classes.concat(newOptions.sizingClasses(options)), [delim], options);
+  const delimSizeMultiplier = newOptions.sizeMultiplier / options.sizeMultiplier;
+  span.height *= delimSizeMultiplier;
+  span.depth *= delimSizeMultiplier;
+  span.maxFontSize = newOptions.sizeMultiplier;
+  return span;
+};
+
+const centerSpan = function centerSpan(span, options, style) {
+  const newOptions = options.havingBaseStyle(style);
+  const shift = (1 - options.sizeMultiplier / newOptions.sizeMultiplier) * options.fontMetrics().axisHeight;
+  span.classes.push("delimcenter");
+  span.style.top = shift + "em";
+  span.height -= shift;
+  span.depth += shift;
+};
+/**
+ * Makes a small delimiter. This is a delimiter that comes in the Main-Regular
+ * font, but is restyled to either be in textstyle, scriptstyle, or
+ * scriptscriptstyle.
+ */
+
+
+const makeSmallDelim = function makeSmallDelim(delim, style, center, options, mode, classes) {
+  const text = buildCommon.makeSymbol(delim, "Main-Regular", mode, options);
+  const span = styleWrap(text, style, options, classes);
+
+  if (center) {
+    centerSpan(span, options, style);
+  }
+
+  return span;
+};
+/**
+ * Builds a symbol in the given font size (note size is an integer)
+ */
+
+
+const mathrmSize = function mathrmSize(value, size, mode, options) {
+  return buildCommon.makeSymbol(value, "Size" + size + "-Regular", mode, options);
+};
+/**
+ * Makes a large delimiter. This is a delimiter that comes in the Size1, Size2,
+ * Size3, or Size4 fonts. It is always rendered in textstyle.
+ */
+
+
+const makeLargeDelim = function makeLargeDelim(delim, size, center, options, mode, classes) {
+  const inner = mathrmSize(delim, size, mode, options);
+  const span = styleWrap(buildCommon.makeSpan(["delimsizing", "size" + size], [inner], options), Style$1.TEXT, options, classes);
+
+  if (center) {
+    centerSpan(span, options, Style$1.TEXT);
+  }
+
+  return span;
+};
+/**
+ * Make an inner span with the given offset and in the given font. This is used
+ * in `makeStackedDelim` to make the stacking pieces for the delimiter.
+ */
+
+
+const makeInner = function makeInner(symbol, font, mode) {
+  let sizeClass; // Apply the correct CSS class to choose the right font.
+
+  if (font === "Size1-Regular") {
+    sizeClass = "delim-size1";
+  } else
+    /* if (font === "Size4-Regular") */
+    {
+      sizeClass = "delim-size4";
+    }
+
+  const inner = buildCommon.makeSpan(["delimsizinginner", sizeClass], [buildCommon.makeSpan([], [buildCommon.makeSymbol(symbol, font, mode)])]); // Since this will be passed into `makeVList` in the end, wrap the element
+  // in the appropriate tag that VList uses.
+
+  return {
+    type: "elem",
+    elem: inner
+  };
+}; // Helper for makeStackedDelim
+
+
+const lap = {
+  type: "kern",
+  size: -0.005
+};
+/**
+ * Make a stacked delimiter out of a given delimiter, with the total height at
+ * least `heightTotal`. This routine is mentioned on page 442 of the TeXbook.
+ */
+
+const makeStackedDelim = function makeStackedDelim(delim, heightTotal, center, options, mode, classes) {
+  // There are four parts, the top, an optional middle, a repeated part, and a
+  // bottom.
+  let top;
+  let middle;
+  let repeat;
+  let bottom;
+  top = repeat = bottom = delim;
+  middle = null; // Also keep track of what font the delimiters are in
+
+  let font = "Size1-Regular"; // We set the parts and font based on the symbol. Note that we use
+  // '\u23d0' instead of '|' and '\u2016' instead of '\\|' for the
+  // repeats of the arrows
+
+  if (delim === "\\uparrow") {
+    repeat = bottom = "\u23d0";
+  } else if (delim === "\\Uparrow") {
+    repeat = bottom = "\u2016";
+  } else if (delim === "\\downarrow") {
+    top = repeat = "\u23d0";
+  } else if (delim === "\\Downarrow") {
+    top = repeat = "\u2016";
+  } else if (delim === "\\updownarrow") {
+    top = "\\uparrow";
+    repeat = "\u23d0";
+    bottom = "\\downarrow";
+  } else if (delim === "\\Updownarrow") {
+    top = "\\Uparrow";
+    repeat = "\u2016";
+    bottom = "\\Downarrow";
+  } else if (delim === "[" || delim === "\\lbrack") {
+    top = "\u23a1";
+    repeat = "\u23a2";
+    bottom = "\u23a3";
+    font = "Size4-Regular";
+  } else if (delim === "]" || delim === "\\rbrack") {
+    top = "\u23a4";
+    repeat = "\u23a5";
+    bottom = "\u23a6";
+    font = "Size4-Regular";
+  } else if (delim === "\\lfloor" || delim === "\u230a") {
+    repeat = top = "\u23a2";
+    bottom = "\u23a3";
+    font = "Size4-Regular";
+  } else if (delim === "\\lceil" || delim === "\u2308") {
+    top = "\u23a1";
+    repeat = bottom = "\u23a2";
+    font = "Size4-Regular";
+  } else if (delim === "\\rfloor" || delim === "\u230b") {
+    repeat = top = "\u23a5";
+    bottom = "\u23a6";
+    font = "Size4-Regular";
+  } else if (delim === "\\rceil" || delim === "\u2309") {
+    top = "\u23a4";
+    repeat = bottom = "\u23a5";
+    font = "Size4-Regular";
+  } else if (delim === "(" || delim === "\\lparen") {
+    top = "\u239b";
+    repeat = "\u239c";
+    bottom = "\u239d";
+    font = "Size4-Regular";
+  } else if (delim === ")" || delim === "\\rparen") {
+    top = "\u239e";
+    repeat = "\u239f";
+    bottom = "\u23a0";
+    font = "Size4-Regular";
+  } else if (delim === "\\{" || delim === "\\lbrace") {
+    top = "\u23a7";
+    middle = "\u23a8";
+    bottom = "\u23a9";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\}" || delim === "\\rbrace") {
+    top = "\u23ab";
+    middle = "\u23ac";
+    bottom = "\u23ad";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\lgroup" || delim === "\u27ee") {
+    top = "\u23a7";
+    bottom = "\u23a9";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\rgroup" || delim === "\u27ef") {
+    top = "\u23ab";
+    bottom = "\u23ad";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\lmoustache" || delim === "\u23b0") {
+    top = "\u23a7";
+    bottom = "\u23ad";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\rmoustache" || delim === "\u23b1") {
+    top = "\u23ab";
+    bottom = "\u23a9";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } // Get the metrics of the four sections
+
+
+  const topMetrics = getMetrics(top, font, mode);
+  const topHeightTotal = topMetrics.height + topMetrics.depth;
+  const repeatMetrics = getMetrics(repeat, font, mode);
+  const repeatHeightTotal = repeatMetrics.height + repeatMetrics.depth;
+  const bottomMetrics = getMetrics(bottom, font, mode);
+  const bottomHeightTotal = bottomMetrics.height + bottomMetrics.depth;
+  let middleHeightTotal = 0;
+  let middleFactor = 1;
+
+  if (middle !== null) {
+    const middleMetrics = getMetrics(middle, font, mode);
+    middleHeightTotal = middleMetrics.height + middleMetrics.depth;
+    middleFactor = 2; // repeat symmetrically above and below middle
+  } // Calcuate the minimal height that the delimiter can have.
+  // It is at least the size of the top, bottom, and optional middle combined.
+
+
+  const minHeight = topHeightTotal + bottomHeightTotal + middleHeightTotal; // Compute the number of copies of the repeat symbol we will need
+
+  const repeatCount = Math.max(0, Math.ceil((heightTotal - minHeight) / (middleFactor * repeatHeightTotal))); // Compute the total height of the delimiter including all the symbols
+
+  const realHeightTotal = minHeight + repeatCount * middleFactor * repeatHeightTotal; // The center of the delimiter is placed at the center of the axis. Note
+  // that in this context, "center" means that the delimiter should be
+  // centered around the axis in the current style, while normally it is
+  // centered around the axis in textstyle.
+
+  let axisHeight = options.fontMetrics().axisHeight;
+
+  if (center) {
+    axisHeight *= options.sizeMultiplier;
+  } // Calculate the depth
+
+
+  const depth = realHeightTotal / 2 - axisHeight; // This function differs from the TeX procedure in one way.
+  // We shift each repeat element downwards by 0.005em, to prevent a gap
+  // due to browser floating point rounding error.
+  // Then, at the last element-to element joint, we add one extra repeat
+  // element to cover the gap created by the shifts.
+  // Find the shift needed to align the upper end of the extra element at a point
+  // 0.005em above the lower end of the top element.
+
+  const shiftOfExtraElement = (repeatCount + 1) * 0.005 - repeatHeightTotal; // Now, we start building the pieces that will go into the vlist
+  // Keep a list of the inner pieces
+
+  const inners = []; // Add the bottom symbol
+
+  inners.push(makeInner(bottom, font, mode));
+
+  if (middle === null) {
+    // Add that many symbols
+    for (let i = 0; i < repeatCount; i++) {
+      inners.push(lap); // overlap
+
+      inners.push(makeInner(repeat, font, mode));
+    }
+  } else {
+    // When there is a middle bit, we need the middle part and two repeated
+    // sections
+    for (let i = 0; i < repeatCount; i++) {
+      inners.push(lap);
+      inners.push(makeInner(repeat, font, mode));
+    } // Insert one extra repeat element.
+
+
+    inners.push({
+      type: "kern",
+      size: shiftOfExtraElement
+    });
+    inners.push(makeInner(repeat, font, mode));
+    inners.push(lap); // Now insert the middle of the brace.
+
+    inners.push(makeInner(middle, font, mode));
+
+    for (let i = 0; i < repeatCount; i++) {
+      inners.push(lap);
+      inners.push(makeInner(repeat, font, mode));
+    }
+  } // To cover the gap create by the overlaps, insert one more repeat element,
+  // at a position that juts 0.005 above the bottom of the top element.
+
+
+  if ((repeat === "\u239c" || repeat === "\u239f") && repeatCount === 0) {
+    // Parentheses need a short repeat element in order to avoid an overrun.
+    // We'll make a 0.3em tall element from a SVG.
+    const overlap = buildCommon.svgData.leftParenInner[2] / 2;
+    inners.push({
+      type: "kern",
+      size: -overlap
+    });
+    const pathName = repeat === "\u239c" ? "leftParenInner" : "rightParenInner";
+    const innerSpan = buildCommon.staticSvg(pathName, options);
+    inners.push({
+      type: "elem",
+      elem: innerSpan
+    });
+    inners.push({
+      type: "kern",
+      size: -overlap
+    });
+  } else {
+    inners.push({
+      type: "kern",
+      size: shiftOfExtraElement
+    });
+    inners.push(makeInner(repeat, font, mode));
+    inners.push(lap);
+  } // Add the top symbol
+
+
+  inners.push(makeInner(top, font, mode)); // Finally, build the vlist
+
+  const newOptions = options.havingBaseStyle(Style$1.TEXT);
+  const inner = buildCommon.makeVList({
+    positionType: "bottom",
+    positionData: depth,
+    children: inners
+  }, newOptions);
+  return styleWrap(buildCommon.makeSpan(["delimsizing", "mult"], [inner], newOptions), Style$1.TEXT, options, classes);
+}; // All surds have 0.08em padding above the viniculum inside the SVG.
+// That keeps browser span height rounding error from pinching the line.
+
+
+const vbPad = 80; // padding above the surd, measured inside the viewBox.
+
+const emPad = 0.08; // padding, in ems, measured in the document.
+
+const sqrtSvg = function sqrtSvg(sqrtName, height, viewBoxHeight, extraViniculum, options) {
+  const path = sqrtPath(sqrtName, extraViniculum, viewBoxHeight);
+  const pathNode = new PathNode(sqrtName, path);
+  const svg = new SvgNode([pathNode], {
+    // Note: 1000:1 ratio of viewBox to document em width.
+    "width": "400em",
+    "height": height + "em",
+    "viewBox": "0 0 400000 " + viewBoxHeight,
+    "preserveAspectRatio": "xMinYMin slice"
+  });
+  return buildCommon.makeSvgSpan(["hide-tail"], [svg], options);
+};
+/**
+ * Make a sqrt image of the given height,
+ */
+
+
+const makeSqrtImage = function makeSqrtImage(height, options) {
+  // Define a newOptions that removes the effect of size changes such as \Huge.
+  // We don't pick different a height surd for \Huge. For it, we scale up.
+  const newOptions = options.havingBaseSizing(); // Pick the desired surd glyph from a sequence of surds.
+
+  const delim = traverseSequence("\\surd", height * newOptions.sizeMultiplier, stackLargeDelimiterSequence, newOptions);
+  let sizeMultiplier = newOptions.sizeMultiplier; // default
+  // The standard sqrt SVGs each have a 0.04em thick viniculum.
+  // If Settings.minRuleThickness is larger than that, we add extraViniculum.
+
+  const extraViniculum = Math.max(0, options.minRuleThickness - options.fontMetrics().sqrtRuleThickness); // Create a span containing an SVG image of a sqrt symbol.
+
+  let span;
+  let spanHeight = 0;
+  let texHeight = 0;
+  let viewBoxHeight = 0;
+  let advanceWidth; // We create viewBoxes with 80 units of "padding" above each surd.
+  // Then browser rounding error on the parent span height will not
+  // encroach on the ink of the viniculum. But that padding is not
+  // included in the TeX-like `height` used for calculation of
+  // vertical alignment. So texHeight = span.height < span.style.height.
+
+  if (delim.type === "small") {
+    // Get an SVG that is derived from glyph U+221A in font KaTeX-Main.
+    // 1000 unit normal glyph height.
+    viewBoxHeight = 1000 + 1000 * extraViniculum + vbPad;
+
+    if (height < 1.0) {
+      sizeMultiplier = 1.0; // mimic a \textfont radical
+    } else if (height < 1.4) {
+      sizeMultiplier = 0.7; // mimic a \scriptfont radical
+    }
+
+    spanHeight = (1.0 + extraViniculum + emPad) / sizeMultiplier;
+    texHeight = (1.00 + extraViniculum) / sizeMultiplier;
+    span = sqrtSvg("sqrtMain", spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "0.853em";
+    advanceWidth = 0.833 / sizeMultiplier; // from the font.
+  } else if (delim.type === "large") {
+    // These SVGs come from fonts: KaTeX_Size1, _Size2, etc.
+    viewBoxHeight = (1000 + vbPad) * sizeToMaxHeight[delim.size];
+    texHeight = (sizeToMaxHeight[delim.size] + extraViniculum) / sizeMultiplier;
+    spanHeight = (sizeToMaxHeight[delim.size] + extraViniculum + emPad) / sizeMultiplier;
+    span = sqrtSvg("sqrtSize" + delim.size, spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "1.02em";
+    advanceWidth = 1.0 / sizeMultiplier; // 1.0 from the font.
+  } else {
+    // Tall sqrt. In TeX, this would be stacked using multiple glyphs.
+    // We'll use a single SVG to accomplish the same thing.
+    spanHeight = height + extraViniculum + emPad;
+    texHeight = height + extraViniculum;
+    viewBoxHeight = Math.floor(1000 * height + extraViniculum) + vbPad;
+    span = sqrtSvg("sqrtTall", spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "0.742em";
+    advanceWidth = 1.056;
+  }
+
+  span.height = texHeight;
+  span.style.height = spanHeight + "em";
+  return {
+    span,
+    advanceWidth,
+    // Calculate the actual line width.
+    // This actually should depend on the chosen font -- e.g. \boldmath
+    // should use the thicker surd symbols from e.g. KaTeX_Main-Bold, and
+    // have thicker rules.
+    ruleWidth: (options.fontMetrics().sqrtRuleThickness + extraViniculum) * sizeMultiplier
+  };
+}; // There are three kinds of delimiters, delimiters that stack when they become
+// too large
+
+
+const stackLargeDelimiters = ["(", "\\lparen", ")", "\\rparen", "[", "\\lbrack", "]", "\\rbrack", "\\{", "\\lbrace", "\\}", "\\rbrace", "\\lfloor", "\\rfloor", "\u230a", "\u230b", "\\lceil", "\\rceil", "\u2308", "\u2309", "\\surd"]; // delimiters that always stack
+
+const stackAlwaysDelimiters = ["\\uparrow", "\\downarrow", "\\updownarrow", "\\Uparrow", "\\Downarrow", "\\Updownarrow", "|", "\\|", "\\vert", "\\Vert", "\\lvert", "\\rvert", "\\lVert", "\\rVert", "\\lgroup", "\\rgroup", "\u27ee", "\u27ef", "\\lmoustache", "\\rmoustache", "\u23b0", "\u23b1"]; // and delimiters that never stack
+
+const stackNeverDelimiters = ["<", ">", "\\langle", "\\rangle", "/", "\\backslash", "\\lt", "\\gt"]; // Metrics of the different sizes. Found by looking at TeX's output of
+// $\bigl| // \Bigl| \biggl| \Biggl| \showlists$
+// Used to create stacked delimiters of appropriate sizes in makeSizedDelim.
+
+const sizeToMaxHeight = [0, 1.2, 1.8, 2.4, 3.0];
+/**
+ * Used to create a delimiter of a specific size, where `size` is 1, 2, 3, or 4.
+ */
+
+const makeSizedDelim = function makeSizedDelim(delim, size, options, mode, classes) {
+  // < and > turn into \langle and \rangle in delimiters
+  if (delim === "<" || delim === "\\lt" || delim === "\u27e8") {
+    delim = "\\langle";
+  } else if (delim === ">" || delim === "\\gt" || delim === "\u27e9") {
+    delim = "\\rangle";
+  } // Sized delimiters are never centered.
+
+
+  if (utils.contains(stackLargeDelimiters, delim) || utils.contains(stackNeverDelimiters, delim)) {
+    return makeLargeDelim(delim, size, false, options, mode, classes);
+  } else if (utils.contains(stackAlwaysDelimiters, delim)) {
+    return makeStackedDelim(delim, sizeToMaxHeight[size], false, options, mode, classes);
+  } else {
+    throw new ParseError("Illegal delimiter: '" + delim + "'");
+  }
+};
+/**
+ * There are three different sequences of delimiter sizes that the delimiters
+ * follow depending on the kind of delimiter. This is used when creating custom
+ * sized delimiters to decide whether to create a small, large, or stacked
+ * delimiter.
+ *
+ * In real TeX, these sequences aren't explicitly defined, but are instead
+ * defined inside the font metrics. Since there are only three sequences that
+ * are possible for the delimiters that TeX defines, it is easier to just encode
+ * them explicitly here.
+ */
+
+
+// Delimiters that never stack try small delimiters and large delimiters only
+const stackNeverDelimiterSequence = [{
+  type: "small",
+  style: Style$1.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: Style$1.SCRIPT
+}, {
+  type: "small",
+  style: Style$1.TEXT
+}, {
+  type: "large",
+  size: 1
+}, {
+  type: "large",
+  size: 2
+}, {
+  type: "large",
+  size: 3
+}, {
+  type: "large",
+  size: 4
+}]; // Delimiters that always stack try the small delimiters first, then stack
+
+const stackAlwaysDelimiterSequence = [{
+  type: "small",
+  style: Style$1.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: Style$1.SCRIPT
+}, {
+  type: "small",
+  style: Style$1.TEXT
+}, {
+  type: "stack"
+}]; // Delimiters that stack when large try the small and then large delimiters, and
+// stack afterwards
+
+const stackLargeDelimiterSequence = [{
+  type: "small",
+  style: Style$1.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: Style$1.SCRIPT
+}, {
+  type: "small",
+  style: Style$1.TEXT
+}, {
+  type: "large",
+  size: 1
+}, {
+  type: "large",
+  size: 2
+}, {
+  type: "large",
+  size: 3
+}, {
+  type: "large",
+  size: 4
+}, {
+  type: "stack"
+}];
+/**
+ * Get the font used in a delimiter based on what kind of delimiter it is.
+ * TODO(#963) Use more specific font family return type once that is introduced.
+ */
+
+const delimTypeToFont = function delimTypeToFont(type) {
+  if (type.type === "small") {
+    return "Main-Regular";
+  } else if (type.type === "large") {
+    return "Size" + type.size + "-Regular";
+  } else if (type.type === "stack") {
+    return "Size4-Regular";
+  } else {
+    throw new Error(`Add support for delim type '${type.type}' here.`);
+  }
+};
+/**
+ * Traverse a sequence of types of delimiters to decide what kind of delimiter
+ * should be used to create a delimiter of the given height+depth.
+ */
+
+
+const traverseSequence = function traverseSequence(delim, height, sequence, options) {
+  // Here, we choose the index we should start at in the sequences. In smaller
+  // sizes (which correspond to larger numbers in style.size) we start earlier
+  // in the sequence. Thus, scriptscript starts at index 3-3=0, script starts
+  // at index 3-2=1, text starts at 3-1=2, and display starts at min(2,3-0)=2
+  const start = Math.min(2, 3 - options.style.size);
+
+  for (let i = start; i < sequence.length; i++) {
+    if (sequence[i].type === "stack") {
+      // This is always the last delimiter, so we just break the loop now.
+      break;
+    }
+
+    const metrics = getMetrics(delim, delimTypeToFont(sequence[i]), "math");
+    let heightDepth = metrics.height + metrics.depth; // Small delimiters are scaled down versions of the same font, so we
+    // account for the style change size.
+
+    if (sequence[i].type === "small") {
+      const newOptions = options.havingBaseStyle(sequence[i].style);
+      heightDepth *= newOptions.sizeMultiplier;
+    } // Check if the delimiter at this size works for the given height.
+
+
+    if (heightDepth > height) {
+      return sequence[i];
+    }
+  } // If we reached the end of the sequence, return the last sequence element.
+
+
+  return sequence[sequence.length - 1];
+};
+/**
+ * Make a delimiter of a given height+depth, with optional centering. Here, we
+ * traverse the sequences, and create a delimiter that the sequence tells us to.
+ */
+
+
+const makeCustomSizedDelim = function makeCustomSizedDelim(delim, height, center, options, mode, classes) {
+  if (delim === "<" || delim === "\\lt" || delim === "\u27e8") {
+    delim = "\\langle";
+  } else if (delim === ">" || delim === "\\gt" || delim === "\u27e9") {
+    delim = "\\rangle";
+  } // Decide what sequence to use
+
+
+  let sequence;
+
+  if (utils.contains(stackNeverDelimiters, delim)) {
+    sequence = stackNeverDelimiterSequence;
+  } else if (utils.contains(stackLargeDelimiters, delim)) {
+    sequence = stackLargeDelimiterSequence;
+  } else {
+    sequence = stackAlwaysDelimiterSequence;
+  } // Look through the sequence
+
+
+  const delimType = traverseSequence(delim, height, sequence, options); // Get the delimiter from font glyphs.
+  // Depending on the sequence element we decided on, call the
+  // appropriate function.
+
+  if (delimType.type === "small") {
+    return makeSmallDelim(delim, delimType.style, center, options, mode, classes);
+  } else if (delimType.type === "large") {
+    return makeLargeDelim(delim, delimType.size, center, options, mode, classes);
+  } else
+    /* if (delimType.type === "stack") */
+    {
+      return makeStackedDelim(delim, height, center, options, mode, classes);
+    }
+};
+/**
+ * Make a delimiter for use with `\left` and `\right`, given a height and depth
+ * of an expression that the delimiters surround.
+ */
+
+
+const makeLeftRightDelim = function makeLeftRightDelim(delim, height, depth, options, mode, classes) {
+  // We always center \left/\right delimiters, so the axis is always shifted
+  const axisHeight = options.fontMetrics().axisHeight * options.sizeMultiplier; // Taken from TeX source, tex.web, function make_left_right
+
+  const delimiterFactor = 901;
+  const delimiterExtend = 5.0 / options.fontMetrics().ptPerEm;
+  const maxDistFromAxis = Math.max(height - axisHeight, depth + axisHeight);
+  const totalHeight = Math.max( // In real TeX, calculations are done using integral values which are
+  // 65536 per pt, or 655360 per em. So, the division here truncates in
+  // TeX but doesn't here, producing different results. If we wanted to
+  // exactly match TeX's calculation, we could do
+  //   Math.floor(655360 * maxDistFromAxis / 500) *
+  //    delimiterFactor / 655360
+  // (To see the difference, compare
+  //    x^{x^{\left(\rule{0.1em}{0.68em}\right)}}
+  // in TeX and KaTeX)
+  maxDistFromAxis / 500 * delimiterFactor, 2 * maxDistFromAxis - delimiterExtend); // Finally, we defer to `makeCustomSizedDelim` with our calculated total
+  // height
+
+  return makeCustomSizedDelim(delim, totalHeight, true, options, mode, classes);
+};
+
+var delimiter = {
+  sqrtImage: makeSqrtImage,
+  sizedDelim: makeSizedDelim,
+  customSizedDelim: makeCustomSizedDelim,
+  leftRightDelim: makeLeftRightDelim
+};
+
+// Extra data needed for the delimiter handler down below
+const delimiterSizes = {
+  "\\bigl": {
+    mclass: "mopen",
+    size: 1
+  },
+  "\\Bigl": {
+    mclass: "mopen",
+    size: 2
+  },
+  "\\biggl": {
+    mclass: "mopen",
+    size: 3
+  },
+  "\\Biggl": {
+    mclass: "mopen",
+    size: 4
+  },
+  "\\bigr": {
+    mclass: "mclose",
+    size: 1
+  },
+  "\\Bigr": {
+    mclass: "mclose",
+    size: 2
+  },
+  "\\biggr": {
+    mclass: "mclose",
+    size: 3
+  },
+  "\\Biggr": {
+    mclass: "mclose",
+    size: 4
+  },
+  "\\bigm": {
+    mclass: "mrel",
+    size: 1
+  },
+  "\\Bigm": {
+    mclass: "mrel",
+    size: 2
+  },
+  "\\biggm": {
+    mclass: "mrel",
+    size: 3
+  },
+  "\\Biggm": {
+    mclass: "mrel",
+    size: 4
+  },
+  "\\big": {
+    mclass: "mord",
+    size: 1
+  },
+  "\\Big": {
+    mclass: "mord",
+    size: 2
+  },
+  "\\bigg": {
+    mclass: "mord",
+    size: 3
+  },
+  "\\Bigg": {
+    mclass: "mord",
+    size: 4
+  }
+};
+const delimiters = ["(", "\\lparen", ")", "\\rparen", "[", "\\lbrack", "]", "\\rbrack", "\\{", "\\lbrace", "\\}", "\\rbrace", "\\lfloor", "\\rfloor", "\u230a", "\u230b", "\\lceil", "\\rceil", "\u2308", "\u2309", "<", ">", "\\langle", "\u27e8", "\\rangle", "\u27e9", "\\lt", "\\gt", "\\lvert", "\\rvert", "\\lVert", "\\rVert", "\\lgroup", "\\rgroup", "\u27ee", "\u27ef", "\\lmoustache", "\\rmoustache", "\u23b0", "\u23b1", "/", "\\backslash", "|", "\\vert", "\\|", "\\Vert", "\\uparrow", "\\Uparrow", "\\downarrow", "\\Downarrow", "\\updownarrow", "\\Updownarrow", "."];
+
+// Delimiter functions
+function checkDelimiter(delim, context) {
+  const symDelim = checkSymbolNodeType(delim);
+
+  if (symDelim && utils.contains(delimiters, symDelim.text)) {
+    return symDelim;
+  } else if (symDelim) {
+    throw new ParseError(`Invalid delimiter '${symDelim.text}' after '${context.funcName}'`, delim);
+  } else {
+    throw new ParseError(`Invalid delimiter type '${delim.type}'`, delim);
+  }
+}
+
+defineFunction({
+  type: "delimsizing",
+  names: ["\\bigl", "\\Bigl", "\\biggl", "\\Biggl", "\\bigr", "\\Bigr", "\\biggr", "\\Biggr", "\\bigm", "\\Bigm", "\\biggm", "\\Biggm", "\\big", "\\Big", "\\bigg", "\\Bigg"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    const delim = checkDelimiter(args[0], context);
+    return {
+      type: "delimsizing",
+      mode: context.parser.mode,
+      size: delimiterSizes[context.funcName].size,
+      mclass: delimiterSizes[context.funcName].mclass,
+      delim: delim.text
+    };
+  },
+  htmlBuilder: (group, options) => {
+    if (group.delim === ".") {
+      // Empty delimiters still count as elements, even though they don't
+      // show anything.
+      return buildCommon.makeSpan([group.mclass]);
+    } // Use delimiter.sizedDelim to generate the delimiter.
+
+
+    return delimiter.sizedDelim(group.delim, group.size, options, group.mode, [group.mclass]);
+  },
+  mathmlBuilder: group => {
+    const children = [];
+
+    if (group.delim !== ".") {
+      children.push(makeText(group.delim, group.mode));
+    }
+
+    const node = new mathMLTree.MathNode("mo", children);
+
+    if (group.mclass === "mopen" || group.mclass === "mclose") {
+      // Only some of the delimsizing functions act as fences, and they
+      // return "mopen" or "mclose" mclass.
+      node.setAttribute("fence", "true");
+    } else {
+      // Explicitly disable fencing if it's not a fence, to override the
+      // defaults.
+      node.setAttribute("fence", "false");
+    }
+
+    return node;
+  }
+});
+
+function assertParsed(group) {
+  if (!group.body) {
+    throw new Error("Bug: The leftright ParseNode wasn't fully parsed.");
+  }
+}
+
+defineFunction({
+  type: "leftright-right",
+  names: ["\\right"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    // \left case below triggers parsing of \right in
+    //   `const right = parser.parseFunction();`
+    // uses this return value.
+    const color = context.parser.gullet.macros.get("\\current@color");
+
+    if (color && typeof color !== "string") {
+      throw new ParseError("\\current@color set to non-string in \\right");
+    }
+
+    return {
+      type: "leftright-right",
+      mode: context.parser.mode,
+      delim: checkDelimiter(args[0], context).text,
+      color // undefined if not set via \color
+
+    };
+  }
+});
+defineFunction({
+  type: "leftright",
+  names: ["\\left"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    const delim = checkDelimiter(args[0], context);
+    const parser = context.parser; // Parse out the implicit body
+
+    ++parser.leftrightDepth; // parseExpression stops before '\\right'
+
+    const body = parser.parseExpression(false);
+    --parser.leftrightDepth; // Check the next token
+
+    parser.expect("\\right", false);
+    const right = assertNodeType(parser.parseFunction(), "leftright-right");
+    return {
+      type: "leftright",
+      mode: parser.mode,
+      body,
+      left: delim.text,
+      right: right.delim,
+      rightColor: right.color
+    };
+  },
+  htmlBuilder: (group, options) => {
+    assertParsed(group); // Build the inner expression
+
+    const inner = buildExpression(group.body, options, true, ["mopen", "mclose"]);
+    let innerHeight = 0;
+    let innerDepth = 0;
+    let hadMiddle = false; // Calculate its height and depth
+
+    for (let i = 0; i < inner.length; i++) {
+      // Property `isMiddle` not defined on `span`. See comment in
+      // "middle"'s htmlBuilder.
+      // $FlowFixMe
+      if (inner[i].isMiddle) {
+        hadMiddle = true;
+      } else {
+        innerHeight = Math.max(inner[i].height, innerHeight);
+        innerDepth = Math.max(inner[i].depth, innerDepth);
+      }
+    } // The size of delimiters is the same, regardless of what style we are
+    // in. Thus, to correctly calculate the size of delimiter we need around
+    // a group, we scale down the inner size based on the size.
+
+
+    innerHeight *= options.sizeMultiplier;
+    innerDepth *= options.sizeMultiplier;
+    let leftDelim;
+
+    if (group.left === ".") {
+      // Empty delimiters in \left and \right make null delimiter spaces.
+      leftDelim = makeNullDelimiter(options, ["mopen"]);
+    } else {
+      // Otherwise, use leftRightDelim to generate the correct sized
+      // delimiter.
+      leftDelim = delimiter.leftRightDelim(group.left, innerHeight, innerDepth, options, group.mode, ["mopen"]);
+    } // Add it to the beginning of the expression
+
+
+    inner.unshift(leftDelim); // Handle middle delimiters
+
+    if (hadMiddle) {
+      for (let i = 1; i < inner.length; i++) {
+        const middleDelim = inner[i]; // Property `isMiddle` not defined on `span`. See comment in
+        // "middle"'s htmlBuilder.
+        // $FlowFixMe
+
+        const isMiddle = middleDelim.isMiddle;
+
+        if (isMiddle) {
+          // Apply the options that were active when \middle was called
+          inner[i] = delimiter.leftRightDelim(isMiddle.delim, innerHeight, innerDepth, isMiddle.options, group.mode, []);
+        }
+      }
+    }
+
+    let rightDelim; // Same for the right delimiter, but using color specified by \color
+
+    if (group.right === ".") {
+      rightDelim = makeNullDelimiter(options, ["mclose"]);
+    } else {
+      const colorOptions = group.rightColor ? options.withColor(group.rightColor) : options;
+      rightDelim = delimiter.leftRightDelim(group.right, innerHeight, innerDepth, colorOptions, group.mode, ["mclose"]);
+    } // Add it to the end of the expression.
+
+
+    inner.push(rightDelim);
+    return buildCommon.makeSpan(["minner"], inner, options);
+  },
+  mathmlBuilder: (group, options) => {
+    assertParsed(group);
+    const inner = buildExpression$1(group.body, options);
+
+    if (group.left !== ".") {
+      const leftNode = new mathMLTree.MathNode("mo", [makeText(group.left, group.mode)]);
+      leftNode.setAttribute("fence", "true");
+      inner.unshift(leftNode);
+    }
+
+    if (group.right !== ".") {
+      const rightNode = new mathMLTree.MathNode("mo", [makeText(group.right, group.mode)]);
+      rightNode.setAttribute("fence", "true");
+
+      if (group.rightColor) {
+        rightNode.setAttribute("mathcolor", group.rightColor);
+      }
+
+      inner.push(rightNode);
+    }
+
+    return makeRow(inner);
+  }
+});
+defineFunction({
+  type: "middle",
+  names: ["\\middle"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    const delim = checkDelimiter(args[0], context);
+
+    if (!context.parser.leftrightDepth) {
+      throw new ParseError("\\middle without preceding \\left", delim);
+    }
+
+    return {
+      type: "middle",
+      mode: context.parser.mode,
+      delim: delim.text
+    };
+  },
+  htmlBuilder: (group, options) => {
+    let middleDelim;
+
+    if (group.delim === ".") {
+      middleDelim = makeNullDelimiter(options, []);
+    } else {
+      middleDelim = delimiter.sizedDelim(group.delim, 1, options, group.mode, []);
+      const isMiddle = {
+        delim: group.delim,
+        options
+      }; // Property `isMiddle` not defined on `span`. It is only used in
+      // this file above.
+      // TODO: Fix this violation of the `span` type and possibly rename
+      // things since `isMiddle` sounds like a boolean, but is a struct.
+      // $FlowFixMe
+
+      middleDelim.isMiddle = isMiddle;
+    }
+
+    return middleDelim;
+  },
+  mathmlBuilder: (group, options) => {
+    // A Firefox \middle will strech a character vertically only if it
+    // is in the fence part of the operator dictionary at:
+    // https://www.w3.org/TR/MathML3/appendixc.html.
+    // So we need to avoid U+2223 and use plain "|" instead.
+    const textNode = group.delim === "\\vert" || group.delim === "|" ? makeText("|", "text") : makeText(group.delim, group.mode);
+    const middleNode = new mathMLTree.MathNode("mo", [textNode]);
+    middleNode.setAttribute("fence", "true"); // MathML gives 5/18em spacing to each <mo> element.
+    // \middle should get delimiter spacing instead.
+
+    middleNode.setAttribute("lspace", "0.05em");
+    middleNode.setAttribute("rspace", "0.05em");
+    return middleNode;
+  }
+});
+
+const htmlBuilder$2 = (group, options) => {
+  // \cancel, \bcancel, \xcancel, \sout, \fbox, \colorbox, \fcolorbox
+  // Some groups can return document fragments.  Handle those by wrapping
+  // them in a span.
+  const inner = buildCommon.wrapFragment(buildGroup(group.body, options), options);
+  const label = group.label.substr(1);
+  const scale = options.sizeMultiplier;
+  let img;
+  let imgShift = 0; // In the LaTeX cancel package, line geometry is slightly different
+  // depending on whether the subject is wider than it is tall, or vice versa.
+  // We don't know the width of a group, so as a proxy, we test if
+  // the subject is a single character. This captures most of the
+  // subjects that should get the "tall" treatment.
+
+  const isSingleChar = utils.isCharacterBox(group.body);
+
+  if (label === "sout") {
+    img = buildCommon.makeSpan(["stretchy", "sout"]);
+    img.height = options.fontMetrics().defaultRuleThickness / scale;
+    imgShift = -0.5 * options.fontMetrics().xHeight;
+  } else {
+    // Add horizontal padding
+    if (/cancel/.test(label)) {
+      if (!isSingleChar) {
+        inner.classes.push("cancel-pad");
+      }
+    } else {
+      inner.classes.push("boxpad");
+    } // Add vertical padding
+
+
+    let vertPad = 0;
+    let ruleThickness = 0; // ref: cancel package: \advance\totalheight2\p@ % "+2"
+
+    if (/box/.test(label)) {
+      ruleThickness = Math.max(options.fontMetrics().fboxrule, // default
+      options.minRuleThickness // User override.
+      );
+      vertPad = options.fontMetrics().fboxsep + (label === "colorbox" ? 0 : ruleThickness);
+    } else {
+      vertPad = isSingleChar ? 0.2 : 0;
+    }
+
+    img = stretchy.encloseSpan(inner, label, vertPad, options);
+
+    if (/fbox|boxed|fcolorbox/.test(label)) {
+      img.style.borderStyle = "solid";
+      img.style.borderWidth = `${ruleThickness}em`;
+    }
+
+    imgShift = inner.depth + vertPad;
+
+    if (group.backgroundColor) {
+      img.style.backgroundColor = group.backgroundColor;
+
+      if (group.borderColor) {
+        img.style.borderColor = group.borderColor;
+      }
+    }
+  }
+
+  let vlist;
+
+  if (group.backgroundColor) {
+    vlist = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [// Put the color background behind inner;
+      {
+        type: "elem",
+        elem: img,
+        shift: imgShift
+      }, {
+        type: "elem",
+        elem: inner,
+        shift: 0
+      }]
+    }, options);
+  } else {
+    vlist = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [// Write the \cancel stroke on top of inner.
+      {
+        type: "elem",
+        elem: inner,
+        shift: 0
+      }, {
+        type: "elem",
+        elem: img,
+        shift: imgShift,
+        wrapperClasses: /cancel/.test(label) ? ["svg-align"] : []
+      }]
+    }, options);
+  }
+
+  if (/cancel/.test(label)) {
+    // The cancel package documentation says that cancel lines add their height
+    // to the expression, but tests show that isn't how it actually works.
+    vlist.height = inner.height;
+    vlist.depth = inner.depth;
+  }
+
+  if (/cancel/.test(label) && !isSingleChar) {
+    // cancel does not create horiz space for its line extension.
+    return buildCommon.makeSpan(["mord", "cancel-lap"], [vlist], options);
+  } else {
+    return buildCommon.makeSpan(["mord"], [vlist], options);
+  }
+};
+
+const mathmlBuilder$2 = (group, options) => {
+  let fboxsep = 0;
+  const node = new mathMLTree.MathNode(group.label.indexOf("colorbox") > -1 ? "mpadded" : "menclose", [buildGroup$1(group.body, options)]);
+
+  switch (group.label) {
+    case "\\cancel":
+      node.setAttribute("notation", "updiagonalstrike");
+      break;
+
+    case "\\bcancel":
+      node.setAttribute("notation", "downdiagonalstrike");
+      break;
+
+    case "\\sout":
+      node.setAttribute("notation", "horizontalstrike");
+      break;
+
+    case "\\fbox":
+      node.setAttribute("notation", "box");
+      break;
+
+    case "\\fcolorbox":
+    case "\\colorbox":
+      // <menclose> doesn't have a good notation option. So use <mpadded>
+      // instead. Set some attributes that come included with <menclose>.
+      fboxsep = options.fontMetrics().fboxsep * options.fontMetrics().ptPerEm;
+      node.setAttribute("width", `+${2 * fboxsep}pt`);
+      node.setAttribute("height", `+${2 * fboxsep}pt`);
+      node.setAttribute("lspace", `${fboxsep}pt`); //
+
+      node.setAttribute("voffset", `${fboxsep}pt`);
+
+      if (group.label === "\\fcolorbox") {
+        const thk = Math.max(options.fontMetrics().fboxrule, // default
+        options.minRuleThickness // user override
+        );
+        node.setAttribute("style", "border: " + thk + "em solid " + String(group.borderColor));
+      }
+
+      break;
+
+    case "\\xcancel":
+      node.setAttribute("notation", "updiagonalstrike downdiagonalstrike");
+      break;
+  }
+
+  if (group.backgroundColor) {
+    node.setAttribute("mathbackground", group.backgroundColor);
+  }
+
+  return node;
+};
+
+defineFunction({
+  type: "enclose",
+  names: ["\\colorbox"],
+  props: {
+    numArgs: 2,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "text"]
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const color = assertNodeType(args[0], "color-token").color;
+    const body = args[1];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      backgroundColor: color,
+      body
+    };
+  },
+
+  htmlBuilder: htmlBuilder$2,
+  mathmlBuilder: mathmlBuilder$2
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\fcolorbox"],
+  props: {
+    numArgs: 3,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "color", "text"]
+  },
+
+  handler(_ref2, args, optArgs) {
+    let parser = _ref2.parser,
+        funcName = _ref2.funcName;
+    const borderColor = assertNodeType(args[0], "color-token").color;
+    const backgroundColor = assertNodeType(args[1], "color-token").color;
+    const body = args[2];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      backgroundColor,
+      borderColor,
+      body
+    };
+  },
+
+  htmlBuilder: htmlBuilder$2,
+  mathmlBuilder: mathmlBuilder$2
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\fbox"],
+  props: {
+    numArgs: 1,
+    argTypes: ["hbox"],
+    allowedInText: true
+  },
+
+  handler(_ref3, args) {
+    let parser = _ref3.parser;
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: "\\fbox",
+      body: args[0]
+    };
+  }
+
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\cancel", "\\bcancel", "\\xcancel", "\\sout"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(_ref4, args, optArgs) {
+    let parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    const body = args[0];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      body
+    };
+  },
+
+  htmlBuilder: htmlBuilder$2,
+  mathmlBuilder: mathmlBuilder$2
+});
+
+/**
+ * All registered environments.
+ * `environments.js` exports this same dictionary again and makes it public.
+ * `Parser.js` requires this dictionary via `environments.js`.
+ */
+const _environments = {};
+function defineEnvironment(_ref) {
+  let type = _ref.type,
+      names = _ref.names,
+      props = _ref.props,
+      handler = _ref.handler,
+      htmlBuilder = _ref.htmlBuilder,
+      mathmlBuilder = _ref.mathmlBuilder;
+  // Set default values of environments.
+  const data = {
+    type,
+    numArgs: props.numArgs || 0,
+    greediness: 1,
+    allowedInText: false,
+    numOptionalArgs: 0,
+    handler
+  };
+
+  for (let i = 0; i < names.length; ++i) {
+    // TODO: The value type of _environments should be a type union of all
+    // possible `EnvSpec<>` possibilities instead of `EnvSpec<*>`, which is
+    // an existential type.
+    // $FlowFixMe
+    _environments[names[i]] = data;
+  }
+
+  if (htmlBuilder) {
+    _htmlGroupBuilders[type] = htmlBuilder;
+  }
+
+  if (mathmlBuilder) {
+    _mathmlGroupBuilders[type] = mathmlBuilder;
+  }
+}
+
+function getHLines(parser) {
+  // Return an array. The array length = number of hlines.
+  // Each element in the array tells if the line is dashed.
+  const hlineInfo = [];
+  parser.consumeSpaces();
+  let nxt = parser.fetch().text;
+
+  while (nxt === "\\hline" || nxt === "\\hdashline") {
+    parser.consume();
+    hlineInfo.push(nxt === "\\hdashline");
+    parser.consumeSpaces();
+    nxt = parser.fetch().text;
+  }
+
+  return hlineInfo;
+}
+/**
+ * Parse the body of the environment, with rows delimited by \\ and
+ * columns delimited by &, and create a nested list in row-major order
+ * with one group per cell.  If given an optional argument style
+ * ("text", "display", etc.), then each cell is cast into that style.
+ */
+
+
+function parseArray(parser, _ref, style) {
+  let hskipBeforeAndAfter = _ref.hskipBeforeAndAfter,
+      addJot = _ref.addJot,
+      cols = _ref.cols,
+      arraystretch = _ref.arraystretch,
+      colSeparationType = _ref.colSeparationType;
+  // Parse body of array with \\ temporarily mapped to \cr
+  parser.gullet.beginGroup();
+  parser.gullet.macros.set("\\\\", "\\cr"); // Get current arraystretch if it's not set by the environment
+
+  if (!arraystretch) {
+    const stretch = parser.gullet.expandMacroAsText("\\arraystretch");
+
+    if (stretch == null) {
+      // Default \arraystretch from lttab.dtx
+      arraystretch = 1;
+    } else {
+      arraystretch = parseFloat(stretch);
+
+      if (!arraystretch || arraystretch < 0) {
+        throw new ParseError(`Invalid \\arraystretch: ${stretch}`);
+      }
+    }
+  } // Start group for first cell
+
+
+  parser.gullet.beginGroup();
+  let row = [];
+  const body = [row];
+  const rowGaps = [];
+  const hLinesBeforeRow = []; // Test for \hline at the top of the array.
+
+  hLinesBeforeRow.push(getHLines(parser));
+
+  while (true) {
+    // eslint-disable-line no-constant-condition
+    // Parse each cell in its own group (namespace)
+    let cell = parser.parseExpression(false, "\\cr");
+    parser.gullet.endGroup();
+    parser.gullet.beginGroup();
+    cell = {
+      type: "ordgroup",
+      mode: parser.mode,
+      body: cell
+    };
+
+    if (style) {
+      cell = {
+        type: "styling",
+        mode: parser.mode,
+        style,
+        body: [cell]
+      };
+    }
+
+    row.push(cell);
+    const next = parser.fetch().text;
+
+    if (next === "&") {
+      parser.consume();
+    } else if (next === "\\end") {
+      // Arrays terminate newlines with `\crcr` which consumes a `\cr` if
+      // the last line is empty.
+      // NOTE: Currently, `cell` is the last item added into `row`.
+      if (row.length === 1 && cell.type === "styling" && cell.body[0].body.length === 0) {
+        body.pop();
+      }
+
+      if (hLinesBeforeRow.length < body.length + 1) {
+        hLinesBeforeRow.push([]);
+      }
+
+      break;
+    } else if (next === "\\cr") {
+      const cr = assertNodeType(parser.parseFunction(), "cr");
+      rowGaps.push(cr.size); // check for \hline(s) following the row separator
+
+      hLinesBeforeRow.push(getHLines(parser));
+      row = [];
+      body.push(row);
+    } else {
+      throw new ParseError("Expected & or \\\\ or \\cr or \\end", parser.nextToken);
+    }
+  } // End cell group
+
+
+  parser.gullet.endGroup(); // End array group defining \\
+
+  parser.gullet.endGroup();
+  return {
+    type: "array",
+    mode: parser.mode,
+    addJot,
+    arraystretch,
+    body,
+    cols,
+    rowGaps,
+    hskipBeforeAndAfter,
+    hLinesBeforeRow,
+    colSeparationType
+  };
+} // Decides on a style for cells in an array according to whether the given
+// environment name starts with the letter 'd'.
+
+
+function dCellStyle(envName) {
+  if (envName.substr(0, 1) === "d") {
+    return "display";
+  } else {
+    return "text";
+  }
+}
+
+const htmlBuilder$3 = function htmlBuilder(group, options) {
+  let r;
+  let c;
+  const nr = group.body.length;
+  const hLinesBeforeRow = group.hLinesBeforeRow;
+  let nc = 0;
+  let body = new Array(nr);
+  const hlines = [];
+  const ruleThickness = Math.max( // From LaTeX \showthe\arrayrulewidth. Equals 0.04 em.
+  options.fontMetrics().arrayRuleWidth, options.minRuleThickness // User override.
+  ); // Horizontal spacing
+
+  const pt = 1 / options.fontMetrics().ptPerEm;
+  let arraycolsep = 5 * pt; // default value, i.e. \arraycolsep in article.cls
+
+  if (group.colSeparationType && group.colSeparationType === "small") {
+    // We're in a {smallmatrix}. Default column space is \thickspace,
+    // i.e. 5/18em = 0.2778em, per amsmath.dtx for {smallmatrix}.
+    // But that needs adjustment because LaTeX applies \scriptstyle to the
+    // entire array, including the colspace, but this function applies
+    // \scriptstyle only inside each element.
+    const localMultiplier = options.havingStyle(Style$1.SCRIPT).sizeMultiplier;
+    arraycolsep = 0.2778 * (localMultiplier / options.sizeMultiplier);
+  } // Vertical spacing
+
+
+  const baselineskip = 12 * pt; // see size10.clo
+  // Default \jot from ltmath.dtx
+  // TODO(edemaine): allow overriding \jot via \setlength (#687)
+
+  const jot = 3 * pt;
+  const arrayskip = group.arraystretch * baselineskip;
+  const arstrutHeight = 0.7 * arrayskip; // \strutbox in ltfsstrc.dtx and
+
+  const arstrutDepth = 0.3 * arrayskip; // \@arstrutbox in lttab.dtx
+
+  let totalHeight = 0; // Set a position for \hline(s) at the top of the array, if any.
+
+  function setHLinePos(hlinesInGap) {
+    for (let i = 0; i < hlinesInGap.length; ++i) {
+      if (i > 0) {
+        totalHeight += 0.25;
+      }
+
+      hlines.push({
+        pos: totalHeight,
+        isDashed: hlinesInGap[i]
+      });
+    }
+  }
+
+  setHLinePos(hLinesBeforeRow[0]);
+
+  for (r = 0; r < group.body.length; ++r) {
+    const inrow = group.body[r];
+    let height = arstrutHeight; // \@array adds an \@arstrut
+
+    let depth = arstrutDepth; // to each tow (via the template)
+
+    if (nc < inrow.length) {
+      nc = inrow.length;
+    }
+
+    const outrow = new Array(inrow.length);
+
+    for (c = 0; c < inrow.length; ++c) {
+      const elt = buildGroup(inrow[c], options);
+
+      if (depth < elt.depth) {
+        depth = elt.depth;
+      }
+
+      if (height < elt.height) {
+        height = elt.height;
+      }
+
+      outrow[c] = elt;
+    }
+
+    const rowGap = group.rowGaps[r];
+    let gap = 0;
+
+    if (rowGap) {
+      gap = calculateSize(rowGap, options);
+
+      if (gap > 0) {
+        // \@argarraycr
+        gap += arstrutDepth;
+
+        if (depth < gap) {
+          depth = gap; // \@xargarraycr
+        }
+
+        gap = 0;
+      }
+    } // In AMS multiline environments such as aligned and gathered, rows
+    // correspond to lines that have additional \jot added to the
+    // \baselineskip via \openup.
+
+
+    if (group.addJot) {
+      depth += jot;
+    }
+
+    outrow.height = height;
+    outrow.depth = depth;
+    totalHeight += height;
+    outrow.pos = totalHeight;
+    totalHeight += depth + gap; // \@yargarraycr
+
+    body[r] = outrow; // Set a position for \hline(s), if any.
+
+    setHLinePos(hLinesBeforeRow[r + 1]);
+  }
+
+  const offset = totalHeight / 2 + options.fontMetrics().axisHeight;
+  const colDescriptions = group.cols || [];
+  const cols = [];
+  let colSep;
+  let colDescrNum;
+
+  for (c = 0, colDescrNum = 0; // Continue while either there are more columns or more column
+  // descriptions, so trailing separators don't get lost.
+  c < nc || colDescrNum < colDescriptions.length; ++c, ++colDescrNum) {
+    let colDescr = colDescriptions[colDescrNum] || {};
+    let firstSeparator = true;
+
+    while (colDescr.type === "separator") {
+      // If there is more than one separator in a row, add a space
+      // between them.
+      if (!firstSeparator) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = options.fontMetrics().doubleRuleSep + "em";
+        cols.push(colSep);
+      }
+
+      if (colDescr.separator === "|" || colDescr.separator === ":") {
+        const lineType = colDescr.separator === "|" ? "solid" : "dashed";
+        const separator = buildCommon.makeSpan(["vertical-separator"], [], options);
+        separator.style.height = totalHeight + "em";
+        separator.style.borderRightWidth = `${ruleThickness}em`;
+        separator.style.borderRightStyle = lineType;
+        separator.style.margin = `0 -${ruleThickness / 2}em`;
+        separator.style.verticalAlign = -(totalHeight - offset) + "em";
+        cols.push(separator);
+      } else {
+        throw new ParseError("Invalid separator type: " + colDescr.separator);
+      }
+
+      colDescrNum++;
+      colDescr = colDescriptions[colDescrNum] || {};
+      firstSeparator = false;
+    }
+
+    if (c >= nc) {
+      continue;
+    }
+
+    let sepwidth;
+
+    if (c > 0 || group.hskipBeforeAndAfter) {
+      sepwidth = utils.deflt(colDescr.pregap, arraycolsep);
+
+      if (sepwidth !== 0) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = sepwidth + "em";
+        cols.push(colSep);
+      }
+    }
+
+    let col = [];
+
+    for (r = 0; r < nr; ++r) {
+      const row = body[r];
+      const elem = row[c];
+
+      if (!elem) {
+        continue;
+      }
+
+      const shift = row.pos - offset;
+      elem.depth = row.depth;
+      elem.height = row.height;
+      col.push({
+        type: "elem",
+        elem: elem,
+        shift: shift
+      });
+    }
+
+    col = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: col
+    }, options);
+    col = buildCommon.makeSpan(["col-align-" + (colDescr.align || "c")], [col]);
+    cols.push(col);
+
+    if (c < nc - 1 || group.hskipBeforeAndAfter) {
+      sepwidth = utils.deflt(colDescr.postgap, arraycolsep);
+
+      if (sepwidth !== 0) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = sepwidth + "em";
+        cols.push(colSep);
+      }
+    }
+  }
+
+  body = buildCommon.makeSpan(["mtable"], cols); // Add \hline(s), if any.
+
+  if (hlines.length > 0) {
+    const line = buildCommon.makeLineSpan("hline", options, ruleThickness);
+    const dashes = buildCommon.makeLineSpan("hdashline", options, ruleThickness);
+    const vListElems = [{
+      type: "elem",
+      elem: body,
+      shift: 0
+    }];
+
+    while (hlines.length > 0) {
+      const hline = hlines.pop();
+      const lineShift = hline.pos - offset;
+
+      if (hline.isDashed) {
+        vListElems.push({
+          type: "elem",
+          elem: dashes,
+          shift: lineShift
+        });
+      } else {
+        vListElems.push({
+          type: "elem",
+          elem: line,
+          shift: lineShift
+        });
+      }
+    }
+
+    body = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: vListElems
+    }, options);
+  }
+
+  return buildCommon.makeSpan(["mord"], [body], options);
+};
+
+const alignMap = {
+  c: "center ",
+  l: "left ",
+  r: "right "
+};
+
+const mathmlBuilder$3 = function mathmlBuilder(group, options) {
+  let table = new mathMLTree.MathNode("mtable", group.body.map(function (row) {
+    return new mathMLTree.MathNode("mtr", row.map(function (cell) {
+      return new mathMLTree.MathNode("mtd", [buildGroup$1(cell, options)]);
+    }));
+  })); // Set column alignment, row spacing, column spacing, and
+  // array lines by setting attributes on the table element.
+  // Set the row spacing. In MathML, we specify a gap distance.
+  // We do not use rowGap[] because MathML automatically increases
+  // cell height with the height/depth of the element content.
+  // LaTeX \arraystretch multiplies the row baseline-to-baseline distance.
+  // We simulate this by adding (arraystretch - 1)em to the gap. This
+  // does a reasonable job of adjusting arrays containing 1 em tall content.
+  // The 0.16 and 0.09 values are found emprically. They produce an array
+  // similar to LaTeX and in which content does not interfere with \hines.
+
+  const gap = group.arraystretch === 0.5 ? 0.1 // {smallmatrix}, {subarray}
+  : 0.16 + group.arraystretch - 1 + (group.addJot ? 0.09 : 0);
+  table.setAttribute("rowspacing", gap + "em"); // MathML table lines go only between cells.
+  // To place a line on an edge we'll use <menclose>, if necessary.
+
+  let menclose = "";
+  let align = "";
+
+  if (group.cols && group.cols.length > 0) {
+    // Find column alignment, column spacing, and  vertical lines.
+    const cols = group.cols;
+    let columnLines = "";
+    let prevTypeWasAlign = false;
+    let iStart = 0;
+    let iEnd = cols.length;
+
+    if (cols[0].type === "separator") {
+      menclose += "top ";
+      iStart = 1;
+    }
+
+    if (cols[cols.length - 1].type === "separator") {
+      menclose += "bottom ";
+      iEnd -= 1;
+    }
+
+    for (let i = iStart; i < iEnd; i++) {
+      if (cols[i].type === "align") {
+        align += alignMap[cols[i].align];
+
+        if (prevTypeWasAlign) {
+          columnLines += "none ";
+        }
+
+        prevTypeWasAlign = true;
+      } else if (cols[i].type === "separator") {
+        // MathML accepts only single lines between cells.
+        // So we read only the first of consecutive separators.
+        if (prevTypeWasAlign) {
+          columnLines += cols[i].separator === "|" ? "solid " : "dashed ";
+          prevTypeWasAlign = false;
+        }
+      }
+    }
+
+    table.setAttribute("columnalign", align.trim());
+
+    if (/[sd]/.test(columnLines)) {
+      table.setAttribute("columnlines", columnLines.trim());
+    }
+  } // Set column spacing.
+
+
+  if (group.colSeparationType === "align") {
+    const cols = group.cols || [];
+    let spacing = "";
+
+    for (let i = 1; i < cols.length; i++) {
+      spacing += i % 2 ? "0em " : "1em ";
+    }
+
+    table.setAttribute("columnspacing", spacing.trim());
+  } else if (group.colSeparationType === "alignat") {
+    table.setAttribute("columnspacing", "0em");
+  } else if (group.colSeparationType === "small") {
+    table.setAttribute("columnspacing", "0.2778em");
+  } else {
+    table.setAttribute("columnspacing", "1em");
+  } // Address \hline and \hdashline
+
+
+  let rowLines = "";
+  const hlines = group.hLinesBeforeRow;
+  menclose += hlines[0].length > 0 ? "left " : "";
+  menclose += hlines[hlines.length - 1].length > 0 ? "right " : "";
+
+  for (let i = 1; i < hlines.length - 1; i++) {
+    rowLines += hlines[i].length === 0 ? "none " // MathML accepts only a single line between rows. Read one element.
+    : hlines[i][0] ? "dashed " : "solid ";
+  }
+
+  if (/[sd]/.test(rowLines)) {
+    table.setAttribute("rowlines", rowLines.trim());
+  }
+
+  if (menclose !== "") {
+    table = new mathMLTree.MathNode("menclose", [table]);
+    table.setAttribute("notation", menclose.trim());
+  }
+
+  if (group.arraystretch && group.arraystretch < 1) {
+    // A small array. Wrap in scriptstyle so row gap is not too large.
+    table = new mathMLTree.MathNode("mstyle", [table]);
+    table.setAttribute("scriptlevel", "1");
+  }
+
+  return table;
+}; // Convenience function for aligned and alignedat environments.
+
+
+const alignedHandler = function alignedHandler(context, args) {
+  const cols = [];
+  const res = parseArray(context.parser, {
+    cols,
+    addJot: true
+  }, "display"); // Determining number of columns.
+  // 1. If the first argument is given, we use it as a number of columns,
+  //    and makes sure that each row doesn't exceed that number.
+  // 2. Otherwise, just count number of columns = maximum number
+  //    of cells in each row ("aligned" mode -- isAligned will be true).
+  //
+  // At the same time, prepend empty group {} at beginning of every second
+  // cell in each row (starting with second cell) so that operators become
+  // binary.  This behavior is implemented in amsmath's \start@aligned.
+
+  let numMaths;
+  let numCols = 0;
+  const emptyGroup = {
+    type: "ordgroup",
+    mode: context.mode,
+    body: []
+  };
+
+  if (args[0] && args[0].type === "ordgroup") {
+    let arg0 = "";
+
+    for (let i = 0; i < args[0].body.length; i++) {
+      const textord = assertNodeType(args[0].body[i], "textord");
+      arg0 += textord.text;
+    }
+
+    numMaths = Number(arg0);
+    numCols = numMaths * 2;
+  }
+
+  const isAligned = !numCols;
+  res.body.forEach(function (row) {
+    for (let i = 1; i < row.length; i += 2) {
+      // Modify ordgroup node within styling node
+      const styling = assertNodeType(row[i], "styling");
+      const ordgroup = assertNodeType(styling.body[0], "ordgroup");
+      ordgroup.body.unshift(emptyGroup);
+    }
+
+    if (!isAligned) {
+      // Case 1
+      const curMaths = row.length / 2;
+
+      if (numMaths < curMaths) {
+        throw new ParseError("Too many math in a row: " + `expected ${numMaths}, but got ${curMaths}`, row[0]);
+      }
+    } else if (numCols < row.length) {
+      // Case 2
+      numCols = row.length;
+    }
+  }); // Adjusting alignment.
+  // In aligned mode, we add one \qquad between columns;
+  // otherwise we add nothing.
+
+  for (let i = 0; i < numCols; ++i) {
+    let align = "r";
+    let pregap = 0;
+
+    if (i % 2 === 1) {
+      align = "l";
+    } else if (i > 0 && isAligned) {
+      // "aligned" mode.
+      pregap = 1; // add one \quad
+    }
+
+    cols[i] = {
+      type: "align",
+      align: align,
+      pregap: pregap,
+      postgap: 0
+    };
+  }
+
+  res.colSeparationType = isAligned ? "align" : "alignat";
+  return res;
+}; // Arrays are part of LaTeX, defined in lttab.dtx so its documentation
+// is part of the source2e.pdf file of LaTeX2e source documentation.
+// {darray} is an {array} environment where cells are set in \displaystyle,
+// as defined in nccmath.sty.
+
+
+defineEnvironment({
+  type: "array",
+  names: ["array", "darray"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(context, args) {
+    // Since no types are specified above, the two possibilities are
+    // - The argument is wrapped in {} or [], in which case Parser's
+    //   parseGroup() returns an "ordgroup" wrapping some symbol node.
+    // - The argument is a bare symbol node.
+    const symNode = checkSymbolNodeType(args[0]);
+    const colalign = symNode ? [args[0]] : assertNodeType(args[0], "ordgroup").body;
+    const cols = colalign.map(function (nde) {
+      const node = assertSymbolNodeType(nde);
+      const ca = node.text;
+
+      if ("lcr".indexOf(ca) !== -1) {
+        return {
+          type: "align",
+          align: ca
+        };
+      } else if (ca === "|") {
+        return {
+          type: "separator",
+          separator: "|"
+        };
+      } else if (ca === ":") {
+        return {
+          type: "separator",
+          separator: ":"
+        };
+      }
+
+      throw new ParseError("Unknown column alignment: " + ca, nde);
+    });
+    const res = {
+      cols,
+      hskipBeforeAndAfter: true // \@preamble in lttab.dtx
+
+    };
+    return parseArray(context.parser, res, dCellStyle(context.envName));
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // The matrix environments of amsmath builds on the array environment
+// of LaTeX, which is discussed above.
+
+defineEnvironment({
+  type: "array",
+  names: ["matrix", "pmatrix", "bmatrix", "Bmatrix", "vmatrix", "Vmatrix"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(context) {
+    const delimiters = {
+      "matrix": null,
+      "pmatrix": ["(", ")"],
+      "bmatrix": ["[", "]"],
+      "Bmatrix": ["\\{", "\\}"],
+      "vmatrix": ["|", "|"],
+      "Vmatrix": ["\\Vert", "\\Vert"]
+    }[context.envName]; // \hskip -\arraycolsep in amsmath
+
+    const payload = {
+      hskipBeforeAndAfter: false
+    };
+    const res = parseArray(context.parser, payload, dCellStyle(context.envName));
+    return delimiters ? {
+      type: "leftright",
+      mode: context.mode,
+      body: [res],
+      left: delimiters[0],
+      right: delimiters[1],
+      rightColor: undefined // \right uninfluenced by \color in array
+
+    } : res;
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+});
+defineEnvironment({
+  type: "array",
+  names: ["smallmatrix"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(context) {
+    const payload = {
+      arraystretch: 0.5
+    };
+    const res = parseArray(context.parser, payload, "script");
+    res.colSeparationType = "small";
+    return res;
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+});
+defineEnvironment({
+  type: "array",
+  names: ["subarray"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(context, args) {
+    // Parsing of {subarray} is similar to {array}
+    const symNode = checkSymbolNodeType(args[0]);
+    const colalign = symNode ? [args[0]] : assertNodeType(args[0], "ordgroup").body;
+    const cols = colalign.map(function (nde) {
+      const node = assertSymbolNodeType(nde);
+      const ca = node.text; // {subarray} only recognizes "l" & "c"
+
+      if ("lc".indexOf(ca) !== -1) {
+        return {
+          type: "align",
+          align: ca
+        };
+      }
+
+      throw new ParseError("Unknown column alignment: " + ca, nde);
+    });
+
+    if (cols.length > 1) {
+      throw new ParseError("{subarray} can contain only one column");
+    }
+
+    let res = {
+      cols,
+      hskipBeforeAndAfter: false,
+      arraystretch: 0.5
+    };
+    res = parseArray(context.parser, res, "script");
+
+    if (res.body.length > 0 && res.body[0].length > 1) {
+      throw new ParseError("{subarray} can contain only one column");
+    }
+
+    return res;
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // A cases environment (in amsmath.sty) is almost equivalent to
+// \def\arraystretch{1.2}%
+// \left\{\begin{array}{@{}l@{\quad}l@{}} … \end{array}\right.
+// {dcases} is a {cases} environment where cells are set in \displaystyle,
+// as defined in mathtools.sty.
+// {rcases} is another mathtools environment. It's brace is on the right side.
+
+defineEnvironment({
+  type: "array",
+  names: ["cases", "dcases", "rcases", "drcases"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(context) {
+    const payload = {
+      arraystretch: 1.2,
+      cols: [{
+        type: "align",
+        align: "l",
+        pregap: 0,
+        // TODO(kevinb) get the current style.
+        // For now we use the metrics for TEXT style which is what we were
+        // doing before.  Before attempting to get the current style we
+        // should look at TeX's behavior especially for \over and matrices.
+        postgap: 1.0
+        /* 1em quad */
+
+      }, {
+        type: "align",
+        align: "l",
+        pregap: 0,
+        postgap: 0
+      }]
+    };
+    const res = parseArray(context.parser, payload, dCellStyle(context.envName));
+    return {
+      type: "leftright",
+      mode: context.mode,
+      body: [res],
+      left: context.envName.indexOf("r") > -1 ? "." : "\\{",
+      right: context.envName.indexOf("r") > -1 ? "\\}" : ".",
+      rightColor: undefined
+    };
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // An aligned environment is like the align* environment
+// except it operates within math mode.
+// Note that we assume \nomallineskiplimit to be zero,
+// so that \strut@ is the same as \strut.
+
+defineEnvironment({
+  type: "array",
+  names: ["aligned"],
+  props: {
+    numArgs: 0
+  },
+  handler: alignedHandler,
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // A gathered environment is like an array environment with one centered
+// column, but where rows are considered lines so get \jot line spacing
+// and contents are set in \displaystyle.
+
+defineEnvironment({
+  type: "array",
+  names: ["gathered"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(context) {
+    const res = {
+      cols: [{
+        type: "align",
+        align: "c"
+      }],
+      addJot: true
+    };
+    return parseArray(context.parser, res, "display");
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // alignat environment is like an align environment, but one must explicitly
+// specify maximum number of columns in each row, and can adjust spacing between
+// each columns.
+
+defineEnvironment({
+  type: "array",
+  names: ["alignedat"],
+  // One for numbered and for unnumbered;
+  // but, KaTeX doesn't supports math numbering yet,
+  // they make no difference for now.
+  props: {
+    numArgs: 1
+  },
+  handler: alignedHandler,
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // Catch \hline outside array environment
+
+defineFunction({
+  type: "text",
+  // Doesn't matter what this is.
+  names: ["\\hline", "\\hdashline"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: true
+  },
+
+  handler(context, args) {
+    throw new ParseError(`${context.funcName} valid only within array environment`);
+  }
+
+});
+
+const environments = _environments;
+
+// defineEnvironment definitions.
+// $FlowFixMe, "environment" handler returns an environment ParseNode
+
+defineFunction({
+  type: "environment",
+  names: ["\\begin", "\\end"],
+  props: {
+    numArgs: 1,
+    argTypes: ["text"]
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const nameGroup = args[0];
+
+    if (nameGroup.type !== "ordgroup") {
+      throw new ParseError("Invalid environment name", nameGroup);
+    }
+
+    let envName = "";
+
+    for (let i = 0; i < nameGroup.body.length; ++i) {
+      envName += assertNodeType(nameGroup.body[i], "textord").text;
+    }
+
+    if (funcName === "\\begin") {
+      // begin...end is similar to left...right
+      if (!environments.hasOwnProperty(envName)) {
+        throw new ParseError("No such environment: " + envName, nameGroup);
+      } // Build the environment object. Arguments and other information will
+      // be made available to the begin and end methods using properties.
+
+
+      const env = environments[envName];
+
+      const _parser$parseArgument = parser.parseArguments("\\begin{" + envName + "}", env),
+            args = _parser$parseArgument.args,
+            optArgs = _parser$parseArgument.optArgs;
+
+      const context = {
+        mode: parser.mode,
+        envName,
+        parser
+      };
+      const result = env.handler(context, args, optArgs);
+      parser.expect("\\end", false);
+      const endNameToken = parser.nextToken;
+      const end = assertNodeType(parser.parseFunction(), "environment");
+
+      if (end.name !== envName) {
+        throw new ParseError(`Mismatch: \\begin{${envName}} matched by \\end{${end.name}}`, endNameToken);
+      }
+
+      return result;
+    }
+
+    return {
+      type: "environment",
+      mode: parser.mode,
+      name: envName,
+      nameGroup
+    };
+  }
+
+});
+
+const makeSpan$2 = buildCommon.makeSpan;
+
+function htmlBuilder$4(group, options) {
+  const elements = buildExpression(group.body, options, true);
+  return makeSpan$2([group.mclass], elements, options);
+}
+
+function mathmlBuilder$4(group, options) {
+  let node;
+  const inner = buildExpression$1(group.body, options);
+
+  if (group.mclass === "minner") {
+    return mathMLTree.newDocumentFragment(inner);
+  } else if (group.mclass === "mord") {
+    if (group.isCharacterBox) {
+      node = inner[0];
+      node.type = "mi";
+    } else {
+      node = new mathMLTree.MathNode("mi", inner);
+    }
+  } else {
+    if (group.isCharacterBox) {
+      node = inner[0];
+      node.type = "mo";
+    } else {
+      node = new mathMLTree.MathNode("mo", inner);
+    } // Set spacing based on what is the most likely adjacent atom type.
+    // See TeXbook p170.
+
+
+    if (group.mclass === "mbin") {
+      node.attributes.lspace = "0.22em"; // medium space
+
+      node.attributes.rspace = "0.22em";
+    } else if (group.mclass === "mpunct") {
+      node.attributes.lspace = "0em";
+      node.attributes.rspace = "0.17em"; // thinspace
+    } else if (group.mclass === "mopen" || group.mclass === "mclose") {
+      node.attributes.lspace = "0em";
+      node.attributes.rspace = "0em";
+    } // MathML <mo> default space is 5/18 em, so <mrel> needs no action.
+    // Ref: https://developer.mozilla.org/en-US/docs/Web/MathML/Element/mo
+
+  }
+
+  return node;
+} // Math class commands except \mathop
+
+
+defineFunction({
+  type: "mclass",
+  names: ["\\mathord", "\\mathbin", "\\mathrel", "\\mathopen", "\\mathclose", "\\mathpunct", "\\mathinner"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: "m" + funcName.substr(5),
+      // TODO(kevinb): don't prefix with 'm'
+      body: ordargument(body),
+      isCharacterBox: utils.isCharacterBox(body)
+    };
+  },
+
+  htmlBuilder: htmlBuilder$4,
+  mathmlBuilder: mathmlBuilder$4
+});
+const binrelClass = arg => {
+  // \binrel@ spacing varies with (bin|rel|ord) of the atom in the argument.
+  // (by rendering separately and with {}s before and after, and measuring
+  // the change in spacing).  We'll do roughly the same by detecting the
+  // atom type directly.
+  const atom = arg.type === "ordgroup" && arg.body.length ? arg.body[0] : arg;
+
+  if (atom.type === "atom" && (atom.family === "bin" || atom.family === "rel")) {
+    return "m" + atom.family;
+  } else {
+    return "mord";
+  }
+}; // \@binrel{x}{y} renders like y but as mbin/mrel/mord if x is mbin/mrel/mord.
+// This is equivalent to \binrel@{x}\binrel@@{y} in AMSTeX.
+
+defineFunction({
+  type: "mclass",
+  names: ["\\@binrel"],
+  props: {
+    numArgs: 2
+  },
+
+  handler(_ref2, args) {
+    let parser = _ref2.parser;
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: binrelClass(args[0]),
+      body: [args[1]],
+      isCharacterBox: utils.isCharacterBox(args[1])
+    };
+  }
+
+}); // Build a relation or stacked op by placing one symbol on top of another
+
+defineFunction({
+  type: "mclass",
+  names: ["\\stackrel", "\\overset", "\\underset"],
+  props: {
+    numArgs: 2
+  },
+
+  handler(_ref3, args) {
+    let parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    const baseArg = args[1];
+    const shiftedArg = args[0];
+    let mclass;
+
+    if (funcName !== "\\stackrel") {
+      // LaTeX applies \binrel spacing to \overset and \underset.
+      mclass = binrelClass(baseArg);
+    } else {
+      mclass = "mrel"; // for \stackrel
+    }
+
+    const baseOp = {
+      type: "op",
+      mode: baseArg.mode,
+      limits: true,
+      alwaysHandleSupSub: true,
+      parentIsSupSub: false,
+      symbol: false,
+      suppressBaseShift: funcName !== "\\stackrel",
+      body: ordargument(baseArg)
+    };
+    const supsub = {
+      type: "supsub",
+      mode: shiftedArg.mode,
+      base: baseOp,
+      sup: funcName === "\\underset" ? null : shiftedArg,
+      sub: funcName === "\\underset" ? shiftedArg : null
+    };
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass,
+      body: [supsub],
+      isCharacterBox: utils.isCharacterBox(supsub)
+    };
+  },
+
+  htmlBuilder: htmlBuilder$4,
+  mathmlBuilder: mathmlBuilder$4
+});
+
+// TODO(kevinb): implement \\sl and \\sc
+
+const htmlBuilder$5 = (group, options) => {
+  const font = group.font;
+  const newOptions = options.withFont(font);
+  return buildGroup(group.body, newOptions);
+};
+
+const mathmlBuilder$5 = (group, options) => {
+  const font = group.font;
+  const newOptions = options.withFont(font);
+  return buildGroup$1(group.body, newOptions);
+};
+
+const fontAliases = {
+  "\\Bbb": "\\mathbb",
+  "\\bold": "\\mathbf",
+  "\\frak": "\\mathfrak",
+  "\\bm": "\\boldsymbol"
+};
+defineFunction({
+  type: "font",
+  names: [// styles, except \boldsymbol defined below
+  "\\mathrm", "\\mathit", "\\mathbf", "\\mathnormal", // families
+  "\\mathbb", "\\mathcal", "\\mathfrak", "\\mathscr", "\\mathsf", "\\mathtt", // aliases, except \bm defined below
+  "\\Bbb", "\\bold", "\\frak"],
+  props: {
+    numArgs: 1,
+    greediness: 2
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    let func = funcName;
+
+    if (func in fontAliases) {
+      func = fontAliases[func];
+    }
+
+    return {
+      type: "font",
+      mode: parser.mode,
+      font: func.slice(1),
+      body
+    };
+  },
+  htmlBuilder: htmlBuilder$5,
+  mathmlBuilder: mathmlBuilder$5
+});
+defineFunction({
+  type: "mclass",
+  names: ["\\boldsymbol", "\\bm"],
+  props: {
+    numArgs: 1,
+    greediness: 2
+  },
+  handler: (_ref2, args) => {
+    let parser = _ref2.parser;
+    const body = args[0];
+    const isCharacterBox = utils.isCharacterBox(body); // amsbsy.sty's \boldsymbol uses \binrel spacing to inherit the
+    // argument's bin|rel|ord status
+
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: binrelClass(body),
+      body: [{
+        type: "font",
+        mode: parser.mode,
+        font: "boldsymbol",
+        body
+      }],
+      isCharacterBox: isCharacterBox
+    };
+  }
+}); // Old font changing functions
+
+defineFunction({
+  type: "font",
+  names: ["\\rm", "\\sf", "\\tt", "\\bf", "\\it", "\\cal"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: (_ref3, args) => {
+    let parser = _ref3.parser,
+        funcName = _ref3.funcName,
+        breakOnTokenText = _ref3.breakOnTokenText;
+    const mode = parser.mode;
+    const body = parser.parseExpression(true, breakOnTokenText);
+    const style = `math${funcName.slice(1)}`;
+    return {
+      type: "font",
+      mode: mode,
+      font: style,
+      body: {
+        type: "ordgroup",
+        mode: parser.mode,
+        body
+      }
+    };
+  },
+  htmlBuilder: htmlBuilder$5,
+  mathmlBuilder: mathmlBuilder$5
+});
+
+const adjustStyle = (size, originalStyle) => {
+  // Figure out what style this fraction should be in based on the
+  // function used
+  let style = originalStyle;
+
+  if (size === "display") {
+    // Get display style as a default.
+    // If incoming style is sub/sup, use style.text() to get correct size.
+    style = style.id >= Style$1.SCRIPT.id ? style.text() : Style$1.DISPLAY;
+  } else if (size === "text" && style.size === Style$1.DISPLAY.size) {
+    // We're in a \tfrac but incoming style is displaystyle, so:
+    style = Style$1.TEXT;
+  } else if (size === "script") {
+    style = Style$1.SCRIPT;
+  } else if (size === "scriptscript") {
+    style = Style$1.SCRIPTSCRIPT;
+  }
+
+  return style;
+};
+
+const htmlBuilder$6 = (group, options) => {
+  // Fractions are handled in the TeXbook on pages 444-445, rules 15(a-e).
+  const style = adjustStyle(group.size, options.style);
+  const nstyle = style.fracNum();
+  const dstyle = style.fracDen();
+  let newOptions;
+  newOptions = options.havingStyle(nstyle);
+  const numerm = buildGroup(group.numer, newOptions, options);
+
+  if (group.continued) {
+    // \cfrac inserts a \strut into the numerator.
+    // Get \strut dimensions from TeXbook page 353.
+    const hStrut = 8.5 / options.fontMetrics().ptPerEm;
+    const dStrut = 3.5 / options.fontMetrics().ptPerEm;
+    numerm.height = numerm.height < hStrut ? hStrut : numerm.height;
+    numerm.depth = numerm.depth < dStrut ? dStrut : numerm.depth;
+  }
+
+  newOptions = options.havingStyle(dstyle);
+  const denomm = buildGroup(group.denom, newOptions, options);
+  let rule;
+  let ruleWidth;
+  let ruleSpacing;
+
+  if (group.hasBarLine) {
+    if (group.barSize) {
+      ruleWidth = calculateSize(group.barSize, options);
+      rule = buildCommon.makeLineSpan("frac-line", options, ruleWidth);
+    } else {
+      rule = buildCommon.makeLineSpan("frac-line", options);
+    }
+
+    ruleWidth = rule.height;
+    ruleSpacing = rule.height;
+  } else {
+    rule = null;
+    ruleWidth = 0;
+    ruleSpacing = options.fontMetrics().defaultRuleThickness;
+  } // Rule 15b
+
+
+  let numShift;
+  let clearance;
+  let denomShift;
+
+  if (style.size === Style$1.DISPLAY.size || group.size === "display") {
+    numShift = options.fontMetrics().num1;
+
+    if (ruleWidth > 0) {
+      clearance = 3 * ruleSpacing;
+    } else {
+      clearance = 7 * ruleSpacing;
+    }
+
+    denomShift = options.fontMetrics().denom1;
+  } else {
+    if (ruleWidth > 0) {
+      numShift = options.fontMetrics().num2;
+      clearance = ruleSpacing;
+    } else {
+      numShift = options.fontMetrics().num3;
+      clearance = 3 * ruleSpacing;
+    }
+
+    denomShift = options.fontMetrics().denom2;
+  }
+
+  let frac;
+
+  if (!rule) {
+    // Rule 15c
+    const candidateClearance = numShift - numerm.depth - (denomm.height - denomShift);
+
+    if (candidateClearance < clearance) {
+      numShift += 0.5 * (clearance - candidateClearance);
+      denomShift += 0.5 * (clearance - candidateClearance);
+    }
+
+    frac = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [{
+        type: "elem",
+        elem: denomm,
+        shift: denomShift
+      }, {
+        type: "elem",
+        elem: numerm,
+        shift: -numShift
+      }]
+    }, options);
+  } else {
+    // Rule 15d
+    const axisHeight = options.fontMetrics().axisHeight;
+
+    if (numShift - numerm.depth - (axisHeight + 0.5 * ruleWidth) < clearance) {
+      numShift += clearance - (numShift - numerm.depth - (axisHeight + 0.5 * ruleWidth));
+    }
+
+    if (axisHeight - 0.5 * ruleWidth - (denomm.height - denomShift) < clearance) {
+      denomShift += clearance - (axisHeight - 0.5 * ruleWidth - (denomm.height - denomShift));
+    }
+
+    const midShift = -(axisHeight - 0.5 * ruleWidth);
+    frac = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [{
+        type: "elem",
+        elem: denomm,
+        shift: denomShift
+      }, {
+        type: "elem",
+        elem: rule,
+        shift: midShift
+      }, {
+        type: "elem",
+        elem: numerm,
+        shift: -numShift
+      }]
+    }, options);
+  } // Since we manually change the style sometimes (with \dfrac or \tfrac),
+  // account for the possible size change here.
+
+
+  newOptions = options.havingStyle(style);
+  frac.height *= newOptions.sizeMultiplier / options.sizeMultiplier;
+  frac.depth *= newOptions.sizeMultiplier / options.sizeMultiplier; // Rule 15e
+
+  let delimSize;
+
+  if (style.size === Style$1.DISPLAY.size) {
+    delimSize = options.fontMetrics().delim1;
+  } else {
+    delimSize = options.fontMetrics().delim2;
+  }
+
+  let leftDelim;
+  let rightDelim;
+
+  if (group.leftDelim == null) {
+    leftDelim = makeNullDelimiter(options, ["mopen"]);
+  } else {
+    leftDelim = delimiter.customSizedDelim(group.leftDelim, delimSize, true, options.havingStyle(style), group.mode, ["mopen"]);
+  }
+
+  if (group.continued) {
+    rightDelim = buildCommon.makeSpan([]); // zero width for \cfrac
+  } else if (group.rightDelim == null) {
+    rightDelim = makeNullDelimiter(options, ["mclose"]);
+  } else {
+    rightDelim = delimiter.customSizedDelim(group.rightDelim, delimSize, true, options.havingStyle(style), group.mode, ["mclose"]);
+  }
+
+  return buildCommon.makeSpan(["mord"].concat(newOptions.sizingClasses(options)), [leftDelim, buildCommon.makeSpan(["mfrac"], [frac]), rightDelim], options);
+};
+
+const mathmlBuilder$6 = (group, options) => {
+  let node = new mathMLTree.MathNode("mfrac", [buildGroup$1(group.numer, options), buildGroup$1(group.denom, options)]);
+
+  if (!group.hasBarLine) {
+    node.setAttribute("linethickness", "0px");
+  } else if (group.barSize) {
+    const ruleWidth = calculateSize(group.barSize, options);
+    node.setAttribute("linethickness", ruleWidth + "em");
+  }
+
+  const style = adjustStyle(group.size, options.style);
+
+  if (style.size !== options.style.size) {
+    node = new mathMLTree.MathNode("mstyle", [node]);
+    const isDisplay = style.size === Style$1.DISPLAY.size ? "true" : "false";
+    node.setAttribute("displaystyle", isDisplay);
+    node.setAttribute("scriptlevel", "0");
+  }
+
+  if (group.leftDelim != null || group.rightDelim != null) {
+    const withDelims = [];
+
+    if (group.leftDelim != null) {
+      const leftOp = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(group.leftDelim.replace("\\", ""))]);
+      leftOp.setAttribute("fence", "true");
+      withDelims.push(leftOp);
+    }
+
+    withDelims.push(node);
+
+    if (group.rightDelim != null) {
+      const rightOp = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(group.rightDelim.replace("\\", ""))]);
+      rightOp.setAttribute("fence", "true");
+      withDelims.push(rightOp);
+    }
+
+    return makeRow(withDelims);
+  }
+
+  return node;
+};
+
+defineFunction({
+  type: "genfrac",
+  names: ["\\cfrac", "\\dfrac", "\\frac", "\\tfrac", "\\dbinom", "\\binom", "\\tbinom", "\\\\atopfrac", // can’t be entered directly
+  "\\\\bracefrac", "\\\\brackfrac"],
+  props: {
+    numArgs: 2,
+    greediness: 2
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const numer = args[0];
+    const denom = args[1];
+    let hasBarLine;
+    let leftDelim = null;
+    let rightDelim = null;
+    let size = "auto";
+
+    switch (funcName) {
+      case "\\cfrac":
+      case "\\dfrac":
+      case "\\frac":
+      case "\\tfrac":
+        hasBarLine = true;
+        break;
+
+      case "\\\\atopfrac":
+        hasBarLine = false;
+        break;
+
+      case "\\dbinom":
+      case "\\binom":
+      case "\\tbinom":
+        hasBarLine = false;
+        leftDelim = "(";
+        rightDelim = ")";
+        break;
+
+      case "\\\\bracefrac":
+        hasBarLine = false;
+        leftDelim = "\\{";
+        rightDelim = "\\}";
+        break;
+
+      case "\\\\brackfrac":
+        hasBarLine = false;
+        leftDelim = "[";
+        rightDelim = "]";
+        break;
+
+      default:
+        throw new Error("Unrecognized genfrac command");
+    }
+
+    switch (funcName) {
+      case "\\cfrac":
+      case "\\dfrac":
+      case "\\dbinom":
+        size = "display";
+        break;
+
+      case "\\tfrac":
+      case "\\tbinom":
+        size = "text";
+        break;
+    }
+
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      continued: funcName === "\\cfrac",
+      numer,
+      denom,
+      hasBarLine,
+      leftDelim,
+      rightDelim,
+      size,
+      barSize: null
+    };
+  },
+  htmlBuilder: htmlBuilder$6,
+  mathmlBuilder: mathmlBuilder$6
+}); // Infix generalized fractions -- these are not rendered directly, but replaced
+// immediately by one of the variants above.
+
+defineFunction({
+  type: "infix",
+  names: ["\\over", "\\choose", "\\atop", "\\brace", "\\brack"],
+  props: {
+    numArgs: 0,
+    infix: true
+  },
+
+  handler(_ref2) {
+    let parser = _ref2.parser,
+        funcName = _ref2.funcName,
+        token = _ref2.token;
+    let replaceWith;
+
+    switch (funcName) {
+      case "\\over":
+        replaceWith = "\\frac";
+        break;
+
+      case "\\choose":
+        replaceWith = "\\binom";
+        break;
+
+      case "\\atop":
+        replaceWith = "\\\\atopfrac";
+        break;
+
+      case "\\brace":
+        replaceWith = "\\\\bracefrac";
+        break;
+
+      case "\\brack":
+        replaceWith = "\\\\brackfrac";
+        break;
+
+      default:
+        throw new Error("Unrecognized infix genfrac command");
+    }
+
+    return {
+      type: "infix",
+      mode: parser.mode,
+      replaceWith,
+      token
+    };
+  }
+
+});
+const stylArray = ["display", "text", "script", "scriptscript"];
+
+const delimFromValue = function delimFromValue(delimString) {
+  let delim = null;
+
+  if (delimString.length > 0) {
+    delim = delimString;
+    delim = delim === "." ? null : delim;
+  }
+
+  return delim;
+};
+
+defineFunction({
+  type: "genfrac",
+  names: ["\\genfrac"],
+  props: {
+    numArgs: 6,
+    greediness: 6,
+    argTypes: ["math", "math", "size", "text", "math", "math"]
+  },
+
+  handler(_ref3, args) {
+    let parser = _ref3.parser;
+    const numer = args[4];
+    const denom = args[5]; // Look into the parse nodes to get the desired delimiters.
+
+    const leftDelim = args[0].type === "atom" && args[0].family === "open" ? delimFromValue(args[0].text) : null;
+    const rightDelim = args[1].type === "atom" && args[1].family === "close" ? delimFromValue(args[1].text) : null;
+    const barNode = assertNodeType(args[2], "size");
+    let hasBarLine;
+    let barSize = null;
+
+    if (barNode.isBlank) {
+      // \genfrac acts differently than \above.
+      // \genfrac treats an empty size group as a signal to use a
+      // standard bar size. \above would see size = 0 and omit the bar.
+      hasBarLine = true;
+    } else {
+      barSize = barNode.value;
+      hasBarLine = barSize.number > 0;
+    } // Find out if we want displaystyle, textstyle, etc.
+
+
+    let size = "auto";
+    let styl = args[3];
+
+    if (styl.type === "ordgroup") {
+      if (styl.body.length > 0) {
+        const textOrd = assertNodeType(styl.body[0], "textord");
+        size = stylArray[Number(textOrd.text)];
+      }
+    } else {
+      styl = assertNodeType(styl, "textord");
+      size = stylArray[Number(styl.text)];
+    }
+
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      numer,
+      denom,
+      continued: false,
+      hasBarLine,
+      barSize,
+      leftDelim,
+      rightDelim,
+      size
+    };
+  },
+
+  htmlBuilder: htmlBuilder$6,
+  mathmlBuilder: mathmlBuilder$6
+}); // \above is an infix fraction that also defines a fraction bar size.
+
+defineFunction({
+  type: "infix",
+  names: ["\\above"],
+  props: {
+    numArgs: 1,
+    argTypes: ["size"],
+    infix: true
+  },
+
+  handler(_ref4, args) {
+    let parser = _ref4.parser,
+        funcName = _ref4.funcName,
+        token = _ref4.token;
+    return {
+      type: "infix",
+      mode: parser.mode,
+      replaceWith: "\\\\abovefrac",
+      size: assertNodeType(args[0], "size").value,
+      token
+    };
+  }
+
+});
+defineFunction({
+  type: "genfrac",
+  names: ["\\\\abovefrac"],
+  props: {
+    numArgs: 3,
+    argTypes: ["math", "size", "math"]
+  },
+  handler: (_ref5, args) => {
+    let parser = _ref5.parser,
+        funcName = _ref5.funcName;
+    const numer = args[0];
+    const barSize = assert(assertNodeType(args[1], "infix").size);
+    const denom = args[2];
+    const hasBarLine = barSize.number > 0;
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      numer,
+      denom,
+      continued: false,
+      hasBarLine,
+      barSize,
+      leftDelim: null,
+      rightDelim: null,
+      size: "auto"
+    };
+  },
+  htmlBuilder: htmlBuilder$6,
+  mathmlBuilder: mathmlBuilder$6
+});
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only "horizBrace", but
+const htmlBuilder$7 = (grp, options) => {
+  const style = options.style; // Pull out the `ParseNode<"horizBrace">` if `grp` is a "supsub" node.
+
+  let supSubGroup;
+  let group;
+
+  if (grp.type === "supsub") {
+    // Ref: LaTeX source2e: }}}}\limits}
+    // i.e. LaTeX treats the brace similar to an op and passes it
+    // with \limits, so we need to assign supsub style.
+    supSubGroup = grp.sup ? buildGroup(grp.sup, options.havingStyle(style.sup()), options) : buildGroup(grp.sub, options.havingStyle(style.sub()), options);
+    group = assertNodeType(grp.base, "horizBrace");
+  } else {
+    group = assertNodeType(grp, "horizBrace");
+  } // Build the base group
+
+
+  const body = buildGroup(group.base, options.havingBaseStyle(Style$1.DISPLAY)); // Create the stretchy element
+
+  const braceBody = stretchy.svgSpan(group, options); // Generate the vlist, with the appropriate kerns        ┏━━━━━━━━┓
+  // This first vlist contains the content and the brace:   equation
+
+  let vlist;
+
+  if (group.isOver) {
+    vlist = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "kern",
+        size: 0.1
+      }, {
+        type: "elem",
+        elem: braceBody
+      }]
+    }, options); // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+    vlist.children[0].children[0].children[1].classes.push("svg-align");
+  } else {
+    vlist = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: body.depth + 0.1 + braceBody.height,
+      children: [{
+        type: "elem",
+        elem: braceBody
+      }, {
+        type: "kern",
+        size: 0.1
+      }, {
+        type: "elem",
+        elem: body
+      }]
+    }, options); // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+    vlist.children[0].children[0].children[0].classes.push("svg-align");
+  }
+
+  if (supSubGroup) {
+    // To write the supsub, wrap the first vlist in another vlist:
+    // They can't all go in the same vlist, because the note might be
+    // wider than the equation. We want the equation to control the
+    // brace width.
+    //      note          long note           long note
+    //   ┏━━━━━━━━┓   or    ┏━━━┓     not    ┏━━━━━━━━━┓
+    //    equation           eqn                 eqn
+    const vSpan = buildCommon.makeSpan(["mord", group.isOver ? "mover" : "munder"], [vlist], options);
+
+    if (group.isOver) {
+      vlist = buildCommon.makeVList({
+        positionType: "firstBaseline",
+        children: [{
+          type: "elem",
+          elem: vSpan
+        }, {
+          type: "kern",
+          size: 0.2
+        }, {
+          type: "elem",
+          elem: supSubGroup
+        }]
+      }, options);
+    } else {
+      vlist = buildCommon.makeVList({
+        positionType: "bottom",
+        positionData: vSpan.depth + 0.2 + supSubGroup.height + supSubGroup.depth,
+        children: [{
+          type: "elem",
+          elem: supSubGroup
+        }, {
+          type: "kern",
+          size: 0.2
+        }, {
+          type: "elem",
+          elem: vSpan
+        }]
+      }, options);
+    }
+  }
+
+  return buildCommon.makeSpan(["mord", group.isOver ? "mover" : "munder"], [vlist], options);
+};
+
+const mathmlBuilder$7 = (group, options) => {
+  const accentNode = stretchy.mathMLnode(group.label);
+  return new mathMLTree.MathNode(group.isOver ? "mover" : "munder", [buildGroup$1(group.base, options), accentNode]);
+}; // Horizontal stretchy braces
+
+
+defineFunction({
+  type: "horizBrace",
+  names: ["\\overbrace", "\\underbrace"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    return {
+      type: "horizBrace",
+      mode: parser.mode,
+      label: funcName,
+      isOver: /^\\over/.test(funcName),
+      base: args[0]
+    };
+  },
+
+  htmlBuilder: htmlBuilder$7,
+  mathmlBuilder: mathmlBuilder$7
+});
+
+defineFunction({
+  type: "href",
+  names: ["\\href"],
+  props: {
+    numArgs: 2,
+    argTypes: ["url", "original"],
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser;
+    const body = args[1];
+    const href = assertNodeType(args[0], "url").url;
+
+    if (!parser.settings.isTrusted({
+      command: "\\href",
+      url: href
+    })) {
+      return parser.formatUnsupportedCmd("\\href");
+    }
+
+    return {
+      type: "href",
+      mode: parser.mode,
+      href,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const elements = buildExpression(group.body, options, false);
+    return buildCommon.makeAnchor(group.href, [], elements, options);
+  },
+  mathmlBuilder: (group, options) => {
+    let math = buildExpressionRow(group.body, options);
+
+    if (!(math instanceof MathNode)) {
+      math = new MathNode("mrow", [math]);
+    }
+
+    math.setAttribute("href", group.href);
+    return math;
+  }
+});
+defineFunction({
+  type: "href",
+  names: ["\\url"],
+  props: {
+    numArgs: 1,
+    argTypes: ["url"],
+    allowedInText: true
+  },
+  handler: (_ref2, args) => {
+    let parser = _ref2.parser;
+    const href = assertNodeType(args[0], "url").url;
+
+    if (!parser.settings.isTrusted({
+      command: "\\url",
+      url: href
+    })) {
+      return parser.formatUnsupportedCmd("\\url");
+    }
+
+    const chars = [];
+
+    for (let i = 0; i < href.length; i++) {
+      let c = href[i];
+
+      if (c === "~") {
+        c = "\\textasciitilde";
+      }
+
+      chars.push({
+        type: "textord",
+        mode: "text",
+        text: c
+      });
+    }
+
+    const body = {
+      type: "text",
+      mode: parser.mode,
+      font: "\\texttt",
+      body: chars
+    };
+    return {
+      type: "href",
+      mode: parser.mode,
+      href,
+      body: ordargument(body)
+    };
+  }
+});
+
+defineFunction({
+  type: "html",
+  names: ["\\htmlClass", "\\htmlId", "\\htmlStyle", "\\htmlData"],
+  props: {
+    numArgs: 2,
+    argTypes: ["raw", "original"],
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName,
+        token = _ref.token;
+    const value = assertNodeType(args[0], "raw").string;
+    const body = args[1];
+
+    if (parser.settings.strict) {
+      parser.settings.reportNonstrict("htmlExtension", "HTML extension is disabled on strict mode");
+    }
+
+    let trustContext;
+    const attributes = {};
+
+    switch (funcName) {
+      case "\\htmlClass":
+        attributes.class = value;
+        trustContext = {
+          command: "\\htmlClass",
+          class: value
+        };
+        break;
+
+      case "\\htmlId":
+        attributes.id = value;
+        trustContext = {
+          command: "\\htmlId",
+          id: value
+        };
+        break;
+
+      case "\\htmlStyle":
+        attributes.style = value;
+        trustContext = {
+          command: "\\htmlStyle",
+          style: value
+        };
+        break;
+
+      case "\\htmlData":
+        {
+          const data = value.split(",");
+
+          for (let i = 0; i < data.length; i++) {
+            const keyVal = data[i].split("=");
+
+            if (keyVal.length !== 2) {
+              throw new ParseError("Error parsing key-value for \\htmlData");
+            }
+
+            attributes["data-" + keyVal[0].trim()] = keyVal[1].trim();
+          }
+
+          trustContext = {
+            command: "\\htmlData",
+            attributes
+          };
+          break;
+        }
+
+      default:
+        throw new Error("Unrecognized html command");
+    }
+
+    if (!parser.settings.isTrusted(trustContext)) {
+      return parser.formatUnsupportedCmd(funcName);
+    }
+
+    return {
+      type: "html",
+      mode: parser.mode,
+      attributes,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const elements = buildExpression(group.body, options, false);
+    const classes = ["enclosing"];
+
+    if (group.attributes.class) {
+      classes.push(...group.attributes.class.trim().split(/\s+/));
+    }
+
+    const span = buildCommon.makeSpan(classes, elements, options);
+
+    for (const attr in group.attributes) {
+      if (attr !== "class" && group.attributes.hasOwnProperty(attr)) {
+        span.setAttribute(attr, group.attributes[attr]);
+      }
+    }
+
+    return span;
+  },
+  mathmlBuilder: (group, options) => {
+    return buildExpressionRow(group.body, options);
+  }
+});
+
+defineFunction({
+  type: "htmlmathml",
+  names: ["\\html@mathml"],
+  props: {
+    numArgs: 2,
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser;
+    return {
+      type: "htmlmathml",
+      mode: parser.mode,
+      html: ordargument(args[0]),
+      mathml: ordargument(args[1])
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const elements = buildExpression(group.html, options, false);
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: (group, options) => {
+    return buildExpressionRow(group.mathml, options);
+  }
+});
+
+const sizeData = function sizeData(str) {
+  if (/^[-+]? *(\d+(\.\d*)?|\.\d+)$/.test(str)) {
+    // str is a number with no unit specified.
+    // default unit is bp, per graphix package.
+    return {
+      number: +str,
+      unit: "bp"
+    };
+  } else {
+    const match = /([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(str);
+
+    if (!match) {
+      throw new ParseError("Invalid size: '" + str + "' in \\includegraphics");
+    }
+
+    const data = {
+      number: +(match[1] + match[2]),
+      // sign + magnitude, cast to number
+      unit: match[3]
+    };
+
+    if (!validUnit(data)) {
+      throw new ParseError("Invalid unit: '" + data.unit + "' in \\includegraphics.");
+    }
+
+    return data;
+  }
+};
+
+defineFunction({
+  type: "includegraphics",
+  names: ["\\includegraphics"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1,
+    argTypes: ["raw", "url"],
+    allowedInText: false
+  },
+  handler: (_ref, args, optArgs) => {
+    let parser = _ref.parser;
+    let width = {
+      number: 0,
+      unit: "em"
+    };
+    let height = {
+      number: 0.9,
+      unit: "em"
+    }; // sorta character sized.
+
+    let totalheight = {
+      number: 0,
+      unit: "em"
+    };
+    let alt = "";
+
+    if (optArgs[0]) {
+      const attributeStr = assertNodeType(optArgs[0], "raw").string; // Parser.js does not parse key/value pairs. We get a string.
+
+      const attributes = attributeStr.split(",");
+
+      for (let i = 0; i < attributes.length; i++) {
+        const keyVal = attributes[i].split("=");
+
+        if (keyVal.length === 2) {
+          const str = keyVal[1].trim();
+
+          switch (keyVal[0].trim()) {
+            case "alt":
+              alt = str;
+              break;
+
+            case "width":
+              width = sizeData(str);
+              break;
+
+            case "height":
+              height = sizeData(str);
+              break;
+
+            case "totalheight":
+              totalheight = sizeData(str);
+              break;
+
+            default:
+              throw new ParseError("Invalid key: '" + keyVal[0] + "' in \\includegraphics.");
+          }
+        }
+      }
+    }
+
+    const src = assertNodeType(args[0], "url").url;
+
+    if (alt === "") {
+      // No alt given. Use the file name. Strip away the path.
+      alt = src;
+      alt = alt.replace(/^.*[\\/]/, '');
+      alt = alt.substring(0, alt.lastIndexOf('.'));
+    }
+
+    if (!parser.settings.isTrusted({
+      command: "\\includegraphics",
+      url: src
+    })) {
+      return parser.formatUnsupportedCmd("\\includegraphics");
+    }
+
+    return {
+      type: "includegraphics",
+      mode: parser.mode,
+      alt: alt,
+      width: width,
+      height: height,
+      totalheight: totalheight,
+      src: src
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const height = calculateSize(group.height, options);
+    let depth = 0;
+
+    if (group.totalheight.number > 0) {
+      depth = calculateSize(group.totalheight, options) - height;
+      depth = Number(depth.toFixed(2));
+    }
+
+    let width = 0;
+
+    if (group.width.number > 0) {
+      width = calculateSize(group.width, options);
+    }
+
+    const style = {
+      height: height + depth + "em"
+    };
+
+    if (width > 0) {
+      style.width = width + "em";
+    }
+
+    if (depth > 0) {
+      style.verticalAlign = -depth + "em";
+    }
+
+    const node = new Img(group.src, group.alt, style);
+    node.height = height;
+    node.depth = depth;
+    return node;
+  },
+  mathmlBuilder: (group, options) => {
+    const node = new mathMLTree.MathNode("mglyph", []);
+    node.setAttribute("alt", group.alt);
+    const height = calculateSize(group.height, options);
+    let depth = 0;
+
+    if (group.totalheight.number > 0) {
+      depth = calculateSize(group.totalheight, options) - height;
+      depth = depth.toFixed(2);
+      node.setAttribute("valign", "-" + depth + "em");
+    }
+
+    node.setAttribute("height", height + depth + "em");
+
+    if (group.width.number > 0) {
+      const width = calculateSize(group.width, options);
+      node.setAttribute("width", width + "em");
+    }
+
+    node.setAttribute("src", group.src);
+    return node;
+  }
+});
+
+// Horizontal spacing commands
+
+defineFunction({
+  type: "kern",
+  names: ["\\kern", "\\mkern", "\\hskip", "\\mskip"],
+  props: {
+    numArgs: 1,
+    argTypes: ["size"],
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const size = assertNodeType(args[0], "size");
+
+    if (parser.settings.strict) {
+      const mathFunction = funcName[1] === 'm'; // \mkern, \mskip
+
+      const muUnit = size.value.unit === 'mu';
+
+      if (mathFunction) {
+        if (!muUnit) {
+          parser.settings.reportNonstrict("mathVsTextUnits", `LaTeX's ${funcName} supports only mu units, ` + `not ${size.value.unit} units`);
+        }
+
+        if (parser.mode !== "math") {
+          parser.settings.reportNonstrict("mathVsTextUnits", `LaTeX's ${funcName} works only in math mode`);
+        }
+      } else {
+        // !mathFunction
+        if (muUnit) {
+          parser.settings.reportNonstrict("mathVsTextUnits", `LaTeX's ${funcName} doesn't support mu units`);
+        }
+      }
+    }
+
+    return {
+      type: "kern",
+      mode: parser.mode,
+      dimension: size.value
+    };
+  },
+
+  htmlBuilder(group, options) {
+    return buildCommon.makeGlue(group.dimension, options);
+  },
+
+  mathmlBuilder(group, options) {
+    const dimension = calculateSize(group.dimension, options);
+    return new mathMLTree.SpaceNode(dimension);
+  }
+
+});
+
+// Horizontal overlap functions
+defineFunction({
+  type: "lap",
+  names: ["\\mathllap", "\\mathrlap", "\\mathclap"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    return {
+      type: "lap",
+      mode: parser.mode,
+      alignment: funcName.slice(5),
+      body
+    };
+  },
+  htmlBuilder: (group, options) => {
+    // mathllap, mathrlap, mathclap
+    let inner;
+
+    if (group.alignment === "clap") {
+      // ref: https://www.math.lsu.edu/~aperlis/publications/mathclap/
+      inner = buildCommon.makeSpan([], [buildGroup(group.body, options)]); // wrap, since CSS will center a .clap > .inner > span
+
+      inner = buildCommon.makeSpan(["inner"], [inner], options);
+    } else {
+      inner = buildCommon.makeSpan(["inner"], [buildGroup(group.body, options)]);
+    }
+
+    const fix = buildCommon.makeSpan(["fix"], []);
+    let node = buildCommon.makeSpan([group.alignment], [inner, fix], options); // At this point, we have correctly set horizontal alignment of the
+    // two items involved in the lap.
+    // Next, use a strut to set the height of the HTML bounding box.
+    // Otherwise, a tall argument may be misplaced.
+    // This code resolved issue #1153
+
+    const strut = buildCommon.makeSpan(["strut"]);
+    strut.style.height = node.height + node.depth + "em";
+    strut.style.verticalAlign = -node.depth + "em";
+    node.children.unshift(strut); // Next, prevent vertical misplacement when next to something tall.
+    // This code resolves issue #1234
+
+    node = buildCommon.makeSpan(["thinbox"], [node], options);
+    return buildCommon.makeSpan(["mord", "vbox"], [node], options);
+  },
+  mathmlBuilder: (group, options) => {
+    // mathllap, mathrlap, mathclap
+    const node = new mathMLTree.MathNode("mpadded", [buildGroup$1(group.body, options)]);
+
+    if (group.alignment !== "rlap") {
+      const offset = group.alignment === "llap" ? "-1" : "-0.5";
+      node.setAttribute("lspace", offset + "width");
+    }
+
+    node.setAttribute("width", "0px");
+    return node;
+  }
+});
+
+defineFunction({
+  type: "styling",
+  names: ["\\(", "$"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: false
+  },
+
+  handler(_ref, args) {
+    let funcName = _ref.funcName,
+        parser = _ref.parser;
+    const outerMode = parser.mode;
+    parser.switchMode("math");
+    const close = funcName === "\\(" ? "\\)" : "$";
+    const body = parser.parseExpression(false, close);
+    parser.expect(close);
+    parser.switchMode(outerMode);
+    return {
+      type: "styling",
+      mode: parser.mode,
+      style: "text",
+      body
+    };
+  }
+
+}); // Check for extra closing math delimiters
+
+defineFunction({
+  type: "text",
+  // Doesn't matter what this is.
+  names: ["\\)", "\\]"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: false
+  },
+
+  handler(context, args) {
+    throw new ParseError(`Mismatched ${context.funcName}`);
+  }
+
+});
+
+const chooseMathStyle = (group, options) => {
+  switch (options.style.size) {
+    case Style$1.DISPLAY.size:
+      return group.display;
+
+    case Style$1.TEXT.size:
+      return group.text;
+
+    case Style$1.SCRIPT.size:
+      return group.script;
+
+    case Style$1.SCRIPTSCRIPT.size:
+      return group.scriptscript;
+
+    default:
+      return group.text;
+  }
+};
+
+defineFunction({
+  type: "mathchoice",
+  names: ["\\mathchoice"],
+  props: {
+    numArgs: 4
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser;
+    return {
+      type: "mathchoice",
+      mode: parser.mode,
+      display: ordargument(args[0]),
+      text: ordargument(args[1]),
+      script: ordargument(args[2]),
+      scriptscript: ordargument(args[3])
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const body = chooseMathStyle(group, options);
+    const elements = buildExpression(body, options, false);
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: (group, options) => {
+    const body = chooseMathStyle(group, options);
+    return buildExpressionRow(body, options);
+  }
+});
+
+// For an operator with limits, assemble the base, sup, and sub into a span.
+const assembleSupSub = (base, supGroup, subGroup, options, style, slant, baseShift) => {
+  base = buildCommon.makeSpan([], [base]);
+  let sub;
+  let sup; // We manually have to handle the superscripts and subscripts. This,
+  // aside from the kern calculations, is copied from supsub.
+
+  if (supGroup) {
+    const elem = buildGroup(supGroup, options.havingStyle(style.sup()), options);
+    sup = {
+      elem,
+      kern: Math.max(options.fontMetrics().bigOpSpacing1, options.fontMetrics().bigOpSpacing3 - elem.depth)
+    };
+  }
+
+  if (subGroup) {
+    const elem = buildGroup(subGroup, options.havingStyle(style.sub()), options);
+    sub = {
+      elem,
+      kern: Math.max(options.fontMetrics().bigOpSpacing2, options.fontMetrics().bigOpSpacing4 - elem.height)
+    };
+  } // Build the final group as a vlist of the possible subscript, base,
+  // and possible superscript.
+
+
+  let finalGroup;
+
+  if (sup && sub) {
+    const bottom = options.fontMetrics().bigOpSpacing5 + sub.elem.height + sub.elem.depth + sub.kern + base.depth + baseShift;
+    finalGroup = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: bottom,
+      children: [{
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }, {
+        type: "elem",
+        elem: sub.elem,
+        marginLeft: -slant + "em"
+      }, {
+        type: "kern",
+        size: sub.kern
+      }, {
+        type: "elem",
+        elem: base
+      }, {
+        type: "kern",
+        size: sup.kern
+      }, {
+        type: "elem",
+        elem: sup.elem,
+        marginLeft: slant + "em"
+      }, {
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }]
+    }, options);
+  } else if (sub) {
+    const top = base.height - baseShift; // Shift the limits by the slant of the symbol. Note
+    // that we are supposed to shift the limits by 1/2 of the slant,
+    // but since we are centering the limits adding a full slant of
+    // margin will shift by 1/2 that.
+
+    finalGroup = buildCommon.makeVList({
+      positionType: "top",
+      positionData: top,
+      children: [{
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }, {
+        type: "elem",
+        elem: sub.elem,
+        marginLeft: -slant + "em"
+      }, {
+        type: "kern",
+        size: sub.kern
+      }, {
+        type: "elem",
+        elem: base
+      }]
+    }, options);
+  } else if (sup) {
+    const bottom = base.depth + baseShift;
+    finalGroup = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: bottom,
+      children: [{
+        type: "elem",
+        elem: base
+      }, {
+        type: "kern",
+        size: sup.kern
+      }, {
+        type: "elem",
+        elem: sup.elem,
+        marginLeft: slant + "em"
+      }, {
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }]
+    }, options);
+  } else {
+    // This case probably shouldn't occur (this would mean the
+    // supsub was sending us a group with no superscript or
+    // subscript) but be safe.
+    return base;
+  }
+
+  return buildCommon.makeSpan(["mop", "op-limits"], [finalGroup], options);
+};
+
+// Limits, symbols
+// Most operators have a large successor symbol, but these don't.
+const noSuccessor = ["\\smallint"]; // NOTE: Unlike most `htmlBuilder`s, this one handles not only "op", but also
+// "supsub" since some of them (like \int) can affect super/subscripting.
+
+const htmlBuilder$8 = (grp, options) => {
+  // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+  let supGroup;
+  let subGroup;
+  let hasLimits = false;
+  let group;
+
+  if (grp.type === "supsub") {
+    // If we have limits, supsub will pass us its group to handle. Pull
+    // out the superscript and subscript and set the group to the op in
+    // its base.
+    supGroup = grp.sup;
+    subGroup = grp.sub;
+    group = assertNodeType(grp.base, "op");
+    hasLimits = true;
+  } else {
+    group = assertNodeType(grp, "op");
+  }
+
+  const style = options.style;
+  let large = false;
+
+  if (style.size === Style$1.DISPLAY.size && group.symbol && !utils.contains(noSuccessor, group.name)) {
+    // Most symbol operators get larger in displaystyle (rule 13)
+    large = true;
+  }
+
+  let base;
+
+  if (group.symbol) {
+    // If this is a symbol, create the symbol.
+    const fontName = large ? "Size2-Regular" : "Size1-Regular";
+    let stash = "";
+
+    if (group.name === "\\oiint" || group.name === "\\oiiint") {
+      // No font glyphs yet, so use a glyph w/o the oval.
+      // TODO: When font glyphs are available, delete this code.
+      stash = group.name.substr(1); // $FlowFixMe
+
+      group.name = stash === "oiint" ? "\\iint" : "\\iiint";
+    }
+
+    base = buildCommon.makeSymbol(group.name, fontName, "math", options, ["mop", "op-symbol", large ? "large-op" : "small-op"]);
+
+    if (stash.length > 0) {
+      // We're in \oiint or \oiiint. Overlay the oval.
+      // TODO: When font glyphs are available, delete this code.
+      const italic = base.italic;
+      const oval = buildCommon.staticSvg(stash + "Size" + (large ? "2" : "1"), options);
+      base = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: base,
+          shift: 0
+        }, {
+          type: "elem",
+          elem: oval,
+          shift: large ? 0.08 : 0
+        }]
+      }, options); // $FlowFixMe
+
+      group.name = "\\" + stash;
+      base.classes.unshift("mop"); // $FlowFixMe
+
+      base.italic = italic;
+    }
+  } else if (group.body) {
+    // If this is a list, compose that list.
+    const inner = buildExpression(group.body, options, true);
+
+    if (inner.length === 1 && inner[0] instanceof SymbolNode) {
+      base = inner[0];
+      base.classes[0] = "mop"; // replace old mclass
+    } else {
+      base = buildCommon.makeSpan(["mop"], buildCommon.tryCombineChars(inner), options);
+    }
+  } else {
+    // Otherwise, this is a text operator. Build the text from the
+    // operator's name.
+    // TODO(emily): Add a space in the middle of some of these
+    // operators, like \limsup
+    const output = [];
+
+    for (let i = 1; i < group.name.length; i++) {
+      output.push(buildCommon.mathsym(group.name[i], group.mode, options));
+    }
+
+    base = buildCommon.makeSpan(["mop"], output, options);
+  } // If content of op is a single symbol, shift it vertically.
+
+
+  let baseShift = 0;
+  let slant = 0;
+
+  if ((base instanceof SymbolNode || group.name === "\\oiint" || group.name === "\\oiiint") && !group.suppressBaseShift) {
+    // We suppress the shift of the base of \overset and \underset. Otherwise,
+    // shift the symbol so its center lies on the axis (rule 13). It
+    // appears that our fonts have the centers of the symbols already
+    // almost on the axis, so these numbers are very small. Note we
+    // don't actually apply this here, but instead it is used either in
+    // the vlist creation or separately when there are no limits.
+    baseShift = (base.height - base.depth) / 2 - options.fontMetrics().axisHeight; // The slant of the symbol is just its italic correction.
+    // $FlowFixMe
+
+    slant = base.italic;
+  }
+
+  if (hasLimits) {
+    return assembleSupSub(base, supGroup, subGroup, options, style, slant, baseShift);
+  } else {
+    if (baseShift) {
+      base.style.position = "relative";
+      base.style.top = baseShift + "em";
+    }
+
+    return base;
+  }
+};
+
+const mathmlBuilder$8 = (group, options) => {
+  let node;
+
+  if (group.symbol) {
+    // This is a symbol. Just add the symbol.
+    node = new MathNode("mo", [makeText(group.name, group.mode)]);
+
+    if (utils.contains(noSuccessor, group.name)) {
+      node.setAttribute("largeop", "false");
+    }
+  } else if (group.body) {
+    // This is an operator with children. Add them.
+    node = new MathNode("mo", buildExpression$1(group.body, options));
+  } else {
+    // This is a text operator. Add all of the characters from the
+    // operator's name.
+    node = new MathNode("mi", [new TextNode(group.name.slice(1))]); // Append an <mo>&ApplyFunction;</mo>.
+    // ref: https://www.w3.org/TR/REC-MathML/chap3_2.html#sec3.2.4
+
+    const operator = new MathNode("mo", [makeText("\u2061", "text")]);
+
+    if (group.parentIsSupSub) {
+      node = new MathNode("mo", [node, operator]);
+    } else {
+      node = newDocumentFragment([node, operator]);
+    }
+  }
+
+  return node;
+};
+
+const singleCharBigOps = {
+  "\u220F": "\\prod",
+  "\u2210": "\\coprod",
+  "\u2211": "\\sum",
+  "\u22c0": "\\bigwedge",
+  "\u22c1": "\\bigvee",
+  "\u22c2": "\\bigcap",
+  "\u22c3": "\\bigcup",
+  "\u2a00": "\\bigodot",
+  "\u2a01": "\\bigoplus",
+  "\u2a02": "\\bigotimes",
+  "\u2a04": "\\biguplus",
+  "\u2a06": "\\bigsqcup"
+};
+defineFunction({
+  type: "op",
+  names: ["\\coprod", "\\bigvee", "\\bigwedge", "\\biguplus", "\\bigcap", "\\bigcup", "\\intop", "\\prod", "\\sum", "\\bigotimes", "\\bigoplus", "\\bigodot", "\\bigsqcup", "\\smallint", "\u220F", "\u2210", "\u2211", "\u22c0", "\u22c1", "\u22c2", "\u22c3", "\u2a00", "\u2a01", "\u2a02", "\u2a04", "\u2a06"],
+  props: {
+    numArgs: 0
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    let fName = funcName;
+
+    if (fName.length === 1) {
+      fName = singleCharBigOps[fName];
+    }
+
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: true,
+      parentIsSupSub: false,
+      symbol: true,
+      name: fName
+    };
+  },
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+}); // Note: calling defineFunction with a type that's already been defined only
+// works because the same htmlBuilder and mathmlBuilder are being used.
+
+defineFunction({
+  type: "op",
+  names: ["\\mathop"],
+  props: {
+    numArgs: 1
+  },
+  handler: (_ref2, args) => {
+    let parser = _ref2.parser;
+    const body = args[0];
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: false,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+}); // There are 2 flags for operators; whether they produce limits in
+// displaystyle, and whether they are symbols and should grow in
+// displaystyle. These four groups cover the four possible choices.
+
+const singleCharIntegrals = {
+  "\u222b": "\\int",
+  "\u222c": "\\iint",
+  "\u222d": "\\iiint",
+  "\u222e": "\\oint",
+  "\u222f": "\\oiint",
+  "\u2230": "\\oiiint"
+}; // No limits, not symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\arcsin", "\\arccos", "\\arctan", "\\arctg", "\\arcctg", "\\arg", "\\ch", "\\cos", "\\cosec", "\\cosh", "\\cot", "\\cotg", "\\coth", "\\csc", "\\ctg", "\\cth", "\\deg", "\\dim", "\\exp", "\\hom", "\\ker", "\\lg", "\\ln", "\\log", "\\sec", "\\sin", "\\sinh", "\\sh", "\\tan", "\\tanh", "\\tg", "\\th"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(_ref3) {
+    let parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: false,
+      name: funcName
+    };
+  },
+
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+}); // Limits, not symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\det", "\\gcd", "\\inf", "\\lim", "\\max", "\\min", "\\Pr", "\\sup"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(_ref4) {
+    let parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: true,
+      parentIsSupSub: false,
+      symbol: false,
+      name: funcName
+    };
+  },
+
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+}); // No limits, symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\int", "\\iint", "\\iiint", "\\oint", "\\oiint", "\\oiiint", "\u222b", "\u222c", "\u222d", "\u222e", "\u222f", "\u2230"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(_ref5) {
+    let parser = _ref5.parser,
+        funcName = _ref5.funcName;
+    let fName = funcName;
+
+    if (fName.length === 1) {
+      fName = singleCharIntegrals[fName];
+    }
+
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: true,
+      name: fName
+    };
+  },
+
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+});
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only
+// "operatorname", but also  "supsub" since \operatorname* can
+const htmlBuilder$9 = (grp, options) => {
+  // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+  let supGroup;
+  let subGroup;
+  let hasLimits = false;
+  let group;
+
+  if (grp.type === "supsub") {
+    // If we have limits, supsub will pass us its group to handle. Pull
+    // out the superscript and subscript and set the group to the op in
+    // its base.
+    supGroup = grp.sup;
+    subGroup = grp.sub;
+    group = assertNodeType(grp.base, "operatorname");
+    hasLimits = true;
+  } else {
+    group = assertNodeType(grp, "operatorname");
+  }
+
+  let base;
+
+  if (group.body.length > 0) {
+    const body = group.body.map(child => {
+      // $FlowFixMe: Check if the node has a string `text` property.
+      const childText = child.text;
+
+      if (typeof childText === "string") {
+        return {
+          type: "textord",
+          mode: child.mode,
+          text: childText
+        };
+      } else {
+        return child;
+      }
+    }); // Consolidate function names into symbol characters.
+
+    const expression = buildExpression(body, options.withFont("mathrm"), true);
+
+    for (let i = 0; i < expression.length; i++) {
+      const child = expression[i];
+
+      if (child instanceof SymbolNode) {
+        // Per amsopn package,
+        // change minus to hyphen and \ast to asterisk
+        child.text = child.text.replace(/\u2212/, "-").replace(/\u2217/, "*");
+      }
+    }
+
+    base = buildCommon.makeSpan(["mop"], expression, options);
+  } else {
+    base = buildCommon.makeSpan(["mop"], [], options);
+  }
+
+  if (hasLimits) {
+    return assembleSupSub(base, supGroup, subGroup, options, options.style, 0, 0);
+  } else {
+    return base;
+  }
+};
+
+const mathmlBuilder$9 = (group, options) => {
+  // The steps taken here are similar to the html version.
+  let expression = buildExpression$1(group.body, options.withFont("mathrm")); // Is expression a string or has it something like a fraction?
+
+  let isAllString = true; // default
+
+  for (let i = 0; i < expression.length; i++) {
+    const node = expression[i];
+
+    if (node instanceof mathMLTree.SpaceNode) ; else if (node instanceof mathMLTree.MathNode) {
+      switch (node.type) {
+        case "mi":
+        case "mn":
+        case "ms":
+        case "mspace":
+        case "mtext":
+          break;
+        // Do nothing yet.
+
+        case "mo":
+          {
+            const child = node.children[0];
+
+            if (node.children.length === 1 && child instanceof mathMLTree.TextNode) {
+              child.text = child.text.replace(/\u2212/, "-").replace(/\u2217/, "*");
+            } else {
+              isAllString = false;
+            }
+
+            break;
+          }
+
+        default:
+          isAllString = false;
+      }
+    } else {
+      isAllString = false;
+    }
+  }
+
+  if (isAllString) {
+    // Write a single TextNode instead of multiple nested tags.
+    const word = expression.map(node => node.toText()).join("");
+    expression = [new mathMLTree.TextNode(word)];
+  }
+
+  const identifier = new mathMLTree.MathNode("mi", expression);
+  identifier.setAttribute("mathvariant", "normal"); // \u2061 is the same as &ApplyFunction;
+  // ref: https://www.w3schools.com/charsets/ref_html_entities_a.asp
+
+  const operator = new mathMLTree.MathNode("mo", [makeText("\u2061", "text")]);
+
+  if (group.parentIsSupSub) {
+    return new mathMLTree.MathNode("mo", [identifier, operator]);
+  } else {
+    return mathMLTree.newDocumentFragment([identifier, operator]);
+  }
+}; // \operatorname
+// amsopn.dtx: \mathop{#1\kern\z@\operator@font#3}\newmcodes@
+
+
+defineFunction({
+  type: "operatorname",
+  names: ["\\operatorname", "\\operatorname*"],
+  props: {
+    numArgs: 1
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    return {
+      type: "operatorname",
+      mode: parser.mode,
+      body: ordargument(body),
+      alwaysHandleSupSub: funcName === "\\operatorname*",
+      limits: false,
+      parentIsSupSub: false
+    };
+  },
+  htmlBuilder: htmlBuilder$9,
+  mathmlBuilder: mathmlBuilder$9
+});
+
+defineFunctionBuilders({
+  type: "ordgroup",
+
+  htmlBuilder(group, options) {
+    if (group.semisimple) {
+      return buildCommon.makeFragment(buildExpression(group.body, options, false));
+    }
+
+    return buildCommon.makeSpan(["mord"], buildExpression(group.body, options, true), options);
+  },
+
+  mathmlBuilder(group, options) {
+    return buildExpressionRow(group.body, options, true);
+  }
+
+});
+
+defineFunction({
+  type: "overline",
+  names: ["\\overline"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    const body = args[0];
+    return {
+      type: "overline",
+      mode: parser.mode,
+      body
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Overlines are handled in the TeXbook pg 443, Rule 9.
+    // Build the inner group in the cramped style.
+    const innerGroup = buildGroup(group.body, options.havingCrampedStyle()); // Create the line above the body
+
+    const line = buildCommon.makeLineSpan("overline-line", options); // Generate the vlist, with the appropriate kerns
+
+    const defaultRuleThickness = options.fontMetrics().defaultRuleThickness;
+    const vlist = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: innerGroup
+      }, {
+        type: "kern",
+        size: 3 * defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: line
+      }, {
+        type: "kern",
+        size: defaultRuleThickness
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "overline"], [vlist], options);
+  },
+
+  mathmlBuilder(group, options) {
+    const operator = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode("\u203e")]);
+    operator.setAttribute("stretchy", "true");
+    const node = new mathMLTree.MathNode("mover", [buildGroup$1(group.body, options), operator]);
+    node.setAttribute("accent", "true");
+    return node;
+  }
+
+});
+
+defineFunction({
+  type: "phantom",
+  names: ["\\phantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser;
+    const body = args[0];
+    return {
+      type: "phantom",
+      mode: parser.mode,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const elements = buildExpression(group.body, options.withPhantom(), false); // \phantom isn't supposed to affect the elements it contains.
+    // See "color" for more details.
+
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: (group, options) => {
+    const inner = buildExpression$1(group.body, options);
+    return new mathMLTree.MathNode("mphantom", inner);
+  }
+});
+defineFunction({
+  type: "hphantom",
+  names: ["\\hphantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref2, args) => {
+    let parser = _ref2.parser;
+    const body = args[0];
+    return {
+      type: "hphantom",
+      mode: parser.mode,
+      body
+    };
+  },
+  htmlBuilder: (group, options) => {
+    let node = buildCommon.makeSpan([], [buildGroup(group.body, options.withPhantom())]);
+    node.height = 0;
+    node.depth = 0;
+
+    if (node.children) {
+      for (let i = 0; i < node.children.length; i++) {
+        node.children[i].height = 0;
+        node.children[i].depth = 0;
+      }
+    } // See smash for comment re: use of makeVList
+
+
+    node = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: node
+      }]
+    }, options); // For spacing, TeX treats \smash as a math group (same spacing as ord).
+
+    return buildCommon.makeSpan(["mord"], [node], options);
+  },
+  mathmlBuilder: (group, options) => {
+    const inner = buildExpression$1(ordargument(group.body), options);
+    const phantom = new mathMLTree.MathNode("mphantom", inner);
+    const node = new mathMLTree.MathNode("mpadded", [phantom]);
+    node.setAttribute("height", "0px");
+    node.setAttribute("depth", "0px");
+    return node;
+  }
+});
+defineFunction({
+  type: "vphantom",
+  names: ["\\vphantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref3, args) => {
+    let parser = _ref3.parser;
+    const body = args[0];
+    return {
+      type: "vphantom",
+      mode: parser.mode,
+      body
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const inner = buildCommon.makeSpan(["inner"], [buildGroup(group.body, options.withPhantom())]);
+    const fix = buildCommon.makeSpan(["fix"], []);
+    return buildCommon.makeSpan(["mord", "rlap"], [inner, fix], options);
+  },
+  mathmlBuilder: (group, options) => {
+    const inner = buildExpression$1(ordargument(group.body), options);
+    const phantom = new mathMLTree.MathNode("mphantom", inner);
+    const node = new mathMLTree.MathNode("mpadded", [phantom]);
+    node.setAttribute("width", "0px");
+    return node;
+  }
+});
+
+defineFunction({
+  type: "raisebox",
+  names: ["\\raisebox"],
+  props: {
+    numArgs: 2,
+    argTypes: ["size", "hbox"],
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    const amount = assertNodeType(args[0], "size").value;
+    const body = args[1];
+    return {
+      type: "raisebox",
+      mode: parser.mode,
+      dy: amount,
+      body
+    };
+  },
+
+  htmlBuilder(group, options) {
+    const body = buildGroup(group.body, options);
+    const dy = calculateSize(group.dy, options);
+    return buildCommon.makeVList({
+      positionType: "shift",
+      positionData: -dy,
+      children: [{
+        type: "elem",
+        elem: body
+      }]
+    }, options);
+  },
+
+  mathmlBuilder(group, options) {
+    const node = new mathMLTree.MathNode("mpadded", [buildGroup$1(group.body, options)]);
+    const dy = group.dy.number + group.dy.unit;
+    node.setAttribute("voffset", dy);
+    return node;
+  }
+
+});
+
+defineFunction({
+  type: "rule",
+  names: ["\\rule"],
+  props: {
+    numArgs: 2,
+    numOptionalArgs: 1,
+    argTypes: ["size", "size", "size"]
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser;
+    const shift = optArgs[0];
+    const width = assertNodeType(args[0], "size");
+    const height = assertNodeType(args[1], "size");
+    return {
+      type: "rule",
+      mode: parser.mode,
+      shift: shift && assertNodeType(shift, "size").value,
+      width: width.value,
+      height: height.value
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Make an empty span for the rule
+    const rule = buildCommon.makeSpan(["mord", "rule"], [], options); // Calculate the shift, width, and height of the rule, and account for units
+
+    const width = calculateSize(group.width, options);
+    const height = calculateSize(group.height, options);
+    const shift = group.shift ? calculateSize(group.shift, options) : 0; // Style the rule to the right size
+
+    rule.style.borderRightWidth = width + "em";
+    rule.style.borderTopWidth = height + "em";
+    rule.style.bottom = shift + "em"; // Record the height and width
+
+    rule.width = width;
+    rule.height = height + shift;
+    rule.depth = -shift; // Font size is the number large enough that the browser will
+    // reserve at least `absHeight` space above the baseline.
+    // The 1.125 factor was empirically determined
+
+    rule.maxFontSize = height * 1.125 * options.sizeMultiplier;
+    return rule;
+  },
+
+  mathmlBuilder(group, options) {
+    const width = calculateSize(group.width, options);
+    const height = calculateSize(group.height, options);
+    const shift = group.shift ? calculateSize(group.shift, options) : 0;
+    const color = options.color && options.getColor() || "black";
+    const rule = new mathMLTree.MathNode("mspace");
+    rule.setAttribute("mathbackground", color);
+    rule.setAttribute("width", width + "em");
+    rule.setAttribute("height", height + "em");
+    const wrapper = new mathMLTree.MathNode("mpadded", [rule]);
+
+    if (shift >= 0) {
+      wrapper.setAttribute("height", "+" + shift + "em");
+    } else {
+      wrapper.setAttribute("height", shift + "em");
+      wrapper.setAttribute("depth", "+" + -shift + "em");
+    }
+
+    wrapper.setAttribute("voffset", shift + "em");
+    return wrapper;
+  }
+
+});
+
+function sizingGroup(value, options, baseOptions) {
+  const inner = buildExpression(value, options, false);
+  const multiplier = options.sizeMultiplier / baseOptions.sizeMultiplier; // Add size-resetting classes to the inner list and set maxFontSize
+  // manually. Handle nested size changes.
+
+  for (let i = 0; i < inner.length; i++) {
+    const pos = inner[i].classes.indexOf("sizing");
+
+    if (pos < 0) {
+      Array.prototype.push.apply(inner[i].classes, options.sizingClasses(baseOptions));
+    } else if (inner[i].classes[pos + 1] === "reset-size" + options.size) {
+      // This is a nested size change: e.g., inner[i] is the "b" in
+      // `\Huge a \small b`. Override the old size (the `reset-` class)
+      // but not the new size.
+      inner[i].classes[pos + 1] = "reset-size" + baseOptions.size;
+    }
+
+    inner[i].height *= multiplier;
+    inner[i].depth *= multiplier;
+  }
+
+  return buildCommon.makeFragment(inner);
+}
+const sizeFuncs = ["\\tiny", "\\sixptsize", "\\scriptsize", "\\footnotesize", "\\small", "\\normalsize", "\\large", "\\Large", "\\LARGE", "\\huge", "\\Huge"];
+const htmlBuilder$a = (group, options) => {
+  // Handle sizing operators like \Huge. Real TeX doesn't actually allow
+  // these functions inside of math expressions, so we do some special
+  // handling.
+  const newOptions = options.havingSize(group.size);
+  return sizingGroup(group.body, newOptions, options);
+};
+defineFunction({
+  type: "sizing",
+  names: sizeFuncs,
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let breakOnTokenText = _ref.breakOnTokenText,
+        funcName = _ref.funcName,
+        parser = _ref.parser;
+    const body = parser.parseExpression(false, breakOnTokenText);
+    return {
+      type: "sizing",
+      mode: parser.mode,
+      // Figure out what size to use based on the list of functions above
+      size: sizeFuncs.indexOf(funcName) + 1,
+      body
+    };
+  },
+  htmlBuilder: htmlBuilder$a,
+  mathmlBuilder: (group, options) => {
+    const newOptions = options.havingSize(group.size);
+    const inner = buildExpression$1(group.body, newOptions);
+    const node = new mathMLTree.MathNode("mstyle", inner); // TODO(emily): This doesn't produce the correct size for nested size
+    // changes, because we don't keep state of what style we're currently
+    // in, so we can't reset the size to normal before changing it.  Now
+    // that we're passing an options parameter we should be able to fix
+    // this.
+
+    node.setAttribute("mathsize", newOptions.sizeMultiplier + "em");
+    return node;
+  }
+});
+
+// smash, with optional [tb], as in AMS
+defineFunction({
+  type: "smash",
+  names: ["\\smash"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref, args, optArgs) => {
+    let parser = _ref.parser;
+    let smashHeight = false;
+    let smashDepth = false;
+    const tbArg = optArgs[0] && assertNodeType(optArgs[0], "ordgroup");
+
+    if (tbArg) {
+      // Optional [tb] argument is engaged.
+      // ref: amsmath: \renewcommand{\smash}[1][tb]{%
+      //               def\mb@t{\ht}\def\mb@b{\dp}\def\mb@tb{\ht\z@\z@\dp}%
+      let letter = "";
+
+      for (let i = 0; i < tbArg.body.length; ++i) {
+        const node = tbArg.body[i]; // $FlowFixMe: Not every node type has a `text` property.
+
+        letter = node.text;
+
+        if (letter === "t") {
+          smashHeight = true;
+        } else if (letter === "b") {
+          smashDepth = true;
+        } else {
+          smashHeight = false;
+          smashDepth = false;
+          break;
+        }
+      }
+    } else {
+      smashHeight = true;
+      smashDepth = true;
+    }
+
+    const body = args[0];
+    return {
+      type: "smash",
+      mode: parser.mode,
+      body,
+      smashHeight,
+      smashDepth
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const node = buildCommon.makeSpan([], [buildGroup(group.body, options)]);
+
+    if (!group.smashHeight && !group.smashDepth) {
+      return node;
+    }
+
+    if (group.smashHeight) {
+      node.height = 0; // In order to influence makeVList, we have to reset the children.
+
+      if (node.children) {
+        for (let i = 0; i < node.children.length; i++) {
+          node.children[i].height = 0;
+        }
+      }
+    }
+
+    if (group.smashDepth) {
+      node.depth = 0;
+
+      if (node.children) {
+        for (let i = 0; i < node.children.length; i++) {
+          node.children[i].depth = 0;
+        }
+      }
+    } // At this point, we've reset the TeX-like height and depth values.
+    // But the span still has an HTML line height.
+    // makeVList applies "display: table-cell", which prevents the browser
+    // from acting on that line height. So we'll call makeVList now.
+
+
+    const smashedNode = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: node
+      }]
+    }, options); // For spacing, TeX treats \hphantom as a math group (same spacing as ord).
+
+    return buildCommon.makeSpan(["mord"], [smashedNode], options);
+  },
+  mathmlBuilder: (group, options) => {
+    const node = new mathMLTree.MathNode("mpadded", [buildGroup$1(group.body, options)]);
+
+    if (group.smashHeight) {
+      node.setAttribute("height", "0px");
+    }
+
+    if (group.smashDepth) {
+      node.setAttribute("depth", "0px");
+    }
+
+    return node;
+  }
+});
+
+defineFunction({
+  type: "sqrt",
+  names: ["\\sqrt"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser;
+    const index = optArgs[0];
+    const body = args[0];
+    return {
+      type: "sqrt",
+      mode: parser.mode,
+      body,
+      index
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Square roots are handled in the TeXbook pg. 443, Rule 11.
+    // First, we do the same steps as in overline to build the inner group
+    // and line
+    let inner = buildGroup(group.body, options.havingCrampedStyle());
+
+    if (inner.height === 0) {
+      // Render a small surd.
+      inner.height = options.fontMetrics().xHeight;
+    } // Some groups can return document fragments.  Handle those by wrapping
+    // them in a span.
+
+
+    inner = buildCommon.wrapFragment(inner, options); // Calculate the minimum size for the \surd delimiter
+
+    const metrics = options.fontMetrics();
+    const theta = metrics.defaultRuleThickness;
+    let phi = theta;
+
+    if (options.style.id < Style$1.TEXT.id) {
+      phi = options.fontMetrics().xHeight;
+    } // Calculate the clearance between the body and line
+
+
+    let lineClearance = theta + phi / 4;
+    const minDelimiterHeight = inner.height + inner.depth + lineClearance + theta; // Create a sqrt SVG of the required minimum size
+
+    const _delimiter$sqrtImage = delimiter.sqrtImage(minDelimiterHeight, options),
+          img = _delimiter$sqrtImage.span,
+          ruleWidth = _delimiter$sqrtImage.ruleWidth,
+          advanceWidth = _delimiter$sqrtImage.advanceWidth;
+
+    const delimDepth = img.height - ruleWidth; // Adjust the clearance based on the delimiter size
+
+    if (delimDepth > inner.height + inner.depth + lineClearance) {
+      lineClearance = (lineClearance + delimDepth - inner.height - inner.depth) / 2;
+    } // Shift the sqrt image
+
+
+    const imgShift = img.height - inner.height - lineClearance - ruleWidth;
+    inner.style.paddingLeft = advanceWidth + "em"; // Overlay the image and the argument.
+
+    const body = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: inner,
+        wrapperClasses: ["svg-align"]
+      }, {
+        type: "kern",
+        size: -(inner.height + imgShift)
+      }, {
+        type: "elem",
+        elem: img
+      }, {
+        type: "kern",
+        size: ruleWidth
+      }]
+    }, options);
+
+    if (!group.index) {
+      return buildCommon.makeSpan(["mord", "sqrt"], [body], options);
+    } else {
+      // Handle the optional root index
+      // The index is always in scriptscript style
+      const newOptions = options.havingStyle(Style$1.SCRIPTSCRIPT);
+      const rootm = buildGroup(group.index, newOptions, options); // The amount the index is shifted by. This is taken from the TeX
+      // source, in the definition of `\r@@t`.
+
+      const toShift = 0.6 * (body.height - body.depth); // Build a VList with the superscript shifted up correctly
+
+      const rootVList = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: -toShift,
+        children: [{
+          type: "elem",
+          elem: rootm
+        }]
+      }, options); // Add a class surrounding it so we can add on the appropriate
+      // kerning
+
+      const rootVListWrap = buildCommon.makeSpan(["root"], [rootVList]);
+      return buildCommon.makeSpan(["mord", "sqrt"], [rootVListWrap, body], options);
+    }
+  },
+
+  mathmlBuilder(group, options) {
+    const body = group.body,
+          index = group.index;
+    return index ? new mathMLTree.MathNode("mroot", [buildGroup$1(body, options), buildGroup$1(index, options)]) : new mathMLTree.MathNode("msqrt", [buildGroup$1(body, options)]);
+  }
+
+});
+
+const styleMap$1 = {
+  "display": Style$1.DISPLAY,
+  "text": Style$1.TEXT,
+  "script": Style$1.SCRIPT,
+  "scriptscript": Style$1.SCRIPTSCRIPT
+};
+defineFunction({
+  type: "styling",
+  names: ["\\displaystyle", "\\textstyle", "\\scriptstyle", "\\scriptscriptstyle"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let breakOnTokenText = _ref.breakOnTokenText,
+        funcName = _ref.funcName,
+        parser = _ref.parser;
+    // parse out the implicit body
+    const body = parser.parseExpression(true, breakOnTokenText); // TODO: Refactor to avoid duplicating styleMap in multiple places (e.g.
+    // here and in buildHTML and de-dupe the enumeration of all the styles).
+    // $FlowFixMe: The names above exactly match the styles.
+
+    const style = funcName.slice(1, funcName.length - 5);
+    return {
+      type: "styling",
+      mode: parser.mode,
+      // Figure out what style to use by pulling out the style from
+      // the function name
+      style,
+      body
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Style changes are handled in the TeXbook on pg. 442, Rule 3.
+    const newStyle = styleMap$1[group.style];
+    const newOptions = options.havingStyle(newStyle).withFont('');
+    return sizingGroup(group.body, newOptions, options);
+  },
+
+  mathmlBuilder(group, options) {
+    // Figure out what style we're changing to.
+    const newStyle = styleMap$1[group.style];
+    const newOptions = options.havingStyle(newStyle);
+    const inner = buildExpression$1(group.body, newOptions);
+    const node = new mathMLTree.MathNode("mstyle", inner);
+    const styleAttributes = {
+      "display": ["0", "true"],
+      "text": ["0", "false"],
+      "script": ["1", "false"],
+      "scriptscript": ["2", "false"]
+    };
+    const attr = styleAttributes[group.style];
+    node.setAttribute("scriptlevel", attr[0]);
+    node.setAttribute("displaystyle", attr[1]);
+    return node;
+  }
+
+});
+
+/**
+ * Sometimes, groups perform special rules when they have superscripts or
+ * subscripts attached to them. This function lets the `supsub` group know that
+ * Sometimes, groups perform special rules when they have superscripts or
+ * its inner element should handle the superscripts and subscripts instead of
+ * handling them itself.
+ */
+const htmlBuilderDelegate = function htmlBuilderDelegate(group, options) {
+  const base = group.base;
+
+  if (!base) {
+    return null;
+  } else if (base.type === "op") {
+    // Operators handle supsubs differently when they have limits
+    // (e.g. `\displaystyle\sum_2^3`)
+    const delegate = base.limits && (options.style.size === Style$1.DISPLAY.size || base.alwaysHandleSupSub);
+    return delegate ? htmlBuilder$8 : null;
+  } else if (base.type === "operatorname") {
+    const delegate = base.alwaysHandleSupSub && (options.style.size === Style$1.DISPLAY.size || base.limits);
+    return delegate ? htmlBuilder$9 : null;
+  } else if (base.type === "accent") {
+    return utils.isCharacterBox(base.base) ? htmlBuilder : null;
+  } else if (base.type === "horizBrace") {
+    const isSup = !group.sub;
+    return isSup === base.isOver ? htmlBuilder$7 : null;
+  } else {
+    return null;
+  }
+}; // Super scripts and subscripts, whose precise placement can depend on other
+// functions that precede them.
+
+
+defineFunctionBuilders({
+  type: "supsub",
+
+  htmlBuilder(group, options) {
+    // Superscript and subscripts are handled in the TeXbook on page
+    // 445-446, rules 18(a-f).
+    // Here is where we defer to the inner group if it should handle
+    // superscripts and subscripts itself.
+    const builderDelegate = htmlBuilderDelegate(group, options);
+
+    if (builderDelegate) {
+      return builderDelegate(group, options);
+    }
+
+    const valueBase = group.base,
+          valueSup = group.sup,
+          valueSub = group.sub;
+    const base = buildGroup(valueBase, options);
+    let supm;
+    let subm;
+    const metrics = options.fontMetrics(); // Rule 18a
+
+    let supShift = 0;
+    let subShift = 0;
+    const isCharacterBox = valueBase && utils.isCharacterBox(valueBase);
+
+    if (valueSup) {
+      const newOptions = options.havingStyle(options.style.sup());
+      supm = buildGroup(valueSup, newOptions, options);
+
+      if (!isCharacterBox) {
+        supShift = base.height - newOptions.fontMetrics().supDrop * newOptions.sizeMultiplier / options.sizeMultiplier;
+      }
+    }
+
+    if (valueSub) {
+      const newOptions = options.havingStyle(options.style.sub());
+      subm = buildGroup(valueSub, newOptions, options);
+
+      if (!isCharacterBox) {
+        subShift = base.depth + newOptions.fontMetrics().subDrop * newOptions.sizeMultiplier / options.sizeMultiplier;
+      }
+    } // Rule 18c
+
+
+    let minSupShift;
+
+    if (options.style === Style$1.DISPLAY) {
+      minSupShift = metrics.sup1;
+    } else if (options.style.cramped) {
+      minSupShift = metrics.sup3;
+    } else {
+      minSupShift = metrics.sup2;
+    } // scriptspace is a font-size-independent size, so scale it
+    // appropriately for use as the marginRight.
+
+
+    const multiplier = options.sizeMultiplier;
+    const marginRight = 0.5 / metrics.ptPerEm / multiplier + "em";
+    let marginLeft = null;
+
+    if (subm) {
+      // Subscripts shouldn't be shifted by the base's italic correction.
+      // Account for that by shifting the subscript back the appropriate
+      // amount. Note we only do this when the base is a single symbol.
+      const isOiint = group.base && group.base.type === "op" && group.base.name && (group.base.name === "\\oiint" || group.base.name === "\\oiiint");
+
+      if (base instanceof SymbolNode || isOiint) {
+        // $FlowFixMe
+        marginLeft = -base.italic + "em";
+      }
+    }
+
+    let supsub;
+
+    if (supm && subm) {
+      supShift = Math.max(supShift, minSupShift, supm.depth + 0.25 * metrics.xHeight);
+      subShift = Math.max(subShift, metrics.sub2);
+      const ruleWidth = metrics.defaultRuleThickness; // Rule 18e
+
+      const maxWidth = 4 * ruleWidth;
+
+      if (supShift - supm.depth - (subm.height - subShift) < maxWidth) {
+        subShift = maxWidth - (supShift - supm.depth) + subm.height;
+        const psi = 0.8 * metrics.xHeight - (supShift - supm.depth);
+
+        if (psi > 0) {
+          supShift += psi;
+          subShift -= psi;
+        }
+      }
+
+      const vlistElem = [{
+        type: "elem",
+        elem: subm,
+        shift: subShift,
+        marginRight,
+        marginLeft
+      }, {
+        type: "elem",
+        elem: supm,
+        shift: -supShift,
+        marginRight
+      }];
+      supsub = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: vlistElem
+      }, options);
+    } else if (subm) {
+      // Rule 18b
+      subShift = Math.max(subShift, metrics.sub1, subm.height - 0.8 * metrics.xHeight);
+      const vlistElem = [{
+        type: "elem",
+        elem: subm,
+        marginLeft,
+        marginRight
+      }];
+      supsub = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: subShift,
+        children: vlistElem
+      }, options);
+    } else if (supm) {
+      // Rule 18c, d
+      supShift = Math.max(supShift, minSupShift, supm.depth + 0.25 * metrics.xHeight);
+      supsub = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: -supShift,
+        children: [{
+          type: "elem",
+          elem: supm,
+          marginRight
+        }]
+      }, options);
+    } else {
+      throw new Error("supsub must have either sup or sub.");
+    } // Wrap the supsub vlist in a span.msupsub to reset text-align.
+
+
+    const mclass = getTypeOfDomTree(base, "right") || "mord";
+    return buildCommon.makeSpan([mclass], [base, buildCommon.makeSpan(["msupsub"], [supsub])], options);
+  },
+
+  mathmlBuilder(group, options) {
+    // Is the inner group a relevant horizonal brace?
+    let isBrace = false;
+    let isOver;
+    let isSup;
+
+    if (group.base && group.base.type === "horizBrace") {
+      isSup = !!group.sup;
+
+      if (isSup === group.base.isOver) {
+        isBrace = true;
+        isOver = group.base.isOver;
+      }
+    }
+
+    if (group.base && (group.base.type === "op" || group.base.type === "operatorname")) {
+      group.base.parentIsSupSub = true;
+    }
+
+    const children = [buildGroup$1(group.base, options)];
+
+    if (group.sub) {
+      children.push(buildGroup$1(group.sub, options));
+    }
+
+    if (group.sup) {
+      children.push(buildGroup$1(group.sup, options));
+    }
+
+    let nodeType;
+
+    if (isBrace) {
+      nodeType = isOver ? "mover" : "munder";
+    } else if (!group.sub) {
+      const base = group.base;
+
+      if (base && base.type === "op" && base.limits && (options.style === Style$1.DISPLAY || base.alwaysHandleSupSub)) {
+        nodeType = "mover";
+      } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub && (base.limits || options.style === Style$1.DISPLAY)) {
+        nodeType = "mover";
+      } else {
+        nodeType = "msup";
+      }
+    } else if (!group.sup) {
+      const base = group.base;
+
+      if (base && base.type === "op" && base.limits && (options.style === Style$1.DISPLAY || base.alwaysHandleSupSub)) {
+        nodeType = "munder";
+      } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub && (base.limits || options.style === Style$1.DISPLAY)) {
+        nodeType = "munder";
+      } else {
+        nodeType = "msub";
+      }
+    } else {
+      const base = group.base;
+
+      if (base && base.type === "op" && base.limits && options.style === Style$1.DISPLAY) {
+        nodeType = "munderover";
+      } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub && (options.style === Style$1.DISPLAY || base.limits)) {
+        nodeType = "munderover";
+      } else {
+        nodeType = "msubsup";
+      }
+    }
+
+    const node = new mathMLTree.MathNode(nodeType, children);
+    return node;
+  }
+
+});
+
+defineFunctionBuilders({
+  type: "atom",
+
+  htmlBuilder(group, options) {
+    return buildCommon.mathsym(group.text, group.mode, options, ["m" + group.family]);
+  },
+
+  mathmlBuilder(group, options) {
+    const node = new mathMLTree.MathNode("mo", [makeText(group.text, group.mode)]);
+
+    if (group.family === "bin") {
+      const variant = getVariant(group, options);
+
+      if (variant === "bold-italic") {
+        node.setAttribute("mathvariant", variant);
+      }
+    } else if (group.family === "punct") {
+      node.setAttribute("separator", "true");
+    } else if (group.family === "open" || group.family === "close") {
+      // Delims built here should not stretch vertically.
+      // See delimsizing.js for stretchy delims.
+      node.setAttribute("stretchy", "false");
+    }
+
+    return node;
+  }
+
+});
+
+// "mathord" and "textord" ParseNodes created in Parser.js from symbol Groups in
+const defaultVariant = {
+  "mi": "italic",
+  "mn": "normal",
+  "mtext": "normal"
+};
+defineFunctionBuilders({
+  type: "mathord",
+
+  htmlBuilder(group, options) {
+    return buildCommon.makeOrd(group, options, "mathord");
+  },
+
+  mathmlBuilder(group, options) {
+    const node = new mathMLTree.MathNode("mi", [makeText(group.text, group.mode, options)]);
+    const variant = getVariant(group, options) || "italic";
+
+    if (variant !== defaultVariant[node.type]) {
+      node.setAttribute("mathvariant", variant);
+    }
+
+    return node;
+  }
+
+});
+defineFunctionBuilders({
+  type: "textord",
+
+  htmlBuilder(group, options) {
+    return buildCommon.makeOrd(group, options, "textord");
+  },
+
+  mathmlBuilder(group, options) {
+    const text = makeText(group.text, group.mode, options);
+    const variant = getVariant(group, options) || "normal";
+    let node;
+
+    if (group.mode === 'text') {
+      node = new mathMLTree.MathNode("mtext", [text]);
+    } else if (/[0-9]/.test(group.text)) {
+      // TODO(kevinb) merge adjacent <mn> nodes
+      // do it as a post processing step
+      node = new mathMLTree.MathNode("mn", [text]);
+    } else if (group.text === "\\prime") {
+      node = new mathMLTree.MathNode("mo", [text]);
+    } else {
+      node = new mathMLTree.MathNode("mi", [text]);
+    }
+
+    if (variant !== defaultVariant[node.type]) {
+      node.setAttribute("mathvariant", variant);
+    }
+
+    return node;
+  }
+
+});
+
+const cssSpace = {
+  "\\nobreak": "nobreak",
+  "\\allowbreak": "allowbreak"
+}; // A lookup table to determine whether a spacing function/symbol should be
+// treated like a regular space character.  If a symbol or command is a key
+// in this table, then it should be a regular space character.  Furthermore,
+// the associated value may have a `className` specifying an extra CSS class
+// to add to the created `span`.
+
+const regularSpace = {
+  " ": {},
+  "\\ ": {},
+  "~": {
+    className: "nobreak"
+  },
+  "\\space": {},
+  "\\nobreakspace": {
+    className: "nobreak"
+  }
+}; // ParseNode<"spacing"> created in Parser.js from the "spacing" symbol Groups in
+// src/symbols.js.
+
+defineFunctionBuilders({
+  type: "spacing",
+
+  htmlBuilder(group, options) {
+    if (regularSpace.hasOwnProperty(group.text)) {
+      const className = regularSpace[group.text].className || ""; // Spaces are generated by adding an actual space. Each of these
+      // things has an entry in the symbols table, so these will be turned
+      // into appropriate outputs.
+
+      if (group.mode === "text") {
+        const ord = buildCommon.makeOrd(group, options, "textord");
+        ord.classes.push(className);
+        return ord;
+      } else {
+        return buildCommon.makeSpan(["mspace", className], [buildCommon.mathsym(group.text, group.mode, options)], options);
+      }
+    } else if (cssSpace.hasOwnProperty(group.text)) {
+      // Spaces based on just a CSS class.
+      return buildCommon.makeSpan(["mspace", cssSpace[group.text]], [], options);
+    } else {
+      throw new ParseError(`Unknown type of space "${group.text}"`);
+    }
+  },
+
+  mathmlBuilder(group, options) {
+    let node;
+
+    if (regularSpace.hasOwnProperty(group.text)) {
+      node = new mathMLTree.MathNode("mtext", [new mathMLTree.TextNode("\u00a0")]);
+    } else if (cssSpace.hasOwnProperty(group.text)) {
+      // CSS-based MathML spaces (\nobreak, \allowbreak) are ignored
+      return new mathMLTree.MathNode("mspace");
+    } else {
+      throw new ParseError(`Unknown type of space "${group.text}"`);
+    }
+
+    return node;
+  }
+
+});
+
+const pad = () => {
+  const padNode = new mathMLTree.MathNode("mtd", []);
+  padNode.setAttribute("width", "50%");
+  return padNode;
+};
+
+defineFunctionBuilders({
+  type: "tag",
+
+  mathmlBuilder(group, options) {
+    const table = new mathMLTree.MathNode("mtable", [new mathMLTree.MathNode("mtr", [pad(), new mathMLTree.MathNode("mtd", [buildExpressionRow(group.body, options)]), pad(), new mathMLTree.MathNode("mtd", [buildExpressionRow(group.tag, options)])])]);
+    table.setAttribute("width", "100%");
+    return table; // TODO: Left-aligned tags.
+    // Currently, the group and options passed here do not contain
+    // enough info to set tag alignment. `leqno` is in Settings but it is
+    // not passed to Options. On the HTML side, leqno is
+    // set by a CSS class applied in buildTree.js. That would have worked
+    // in MathML if browsers supported <mlabeledtr>. Since they don't, we
+    // need to rewrite the way this function is called.
+  }
+
+});
+
+const textFontFamilies = {
+  "\\text": undefined,
+  "\\textrm": "textrm",
+  "\\textsf": "textsf",
+  "\\texttt": "texttt",
+  "\\textnormal": "textrm"
+};
+const textFontWeights = {
+  "\\textbf": "textbf",
+  "\\textmd": "textmd"
+};
+const textFontShapes = {
+  "\\textit": "textit",
+  "\\textup": "textup"
+};
+
+const optionsWithFont = (group, options) => {
+  const font = group.font; // Checks if the argument is a font family or a font style.
+
+  if (!font) {
+    return options;
+  } else if (textFontFamilies[font]) {
+    return options.withTextFontFamily(textFontFamilies[font]);
+  } else if (textFontWeights[font]) {
+    return options.withTextFontWeight(textFontWeights[font]);
+  } else {
+    return options.withTextFontShape(textFontShapes[font]);
+  }
+};
+
+defineFunction({
+  type: "text",
+  names: [// Font families
+  "\\text", "\\textrm", "\\textsf", "\\texttt", "\\textnormal", // Font weights
+  "\\textbf", "\\textmd", // Font Shapes
+  "\\textit", "\\textup"],
+  props: {
+    numArgs: 1,
+    argTypes: ["text"],
+    greediness: 2,
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    return {
+      type: "text",
+      mode: parser.mode,
+      body: ordargument(body),
+      font: funcName
+    };
+  },
+
+  htmlBuilder(group, options) {
+    const newOptions = optionsWithFont(group, options);
+    const inner = buildExpression(group.body, newOptions, true);
+    return buildCommon.makeSpan(["mord", "text"], buildCommon.tryCombineChars(inner), newOptions);
+  },
+
+  mathmlBuilder(group, options) {
+    const newOptions = optionsWithFont(group, options);
+    return buildExpressionRow(group.body, newOptions);
+  }
+
+});
+
+defineFunction({
+  type: "underline",
+  names: ["\\underline"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    return {
+      type: "underline",
+      mode: parser.mode,
+      body: args[0]
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Underlines are handled in the TeXbook pg 443, Rule 10.
+    // Build the inner group.
+    const innerGroup = buildGroup(group.body, options); // Create the line to go below the body
+
+    const line = buildCommon.makeLineSpan("underline-line", options); // Generate the vlist, with the appropriate kerns
+
+    const defaultRuleThickness = options.fontMetrics().defaultRuleThickness;
+    const vlist = buildCommon.makeVList({
+      positionType: "top",
+      positionData: innerGroup.height,
+      children: [{
+        type: "kern",
+        size: defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: line
+      }, {
+        type: "kern",
+        size: 3 * defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: innerGroup
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "underline"], [vlist], options);
+  },
+
+  mathmlBuilder(group, options) {
+    const operator = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode("\u203e")]);
+    operator.setAttribute("stretchy", "true");
+    const node = new mathMLTree.MathNode("munder", [buildGroup$1(group.body, options), operator]);
+    node.setAttribute("accentunder", "true");
+    return node;
+  }
+
+});
+
+defineFunction({
+  type: "verb",
+  names: ["\\verb"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(context, args, optArgs) {
+    // \verb and \verb* are dealt with directly in Parser.js.
+    // If we end up here, it's because of a failure to match the two delimiters
+    // in the regex in Lexer.js.  LaTeX raises the following error when \verb is
+    // terminated by end of line (or file).
+    throw new ParseError("\\verb ended by end of line instead of matching delimiter");
+  },
+
+  htmlBuilder(group, options) {
+    const text = makeVerb(group);
+    const body = []; // \verb enters text mode and therefore is sized like \textstyle
+
+    const newOptions = options.havingStyle(options.style.text());
+
+    for (let i = 0; i < text.length; i++) {
+      let c = text[i];
+
+      if (c === '~') {
+        c = '\\textasciitilde';
+      }
+
+      body.push(buildCommon.makeSymbol(c, "Typewriter-Regular", group.mode, newOptions, ["mord", "texttt"]));
+    }
+
+    return buildCommon.makeSpan(["mord", "text"].concat(newOptions.sizingClasses(options)), buildCommon.tryCombineChars(body), newOptions);
+  },
+
+  mathmlBuilder(group, options) {
+    const text = new mathMLTree.TextNode(makeVerb(group));
+    const node = new mathMLTree.MathNode("mtext", [text]);
+    node.setAttribute("mathvariant", "monospace");
+    return node;
+  }
+
+});
+/**
+ * Converts verb group into body string.
+ *
+ * \verb* replaces each space with an open box \u2423
+ * \verb replaces each space with a no-break space \xA0
+ */
+
+const makeVerb = group => group.body.replace(/ /g, group.star ? '\u2423' : '\xA0');
+
+/** Include this to ensure that all functions are defined. */
+const functions = _functions;
+
+/**
+ * The Lexer class handles tokenizing the input in various ways. Since our
+ * parser expects us to be able to backtrack, the lexer allows lexing from any
+ * given starting point.
+ *
+ * Its main exposed function is the `lex` function, which takes a position to
+ * lex from and a type of token to lex. It defers to the appropriate `_innerLex`
+ * function.
+ *
+ * The various `_innerLex` functions perform the actual lexing of different
+ * kinds.
+ */
+
+/* The following tokenRegex
+ * - matches typical whitespace (but not NBSP etc.) using its first group
+ * - does not match any control character \x00-\x1f except whitespace
+ * - does not match a bare backslash
+ * - matches any ASCII character except those just mentioned
+ * - does not match the BMP private use area \uE000-\uF8FF
+ * - does not match bare surrogate code units
+ * - matches any BMP character except for those just described
+ * - matches any valid Unicode surrogate pair
+ * - matches a backslash followed by one or more letters
+ * - matches a backslash followed by any BMP character, including newline
+ * Just because the Lexer matches something doesn't mean it's valid input:
+ * If there is no matching function or symbol definition, the Parser will
+ * still reject the input.
+ */
+const spaceRegexString = "[ \r\n\t]";
+const controlWordRegexString = "\\\\[a-zA-Z@]+";
+const controlSymbolRegexString = "\\\\[^\uD800-\uDFFF]";
+const controlWordWhitespaceRegexString = `${controlWordRegexString}${spaceRegexString}*`;
+const controlWordWhitespaceRegex = new RegExp(`^(${controlWordRegexString})${spaceRegexString}*$`);
+const combiningDiacriticalMarkString = "[\u0300-\u036f]";
+const combiningDiacriticalMarksEndRegex = new RegExp(`${combiningDiacriticalMarkString}+$`);
+const tokenRegexString = `(${spaceRegexString}+)|` + // whitespace
+"([!-\\[\\]-\u2027\u202A-\uD7FF\uF900-\uFFFF]" + // single codepoint
+`${combiningDiacriticalMarkString}*` + // ...plus accents
+"|[\uD800-\uDBFF][\uDC00-\uDFFF]" + // surrogate pair
+`${combiningDiacriticalMarkString}*` + // ...plus accents
+"|\\\\verb\\*([^]).*?\\3" + // \verb*
+"|\\\\verb([^*a-zA-Z]).*?\\4" + // \verb unstarred
+"|\\\\operatorname\\*" + // \operatorname*
+`|${controlWordWhitespaceRegexString}` + // \macroName + spaces
+`|${controlSymbolRegexString})`; // \\, \', etc.
+
+/** Main Lexer class */
+
+class Lexer {
+  // category codes, only supports comment characters (14) for now
+  constructor(input, settings) {
+    this.input = void 0;
+    this.settings = void 0;
+    this.tokenRegex = void 0;
+    this.catcodes = void 0;
+    // Separate accents from characters
+    this.input = input;
+    this.settings = settings;
+    this.tokenRegex = new RegExp(tokenRegexString, 'g');
+    this.catcodes = {
+      "%": 14 // comment character
+
+    };
+  }
+
+  setCatcode(char, code) {
+    this.catcodes[char] = code;
+  }
+  /**
+   * This function lexes a single token.
+   */
+
+
+  lex() {
+    const input = this.input;
+    const pos = this.tokenRegex.lastIndex;
+
+    if (pos === input.length) {
+      return new Token("EOF", new SourceLocation(this, pos, pos));
+    }
+
+    const match = this.tokenRegex.exec(input);
+
+    if (match === null || match.index !== pos) {
+      throw new ParseError(`Unexpected character: '${input[pos]}'`, new Token(input[pos], new SourceLocation(this, pos, pos + 1)));
+    }
+
+    let text = match[2] || " ";
+
+    if (this.catcodes[text] === 14) {
+      // comment character
+      const nlIndex = input.indexOf('\n', this.tokenRegex.lastIndex);
+
+      if (nlIndex === -1) {
+        this.tokenRegex.lastIndex = input.length; // EOF
+
+        this.settings.reportNonstrict("commentAtEnd", "% comment has no terminating newline; LaTeX would " + "fail because of commenting the end of math mode (e.g. $)");
+      } else {
+        this.tokenRegex.lastIndex = nlIndex + 1;
+      }
+
+      return this.lex();
+    } // Trim any trailing whitespace from control word match
+
+
+    const controlMatch = text.match(controlWordWhitespaceRegex);
+
+    if (controlMatch) {
+      text = controlMatch[1];
+    }
+
+    return new Token(text, new SourceLocation(this, pos, this.tokenRegex.lastIndex));
+  }
+
+}
+
+/**
+ * A `Namespace` refers to a space of nameable things like macros or lengths,
+ * which can be `set` either globally or local to a nested group, using an
+ * undo stack similar to how TeX implements this functionality.
+ * Performance-wise, `get` and local `set` take constant time, while global
+ * `set` takes time proportional to the depth of group nesting.
+ */
+class Namespace {
+  /**
+   * Both arguments are optional.  The first argument is an object of
+   * built-in mappings which never change.  The second argument is an object
+   * of initial (global-level) mappings, which will constantly change
+   * according to any global/top-level `set`s done.
+   */
+  constructor(builtins, globalMacros) {
+    if (builtins === void 0) {
+      builtins = {};
+    }
+
+    if (globalMacros === void 0) {
+      globalMacros = {};
+    }
+
+    this.current = void 0;
+    this.builtins = void 0;
+    this.undefStack = void 0;
+    this.current = globalMacros;
+    this.builtins = builtins;
+    this.undefStack = [];
+  }
+  /**
+   * Start a new nested group, affecting future local `set`s.
+   */
+
+
+  beginGroup() {
+    this.undefStack.push({});
+  }
+  /**
+   * End current nested group, restoring values before the group began.
+   */
+
+
+  endGroup() {
+    if (this.undefStack.length === 0) {
+      throw new ParseError("Unbalanced namespace destruction: attempt " + "to pop global namespace; please report this as a bug");
+    }
+
+    const undefs = this.undefStack.pop();
+
+    for (const undef in undefs) {
+      if (undefs.hasOwnProperty(undef)) {
+        if (undefs[undef] === undefined) {
+          delete this.current[undef];
+        } else {
+          this.current[undef] = undefs[undef];
+        }
+      }
+    }
+  }
+  /**
+   * Detect whether `name` has a definition.  Equivalent to
+   * `get(name) != null`.
+   */
+
+
+  has(name) {
+    return this.current.hasOwnProperty(name) || this.builtins.hasOwnProperty(name);
+  }
+  /**
+   * Get the current value of a name, or `undefined` if there is no value.
+   *
+   * Note: Do not use `if (namespace.get(...))` to detect whether a macro
+   * is defined, as the definition may be the empty string which evaluates
+   * to `false` in JavaScript.  Use `if (namespace.get(...) != null)` or
+   * `if (namespace.has(...))`.
+   */
+
+
+  get(name) {
+    if (this.current.hasOwnProperty(name)) {
+      return this.current[name];
+    } else {
+      return this.builtins[name];
+    }
+  }
+  /**
+   * Set the current value of a name, and optionally set it globally too.
+   * Local set() sets the current value and (when appropriate) adds an undo
+   * operation to the undo stack.  Global set() may change the undo
+   * operation at every level, so takes time linear in their number.
+   */
+
+
+  set(name, value, global) {
+    if (global === void 0) {
+      global = false;
+    }
+
+    if (global) {
+      // Global set is equivalent to setting in all groups.  Simulate this
+      // by destroying any undos currently scheduled for this name,
+      // and adding an undo with the *new* value (in case it later gets
+      // locally reset within this environment).
+      for (let i = 0; i < this.undefStack.length; i++) {
+        delete this.undefStack[i][name];
+      }
+
+      if (this.undefStack.length > 0) {
+        this.undefStack[this.undefStack.length - 1][name] = value;
+      }
+    } else {
+      // Undo this set at end of this group (possibly to `undefined`),
+      // unless an undo is already in place, in which case that older
+      // value is the correct one.
+      const top = this.undefStack[this.undefStack.length - 1];
+
+      if (top && !top.hasOwnProperty(name)) {
+        top[name] = this.current[name];
+      }
+    }
+
+    this.current[name] = value;
+  }
+
+}
+
+/**
+ * Predefined macros for KaTeX.
+ * This can be used to define some commands in terms of others.
+ */
+const builtinMacros = {};
+
+function defineMacro(name, body) {
+  builtinMacros[name] = body;
+} //////////////////////////////////////////////////////////////////////
+// macro tools
+
+defineMacro("\\noexpand", function (context) {
+  // The expansion is the token itself; but that token is interpreted
+  // as if its meaning were ‘\relax’ if it is a control sequence that
+  // would ordinarily be expanded by TeX’s expansion rules.
+  const t = context.popToken();
+
+  if (context.isExpandable(t.text)) {
+    t.noexpand = true;
+    t.treatAsRelax = true;
+  }
+
+  return {
+    tokens: [t],
+    numArgs: 0
+  };
+});
+defineMacro("\\expandafter", function (context) {
+  // TeX first reads the token that comes immediately after \expandafter,
+  // without expanding it; let’s call this token t. Then TeX reads the
+  // token that comes after t (and possibly more tokens, if that token
+  // has an argument), replacing it by its expansion. Finally TeX puts
+  // t back in front of that expansion.
+  const t = context.popToken();
+  context.expandOnce(true); // expand only an expandable token
+
+  return {
+    tokens: [t],
+    numArgs: 0
+  };
+}); // LaTeX's \@firstoftwo{#1}{#2} expands to #1, skipping #2
+// TeX source: \long\def\@firstoftwo#1#2{#1}
+
+defineMacro("\\@firstoftwo", function (context) {
+  const args = context.consumeArgs(2);
+  return {
+    tokens: args[0],
+    numArgs: 0
+  };
+}); // LaTeX's \@secondoftwo{#1}{#2} expands to #2, skipping #1
+// TeX source: \long\def\@secondoftwo#1#2{#2}
+
+defineMacro("\\@secondoftwo", function (context) {
+  const args = context.consumeArgs(2);
+  return {
+    tokens: args[1],
+    numArgs: 0
+  };
+}); // LaTeX's \@ifnextchar{#1}{#2}{#3} looks ahead to the next (unexpanded)
+// symbol that isn't a space, consuming any spaces but not consuming the
+// first nonspace character.  If that nonspace character matches #1, then
+// the macro expands to #2; otherwise, it expands to #3.
+
+defineMacro("\\@ifnextchar", function (context) {
+  const args = context.consumeArgs(3); // symbol, if, else
+
+  context.consumeSpaces();
+  const nextToken = context.future();
+
+  if (args[0].length === 1 && args[0][0].text === nextToken.text) {
+    return {
+      tokens: args[1],
+      numArgs: 0
+    };
+  } else {
+    return {
+      tokens: args[2],
+      numArgs: 0
+    };
+  }
+}); // LaTeX's \@ifstar{#1}{#2} looks ahead to the next (unexpanded) symbol.
+// If it is `*`, then it consumes the symbol, and the macro expands to #1;
+// otherwise, the macro expands to #2 (without consuming the symbol).
+// TeX source: \def\@ifstar#1{\@ifnextchar *{\@firstoftwo{#1}}}
+
+defineMacro("\\@ifstar", "\\@ifnextchar *{\\@firstoftwo{#1}}"); // LaTeX's \TextOrMath{#1}{#2} expands to #1 in text mode, #2 in math mode
+
+defineMacro("\\TextOrMath", function (context) {
+  const args = context.consumeArgs(2);
+
+  if (context.mode === 'text') {
+    return {
+      tokens: args[0],
+      numArgs: 0
+    };
+  } else {
+    return {
+      tokens: args[1],
+      numArgs: 0
+    };
+  }
+}); // Lookup table for parsing numbers in base 8 through 16
+
+const digitToNumber = {
+  "0": 0,
+  "1": 1,
+  "2": 2,
+  "3": 3,
+  "4": 4,
+  "5": 5,
+  "6": 6,
+  "7": 7,
+  "8": 8,
+  "9": 9,
+  "a": 10,
+  "A": 10,
+  "b": 11,
+  "B": 11,
+  "c": 12,
+  "C": 12,
+  "d": 13,
+  "D": 13,
+  "e": 14,
+  "E": 14,
+  "f": 15,
+  "F": 15
+}; // TeX \char makes a literal character (catcode 12) using the following forms:
+// (see The TeXBook, p. 43)
+//   \char123  -- decimal
+//   \char'123 -- octal
+//   \char"123 -- hex
+//   \char`x   -- character that can be written (i.e. isn't active)
+//   \char`\x  -- character that cannot be written (e.g. %)
+// These all refer to characters from the font, so we turn them into special
+// calls to a function \@char dealt with in the Parser.
+
+defineMacro("\\char", function (context) {
+  let token = context.popToken();
+  let base;
+  let number = '';
+
+  if (token.text === "'") {
+    base = 8;
+    token = context.popToken();
+  } else if (token.text === '"') {
+    base = 16;
+    token = context.popToken();
+  } else if (token.text === "`") {
+    token = context.popToken();
+
+    if (token.text[0] === "\\") {
+      number = token.text.charCodeAt(1);
+    } else if (token.text === "EOF") {
+      throw new ParseError("\\char` missing argument");
+    } else {
+      number = token.text.charCodeAt(0);
+    }
+  } else {
+    base = 10;
+  }
+
+  if (base) {
+    // Parse a number in the given base, starting with first `token`.
+    number = digitToNumber[token.text];
+
+    if (number == null || number >= base) {
+      throw new ParseError(`Invalid base-${base} digit ${token.text}`);
+    }
+
+    let digit;
+
+    while ((digit = digitToNumber[context.future().text]) != null && digit < base) {
+      number *= base;
+      number += digit;
+      context.popToken();
+    }
+  }
+
+  return `\\@char{${number}}`;
+}); // \newcommand{\macro}[args]{definition}
+// \renewcommand{\macro}[args]{definition}
+// TODO: Optional arguments: \newcommand{\macro}[args][default]{definition}
+
+const newcommand = (context, existsOK, nonexistsOK) => {
+  let arg = context.consumeArgs(1)[0];
+
+  if (arg.length !== 1) {
+    throw new ParseError("\\newcommand's first argument must be a macro name");
+  }
+
+  const name = arg[0].text;
+  const exists = context.isDefined(name);
+
+  if (exists && !existsOK) {
+    throw new ParseError(`\\newcommand{${name}} attempting to redefine ` + `${name}; use \\renewcommand`);
+  }
+
+  if (!exists && !nonexistsOK) {
+    throw new ParseError(`\\renewcommand{${name}} when command ${name} ` + `does not yet exist; use \\newcommand`);
+  }
+
+  let numArgs = 0;
+  arg = context.consumeArgs(1)[0];
+
+  if (arg.length === 1 && arg[0].text === "[") {
+    let argText = '';
+    let token = context.expandNextToken();
+
+    while (token.text !== "]" && token.text !== "EOF") {
+      // TODO: Should properly expand arg, e.g., ignore {}s
+      argText += token.text;
+      token = context.expandNextToken();
+    }
+
+    if (!argText.match(/^\s*[0-9]+\s*$/)) {
+      throw new ParseError(`Invalid number of arguments: ${argText}`);
+    }
+
+    numArgs = parseInt(argText);
+    arg = context.consumeArgs(1)[0];
+  } // Final arg is the expansion of the macro
+
+
+  context.macros.set(name, {
+    tokens: arg,
+    numArgs
+  });
+  return '';
+};
+
+defineMacro("\\newcommand", context => newcommand(context, false, true));
+defineMacro("\\renewcommand", context => newcommand(context, true, false));
+defineMacro("\\providecommand", context => newcommand(context, true, true)); // terminal (console) tools
+
+defineMacro("\\message", context => {
+  const arg = context.consumeArgs(1)[0]; // eslint-disable-next-line no-console
+
+  console.log(arg.reverse().map(token => token.text).join(""));
+  return '';
+});
+defineMacro("\\errmessage", context => {
+  const arg = context.consumeArgs(1)[0]; // eslint-disable-next-line no-console
+
+  console.error(arg.reverse().map(token => token.text).join(""));
+  return '';
+});
+defineMacro("\\show", context => {
+  const tok = context.popToken();
+  const name = tok.text; // eslint-disable-next-line no-console
+
+  console.log(tok, context.macros.get(name), functions[name], symbols.math[name], symbols.text[name]);
+  return '';
+}); //////////////////////////////////////////////////////////////////////
+// Grouping
+// \let\bgroup={ \let\egroup=}
+
+defineMacro("\\bgroup", "{");
+defineMacro("\\egroup", "}"); // Symbols from latex.ltx:
+// \def\lq{`}
+// \def\rq{'}
+// \def \aa {\r a}
+// \def \AA {\r A}
+
+defineMacro("\\lq", "`");
+defineMacro("\\rq", "'");
+defineMacro("\\aa", "\\r a");
+defineMacro("\\AA", "\\r A"); // Copyright (C) and registered (R) symbols. Use raw symbol in MathML.
+// \DeclareTextCommandDefault{\textcopyright}{\textcircled{c}}
+// \DeclareTextCommandDefault{\textregistered}{\textcircled{%
+//      \check@mathfonts\fontsize\sf@size\z@\math@fontsfalse\selectfont R}}
+// \DeclareRobustCommand{\copyright}{%
+//    \ifmmode{\nfss@text{\textcopyright}}\else\textcopyright\fi}
+
+defineMacro("\\textcopyright", "\\html@mathml{\\textcircled{c}}{\\char`©}");
+defineMacro("\\copyright", "\\TextOrMath{\\textcopyright}{\\text{\\textcopyright}}");
+defineMacro("\\textregistered", "\\html@mathml{\\textcircled{\\scriptsize R}}{\\char`®}"); // Characters omitted from Unicode range 1D400–1D7FF
+
+defineMacro("\u212C", "\\mathscr{B}"); // script
+
+defineMacro("\u2130", "\\mathscr{E}");
+defineMacro("\u2131", "\\mathscr{F}");
+defineMacro("\u210B", "\\mathscr{H}");
+defineMacro("\u2110", "\\mathscr{I}");
+defineMacro("\u2112", "\\mathscr{L}");
+defineMacro("\u2133", "\\mathscr{M}");
+defineMacro("\u211B", "\\mathscr{R}");
+defineMacro("\u212D", "\\mathfrak{C}"); // Fraktur
+
+defineMacro("\u210C", "\\mathfrak{H}");
+defineMacro("\u2128", "\\mathfrak{Z}"); // Define \Bbbk with a macro that works in both HTML and MathML.
+
+defineMacro("\\Bbbk", "\\Bbb{k}"); // Unicode middle dot
+// The KaTeX fonts do not contain U+00B7. Instead, \cdotp displays
+// the dot at U+22C5 and gives it punct spacing.
+
+defineMacro("\u00b7", "\\cdotp"); // \llap and \rlap render their contents in text mode
+
+defineMacro("\\llap", "\\mathllap{\\textrm{#1}}");
+defineMacro("\\rlap", "\\mathrlap{\\textrm{#1}}");
+defineMacro("\\clap", "\\mathclap{\\textrm{#1}}"); // \not is defined by base/fontmath.ltx via
+// \DeclareMathSymbol{\not}{\mathrel}{symbols}{"36}
+// It's thus treated like a \mathrel, but defined by a symbol that has zero
+// width but extends to the right.  We use \rlap to get that spacing.
+// For MathML we write U+0338 here. buildMathML.js will then do the overlay.
+
+defineMacro("\\not", '\\html@mathml{\\mathrel{\\mathrlap\\@not}}{\\char"338}'); // Negated symbols from base/fontmath.ltx:
+// \def\neq{\not=} \let\ne=\neq
+// \DeclareRobustCommand
+//   \notin{\mathrel{\m@th\mathpalette\c@ncel\in}}
+// \def\c@ncel#1#2{\m@th\ooalign{$\hfil#1\mkern1mu/\hfil$\crcr$#1#2$}}
+
+defineMacro("\\neq", "\\html@mathml{\\mathrel{\\not=}}{\\mathrel{\\char`≠}}");
+defineMacro("\\ne", "\\neq");
+defineMacro("\u2260", "\\neq");
+defineMacro("\\notin", "\\html@mathml{\\mathrel{{\\in}\\mathllap{/\\mskip1mu}}}" + "{\\mathrel{\\char`∉}}");
+defineMacro("\u2209", "\\notin"); // Unicode stacked relations
+
+defineMacro("\u2258", "\\html@mathml{" + "\\mathrel{=\\kern{-1em}\\raisebox{0.4em}{$\\scriptsize\\frown$}}" + "}{\\mathrel{\\char`\u2258}}");
+defineMacro("\u2259", "\\html@mathml{\\stackrel{\\tiny\\wedge}{=}}{\\mathrel{\\char`\u2258}}");
+defineMacro("\u225A", "\\html@mathml{\\stackrel{\\tiny\\vee}{=}}{\\mathrel{\\char`\u225A}}");
+defineMacro("\u225B", "\\html@mathml{\\stackrel{\\scriptsize\\star}{=}}" + "{\\mathrel{\\char`\u225B}}");
+defineMacro("\u225D", "\\html@mathml{\\stackrel{\\tiny\\mathrm{def}}{=}}" + "{\\mathrel{\\char`\u225D}}");
+defineMacro("\u225E", "\\html@mathml{\\stackrel{\\tiny\\mathrm{m}}{=}}" + "{\\mathrel{\\char`\u225E}}");
+defineMacro("\u225F", "\\html@mathml{\\stackrel{\\tiny?}{=}}{\\mathrel{\\char`\u225F}}"); // Misc Unicode
+
+defineMacro("\u27C2", "\\perp");
+defineMacro("\u203C", "\\mathclose{!\\mkern-0.8mu!}");
+defineMacro("\u220C", "\\notni");
+defineMacro("\u231C", "\\ulcorner");
+defineMacro("\u231D", "\\urcorner");
+defineMacro("\u231E", "\\llcorner");
+defineMacro("\u231F", "\\lrcorner");
+defineMacro("\u00A9", "\\copyright");
+defineMacro("\u00AE", "\\textregistered");
+defineMacro("\uFE0F", "\\textregistered"); // The KaTeX fonts have corners at codepoints that don't match Unicode.
+// For MathML purposes, use the Unicode code point.
+
+defineMacro("\\ulcorner", "\\html@mathml{\\@ulcorner}{\\mathop{\\char\"231c}}");
+defineMacro("\\urcorner", "\\html@mathml{\\@urcorner}{\\mathop{\\char\"231d}}");
+defineMacro("\\llcorner", "\\html@mathml{\\@llcorner}{\\mathop{\\char\"231e}}");
+defineMacro("\\lrcorner", "\\html@mathml{\\@lrcorner}{\\mathop{\\char\"231f}}"); //////////////////////////////////////////////////////////////////////
+// LaTeX_2ε
+// \vdots{\vbox{\baselineskip4\p@  \lineskiplimit\z@
+// \kern6\p@\hbox{.}\hbox{.}\hbox{.}}}
+// We'll call \varvdots, which gets a glyph from symbols.js.
+// The zero-width rule gets us an equivalent to the vertical 6pt kern.
+
+defineMacro("\\vdots", "\\mathord{\\varvdots\\rule{0pt}{15pt}}");
+defineMacro("\u22ee", "\\vdots"); //////////////////////////////////////////////////////////////////////
+// amsmath.sty
+// http://mirrors.concertpass.com/tex-archive/macros/latex/required/amsmath/amsmath.pdf
+// Italic Greek capital letters.  AMS defines these with \DeclareMathSymbol,
+// but they are equivalent to \mathit{\Letter}.
+
+defineMacro("\\varGamma", "\\mathit{\\Gamma}");
+defineMacro("\\varDelta", "\\mathit{\\Delta}");
+defineMacro("\\varTheta", "\\mathit{\\Theta}");
+defineMacro("\\varLambda", "\\mathit{\\Lambda}");
+defineMacro("\\varXi", "\\mathit{\\Xi}");
+defineMacro("\\varPi", "\\mathit{\\Pi}");
+defineMacro("\\varSigma", "\\mathit{\\Sigma}");
+defineMacro("\\varUpsilon", "\\mathit{\\Upsilon}");
+defineMacro("\\varPhi", "\\mathit{\\Phi}");
+defineMacro("\\varPsi", "\\mathit{\\Psi}");
+defineMacro("\\varOmega", "\\mathit{\\Omega}"); //\newcommand{\substack}[1]{\subarray{c}#1\endsubarray}
+
+defineMacro("\\substack", "\\begin{subarray}{c}#1\\end{subarray}"); // \renewcommand{\colon}{\nobreak\mskip2mu\mathpunct{}\nonscript
+// \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
+
+defineMacro("\\colon", "\\nobreak\\mskip2mu\\mathpunct{}" + "\\mathchoice{\\mkern-3mu}{\\mkern-3mu}{}{}{:}\\mskip6mu"); // \newcommand{\boxed}[1]{\fbox{\m@th$\displaystyle#1$}}
+
+defineMacro("\\boxed", "\\fbox{$\\displaystyle{#1}$}"); // \def\iff{\DOTSB\;\Longleftrightarrow\;}
+// \def\implies{\DOTSB\;\Longrightarrow\;}
+// \def\impliedby{\DOTSB\;\Longleftarrow\;}
+
+defineMacro("\\iff", "\\DOTSB\\;\\Longleftrightarrow\\;");
+defineMacro("\\implies", "\\DOTSB\\;\\Longrightarrow\\;");
+defineMacro("\\impliedby", "\\DOTSB\\;\\Longleftarrow\\;"); // AMSMath's automatic \dots, based on \mdots@@ macro.
+
+const dotsByToken = {
+  ',': '\\dotsc',
+  '\\not': '\\dotsb',
+  // \keybin@ checks for the following:
+  '+': '\\dotsb',
+  '=': '\\dotsb',
+  '<': '\\dotsb',
+  '>': '\\dotsb',
+  '-': '\\dotsb',
+  '*': '\\dotsb',
+  ':': '\\dotsb',
+  // Symbols whose definition starts with \DOTSB:
+  '\\DOTSB': '\\dotsb',
+  '\\coprod': '\\dotsb',
+  '\\bigvee': '\\dotsb',
+  '\\bigwedge': '\\dotsb',
+  '\\biguplus': '\\dotsb',
+  '\\bigcap': '\\dotsb',
+  '\\bigcup': '\\dotsb',
+  '\\prod': '\\dotsb',
+  '\\sum': '\\dotsb',
+  '\\bigotimes': '\\dotsb',
+  '\\bigoplus': '\\dotsb',
+  '\\bigodot': '\\dotsb',
+  '\\bigsqcup': '\\dotsb',
+  '\\And': '\\dotsb',
+  '\\longrightarrow': '\\dotsb',
+  '\\Longrightarrow': '\\dotsb',
+  '\\longleftarrow': '\\dotsb',
+  '\\Longleftarrow': '\\dotsb',
+  '\\longleftrightarrow': '\\dotsb',
+  '\\Longleftrightarrow': '\\dotsb',
+  '\\mapsto': '\\dotsb',
+  '\\longmapsto': '\\dotsb',
+  '\\hookrightarrow': '\\dotsb',
+  '\\doteq': '\\dotsb',
+  // Symbols whose definition starts with \mathbin:
+  '\\mathbin': '\\dotsb',
+  // Symbols whose definition starts with \mathrel:
+  '\\mathrel': '\\dotsb',
+  '\\relbar': '\\dotsb',
+  '\\Relbar': '\\dotsb',
+  '\\xrightarrow': '\\dotsb',
+  '\\xleftarrow': '\\dotsb',
+  // Symbols whose definition starts with \DOTSI:
+  '\\DOTSI': '\\dotsi',
+  '\\int': '\\dotsi',
+  '\\oint': '\\dotsi',
+  '\\iint': '\\dotsi',
+  '\\iiint': '\\dotsi',
+  '\\iiiint': '\\dotsi',
+  '\\idotsint': '\\dotsi',
+  // Symbols whose definition starts with \DOTSX:
+  '\\DOTSX': '\\dotsx'
+};
+defineMacro("\\dots", function (context) {
+  // TODO: If used in text mode, should expand to \textellipsis.
+  // However, in KaTeX, \textellipsis and \ldots behave the same
+  // (in text mode), and it's unlikely we'd see any of the math commands
+  // that affect the behavior of \dots when in text mode.  So fine for now
+  // (until we support \ifmmode ... \else ... \fi).
+  let thedots = '\\dotso';
+  const next = context.expandAfterFuture().text;
+
+  if (next in dotsByToken) {
+    thedots = dotsByToken[next];
+  } else if (next.substr(0, 4) === '\\not') {
+    thedots = '\\dotsb';
+  } else if (next in symbols.math) {
+    if (utils.contains(['bin', 'rel'], symbols.math[next].group)) {
+      thedots = '\\dotsb';
+    }
+  }
+
+  return thedots;
+});
+const spaceAfterDots = {
+  // \rightdelim@ checks for the following:
+  ')': true,
+  ']': true,
+  '\\rbrack': true,
+  '\\}': true,
+  '\\rbrace': true,
+  '\\rangle': true,
+  '\\rceil': true,
+  '\\rfloor': true,
+  '\\rgroup': true,
+  '\\rmoustache': true,
+  '\\right': true,
+  '\\bigr': true,
+  '\\biggr': true,
+  '\\Bigr': true,
+  '\\Biggr': true,
+  // \extra@ also tests for the following:
+  '$': true,
+  // \extrap@ checks for the following:
+  ';': true,
+  '.': true,
+  ',': true
+};
+defineMacro("\\dotso", function (context) {
+  const next = context.future().text;
+
+  if (next in spaceAfterDots) {
+    return "\\ldots\\,";
+  } else {
+    return "\\ldots";
+  }
+});
+defineMacro("\\dotsc", function (context) {
+  const next = context.future().text; // \dotsc uses \extra@ but not \extrap@, instead specially checking for
+  // ';' and '.', but doesn't check for ','.
+
+  if (next in spaceAfterDots && next !== ',') {
+    return "\\ldots\\,";
+  } else {
+    return "\\ldots";
+  }
+});
+defineMacro("\\cdots", function (context) {
+  const next = context.future().text;
+
+  if (next in spaceAfterDots) {
+    return "\\@cdots\\,";
+  } else {
+    return "\\@cdots";
+  }
+});
+defineMacro("\\dotsb", "\\cdots");
+defineMacro("\\dotsm", "\\cdots");
+defineMacro("\\dotsi", "\\!\\cdots"); // amsmath doesn't actually define \dotsx, but \dots followed by a macro
+// starting with \DOTSX implies \dotso, and then \extra@ detects this case
+// and forces the added `\,`.
+
+defineMacro("\\dotsx", "\\ldots\\,"); // \let\DOTSI\relax
+// \let\DOTSB\relax
+// \let\DOTSX\relax
+
+defineMacro("\\DOTSI", "\\relax");
+defineMacro("\\DOTSB", "\\relax");
+defineMacro("\\DOTSX", "\\relax"); // Spacing, based on amsmath.sty's override of LaTeX defaults
+// \DeclareRobustCommand{\tmspace}[3]{%
+//   \ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
+
+defineMacro("\\tmspace", "\\TextOrMath{\\kern#1#3}{\\mskip#1#2}\\relax"); // \renewcommand{\,}{\tmspace+\thinmuskip{.1667em}}
+// TODO: math mode should use \thinmuskip
+
+defineMacro("\\,", "\\tmspace+{3mu}{.1667em}"); // \let\thinspace\,
+
+defineMacro("\\thinspace", "\\,"); // \def\>{\mskip\medmuskip}
+// \renewcommand{\:}{\tmspace+\medmuskip{.2222em}}
+// TODO: \> and math mode of \: should use \medmuskip = 4mu plus 2mu minus 4mu
+
+defineMacro("\\>", "\\mskip{4mu}");
+defineMacro("\\:", "\\tmspace+{4mu}{.2222em}"); // \let\medspace\:
+
+defineMacro("\\medspace", "\\:"); // \renewcommand{\;}{\tmspace+\thickmuskip{.2777em}}
+// TODO: math mode should use \thickmuskip = 5mu plus 5mu
+
+defineMacro("\\;", "\\tmspace+{5mu}{.2777em}"); // \let\thickspace\;
+
+defineMacro("\\thickspace", "\\;"); // \renewcommand{\!}{\tmspace-\thinmuskip{.1667em}}
+// TODO: math mode should use \thinmuskip
+
+defineMacro("\\!", "\\tmspace-{3mu}{.1667em}"); // \let\negthinspace\!
+
+defineMacro("\\negthinspace", "\\!"); // \newcommand{\negmedspace}{\tmspace-\medmuskip{.2222em}}
+// TODO: math mode should use \medmuskip
+
+defineMacro("\\negmedspace", "\\tmspace-{4mu}{.2222em}"); // \newcommand{\negthickspace}{\tmspace-\thickmuskip{.2777em}}
+// TODO: math mode should use \thickmuskip
+
+defineMacro("\\negthickspace", "\\tmspace-{5mu}{.277em}"); // \def\enspace{\kern.5em }
+
+defineMacro("\\enspace", "\\kern.5em "); // \def\enskip{\hskip.5em\relax}
+
+defineMacro("\\enskip", "\\hskip.5em\\relax"); // \def\quad{\hskip1em\relax}
+
+defineMacro("\\quad", "\\hskip1em\\relax"); // \def\qquad{\hskip2em\relax}
+
+defineMacro("\\qquad", "\\hskip2em\\relax"); // \tag@in@display form of \tag
+
+defineMacro("\\tag", "\\@ifstar\\tag@literal\\tag@paren");
+defineMacro("\\tag@paren", "\\tag@literal{({#1})}");
+defineMacro("\\tag@literal", context => {
+  if (context.macros.get("\\df@tag")) {
+    throw new ParseError("Multiple \\tag");
+  }
+
+  return "\\gdef\\df@tag{\\text{#1}}";
+}); // \renewcommand{\bmod}{\nonscript\mskip-\medmuskip\mkern5mu\mathbin
+//   {\operator@font mod}\penalty900
+//   \mkern5mu\nonscript\mskip-\medmuskip}
+// \newcommand{\pod}[1]{\allowbreak
+//   \if@display\mkern18mu\else\mkern8mu\fi(#1)}
+// \renewcommand{\pmod}[1]{\pod{{\operator@font mod}\mkern6mu#1}}
+// \newcommand{\mod}[1]{\allowbreak\if@display\mkern18mu
+//   \else\mkern12mu\fi{\operator@font mod}\,\,#1}
+// TODO: math mode should use \medmuskip = 4mu plus 2mu minus 4mu
+
+defineMacro("\\bmod", "\\mathchoice{\\mskip1mu}{\\mskip1mu}{\\mskip5mu}{\\mskip5mu}" + "\\mathbin{\\rm mod}" + "\\mathchoice{\\mskip1mu}{\\mskip1mu}{\\mskip5mu}{\\mskip5mu}");
+defineMacro("\\pod", "\\allowbreak" + "\\mathchoice{\\mkern18mu}{\\mkern8mu}{\\mkern8mu}{\\mkern8mu}(#1)");
+defineMacro("\\pmod", "\\pod{{\\rm mod}\\mkern6mu#1}");
+defineMacro("\\mod", "\\allowbreak" + "\\mathchoice{\\mkern18mu}{\\mkern12mu}{\\mkern12mu}{\\mkern12mu}" + "{\\rm mod}\\,\\,#1"); // \pmb    --   A simulation of bold.
+// The version in ambsy.sty works by typesetting three copies of the argument
+// with small offsets. We use two copies. We omit the vertical offset because
+// of rendering problems that makeVList encounters in Safari.
+
+defineMacro("\\pmb", "\\html@mathml{" + "\\@binrel{#1}{\\mathrlap{#1}\\kern0.5px#1}}" + "{\\mathbf{#1}}"); //////////////////////////////////////////////////////////////////////
+// LaTeX source2e
+// \\ defaults to \newline, but changes to \cr within array environment
+
+defineMacro("\\\\", "\\newline"); // \def\TeX{T\kern-.1667em\lower.5ex\hbox{E}\kern-.125emX\@}
+// TODO: Doesn't normally work in math mode because \@ fails.  KaTeX doesn't
+// support \@ yet, so that's omitted, and we add \text so that the result
+// doesn't look funny in math mode.
+
+defineMacro("\\TeX", "\\textrm{\\html@mathml{" + "T\\kern-.1667em\\raisebox{-.5ex}{E}\\kern-.125emX" + "}{TeX}}"); // \DeclareRobustCommand{\LaTeX}{L\kern-.36em%
+//         {\sbox\z@ T%
+//          \vbox to\ht\z@{\hbox{\check@mathfonts
+//                               \fontsize\sf@size\z@
+//                               \math@fontsfalse\selectfont
+//                               A}%
+//                         \vss}%
+//         }%
+//         \kern-.15em%
+//         \TeX}
+// This code aligns the top of the A with the T (from the perspective of TeX's
+// boxes, though visually the A appears to extend above slightly).
+// We compute the corresponding \raisebox when A is rendered in \normalsize
+// \scriptstyle, which has a scale factor of 0.7 (see Options.js).
+
+const latexRaiseA = metricMap['Main-Regular']["T".charCodeAt(0)][1] - 0.7 * metricMap['Main-Regular']["A".charCodeAt(0)][1] + "em";
+defineMacro("\\LaTeX", "\\textrm{\\html@mathml{" + `L\\kern-.36em\\raisebox{${latexRaiseA}}{\\scriptstyle A}` + "\\kern-.15em\\TeX}{LaTeX}}"); // New KaTeX logo based on tweaking LaTeX logo
+
+defineMacro("\\KaTeX", "\\textrm{\\html@mathml{" + `K\\kern-.17em\\raisebox{${latexRaiseA}}{\\scriptstyle A}` + "\\kern-.15em\\TeX}{KaTeX}}"); // \DeclareRobustCommand\hspace{\@ifstar\@hspacer\@hspace}
+// \def\@hspace#1{\hskip  #1\relax}
+// \def\@hspacer#1{\vrule \@width\z@\nobreak
+//                 \hskip #1\hskip \z@skip}
+
+defineMacro("\\hspace", "\\@ifstar\\@hspacer\\@hspace");
+defineMacro("\\@hspace", "\\hskip #1\\relax");
+defineMacro("\\@hspacer", "\\rule{0pt}{0pt}\\hskip #1\\relax"); //////////////////////////////////////////////////////////////////////
+// mathtools.sty
+//\providecommand\ordinarycolon{:}
+
+defineMacro("\\ordinarycolon", ":"); //\def\vcentcolon{\mathrel{\mathop\ordinarycolon}}
+//TODO(edemaine): Not yet centered. Fix via \raisebox or #726
+
+defineMacro("\\vcentcolon", "\\mathrel{\\mathop\\ordinarycolon}"); // \providecommand*\dblcolon{\vcentcolon\mathrel{\mkern-.9mu}\vcentcolon}
+
+defineMacro("\\dblcolon", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-.9mu}\\vcentcolon}}" + "{\\mathop{\\char\"2237}}"); // \providecommand*\coloneqq{\vcentcolon\mathrel{\mkern-1.2mu}=}
+
+defineMacro("\\coloneqq", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}=}}" + "{\\mathop{\\char\"2254}}"); // ≔
+// \providecommand*\Coloneqq{\dblcolon\mathrel{\mkern-1.2mu}=}
+
+defineMacro("\\Coloneqq", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}=}}" + "{\\mathop{\\char\"2237\\char\"3d}}"); // \providecommand*\coloneq{\vcentcolon\mathrel{\mkern-1.2mu}\mathrel{-}}
+
+defineMacro("\\coloneq", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\mathrel{-}}}" + "{\\mathop{\\char\"3a\\char\"2212}}"); // \providecommand*\Coloneq{\dblcolon\mathrel{\mkern-1.2mu}\mathrel{-}}
+
+defineMacro("\\Coloneq", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\mathrel{-}}}" + "{\\mathop{\\char\"2237\\char\"2212}}"); // \providecommand*\eqqcolon{=\mathrel{\mkern-1.2mu}\vcentcolon}
+
+defineMacro("\\eqqcolon", "\\html@mathml{" + "\\mathrel{=\\mathrel{\\mkern-1.2mu}\\vcentcolon}}" + "{\\mathop{\\char\"2255}}"); // ≕
+// \providecommand*\Eqqcolon{=\mathrel{\mkern-1.2mu}\dblcolon}
+
+defineMacro("\\Eqqcolon", "\\html@mathml{" + "\\mathrel{=\\mathrel{\\mkern-1.2mu}\\dblcolon}}" + "{\\mathop{\\char\"3d\\char\"2237}}"); // \providecommand*\eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\vcentcolon}
+
+defineMacro("\\eqcolon", "\\html@mathml{" + "\\mathrel{\\mathrel{-}\\mathrel{\\mkern-1.2mu}\\vcentcolon}}" + "{\\mathop{\\char\"2239}}"); // \providecommand*\Eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\dblcolon}
+
+defineMacro("\\Eqcolon", "\\html@mathml{" + "\\mathrel{\\mathrel{-}\\mathrel{\\mkern-1.2mu}\\dblcolon}}" + "{\\mathop{\\char\"2212\\char\"2237}}"); // \providecommand*\colonapprox{\vcentcolon\mathrel{\mkern-1.2mu}\approx}
+
+defineMacro("\\colonapprox", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\approx}}" + "{\\mathop{\\char\"3a\\char\"2248}}"); // \providecommand*\Colonapprox{\dblcolon\mathrel{\mkern-1.2mu}\approx}
+
+defineMacro("\\Colonapprox", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\approx}}" + "{\\mathop{\\char\"2237\\char\"2248}}"); // \providecommand*\colonsim{\vcentcolon\mathrel{\mkern-1.2mu}\sim}
+
+defineMacro("\\colonsim", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\sim}}" + "{\\mathop{\\char\"3a\\char\"223c}}"); // \providecommand*\Colonsim{\dblcolon\mathrel{\mkern-1.2mu}\sim}
+
+defineMacro("\\Colonsim", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\sim}}" + "{\\mathop{\\char\"2237\\char\"223c}}"); // Some Unicode characters are implemented with macros to mathtools functions.
+
+defineMacro("\u2237", "\\dblcolon"); // ::
+
+defineMacro("\u2239", "\\eqcolon"); // -:
+
+defineMacro("\u2254", "\\coloneqq"); // :=
+
+defineMacro("\u2255", "\\eqqcolon"); // =:
+
+defineMacro("\u2A74", "\\Coloneqq"); // ::=
+//////////////////////////////////////////////////////////////////////
+// colonequals.sty
+// Alternate names for mathtools's macros:
+
+defineMacro("\\ratio", "\\vcentcolon");
+defineMacro("\\coloncolon", "\\dblcolon");
+defineMacro("\\colonequals", "\\coloneqq");
+defineMacro("\\coloncolonequals", "\\Coloneqq");
+defineMacro("\\equalscolon", "\\eqqcolon");
+defineMacro("\\equalscoloncolon", "\\Eqqcolon");
+defineMacro("\\colonminus", "\\coloneq");
+defineMacro("\\coloncolonminus", "\\Coloneq");
+defineMacro("\\minuscolon", "\\eqcolon");
+defineMacro("\\minuscoloncolon", "\\Eqcolon"); // \colonapprox name is same in mathtools and colonequals.
+
+defineMacro("\\coloncolonapprox", "\\Colonapprox"); // \colonsim name is same in mathtools and colonequals.
+
+defineMacro("\\coloncolonsim", "\\Colonsim"); // Additional macros, implemented by analogy with mathtools definitions:
+
+defineMacro("\\simcolon", "\\mathrel{\\sim\\mathrel{\\mkern-1.2mu}\\vcentcolon}");
+defineMacro("\\simcoloncolon", "\\mathrel{\\sim\\mathrel{\\mkern-1.2mu}\\dblcolon}");
+defineMacro("\\approxcolon", "\\mathrel{\\approx\\mathrel{\\mkern-1.2mu}\\vcentcolon}");
+defineMacro("\\approxcoloncolon", "\\mathrel{\\approx\\mathrel{\\mkern-1.2mu}\\dblcolon}"); // Present in newtxmath, pxfonts and txfonts
+
+defineMacro("\\notni", "\\html@mathml{\\not\\ni}{\\mathrel{\\char`\u220C}}");
+defineMacro("\\limsup", "\\DOTSB\\operatorname*{lim\\,sup}");
+defineMacro("\\liminf", "\\DOTSB\\operatorname*{lim\\,inf}"); //////////////////////////////////////////////////////////////////////
+// MathML alternates for KaTeX glyphs in the Unicode private area
+
+defineMacro("\\gvertneqq", "\\html@mathml{\\@gvertneqq}{\u2269}");
+defineMacro("\\lvertneqq", "\\html@mathml{\\@lvertneqq}{\u2268}");
+defineMacro("\\ngeqq", "\\html@mathml{\\@ngeqq}{\u2271}");
+defineMacro("\\ngeqslant", "\\html@mathml{\\@ngeqslant}{\u2271}");
+defineMacro("\\nleqq", "\\html@mathml{\\@nleqq}{\u2270}");
+defineMacro("\\nleqslant", "\\html@mathml{\\@nleqslant}{\u2270}");
+defineMacro("\\nshortmid", "\\html@mathml{\\@nshortmid}{∤}");
+defineMacro("\\nshortparallel", "\\html@mathml{\\@nshortparallel}{∦}");
+defineMacro("\\nsubseteqq", "\\html@mathml{\\@nsubseteqq}{\u2288}");
+defineMacro("\\nsupseteqq", "\\html@mathml{\\@nsupseteqq}{\u2289}");
+defineMacro("\\varsubsetneq", "\\html@mathml{\\@varsubsetneq}{⊊}");
+defineMacro("\\varsubsetneqq", "\\html@mathml{\\@varsubsetneqq}{⫋}");
+defineMacro("\\varsupsetneq", "\\html@mathml{\\@varsupsetneq}{⊋}");
+defineMacro("\\varsupsetneqq", "\\html@mathml{\\@varsupsetneqq}{⫌}");
+defineMacro("\\imath", "\\html@mathml{\\@imath}{\u0131}");
+defineMacro("\\jmath", "\\html@mathml{\\@jmath}{\u0237}"); //////////////////////////////////////////////////////////////////////
+// stmaryrd and semantic
+// The stmaryrd and semantic packages render the next four items by calling a
+// glyph. Those glyphs do not exist in the KaTeX fonts. Hence the macros.
+
+defineMacro("\\llbracket", "\\html@mathml{" + "\\mathopen{[\\mkern-3.2mu[}}" + "{\\mathopen{\\char`\u27e6}}");
+defineMacro("\\rrbracket", "\\html@mathml{" + "\\mathclose{]\\mkern-3.2mu]}}" + "{\\mathclose{\\char`\u27e7}}");
+defineMacro("\u27e6", "\\llbracket"); // blackboard bold [
+
+defineMacro("\u27e7", "\\rrbracket"); // blackboard bold ]
+
+defineMacro("\\lBrace", "\\html@mathml{" + "\\mathopen{\\{\\mkern-3.2mu[}}" + "{\\mathopen{\\char`\u2983}}");
+defineMacro("\\rBrace", "\\html@mathml{" + "\\mathclose{]\\mkern-3.2mu\\}}}" + "{\\mathclose{\\char`\u2984}}");
+defineMacro("\u2983", "\\lBrace"); // blackboard bold {
+
+defineMacro("\u2984", "\\rBrace"); // blackboard bold }
+// TODO: Create variable sized versions of the last two items. I believe that
+// will require new font glyphs.
+// The stmaryrd function `\minuso` provides a "Plimsoll" symbol that
+// superimposes the characters \circ and \mathminus. Used in chemistry.
+
+defineMacro("\\minuso", "\\mathbin{\\html@mathml{" + "{\\mathrlap{\\mathchoice{\\kern{0.145em}}{\\kern{0.145em}}" + "{\\kern{0.1015em}}{\\kern{0.0725em}}\\circ}{-}}}" + "{\\char`⦵}}");
+defineMacro("⦵", "\\minuso"); //////////////////////////////////////////////////////////////////////
+// texvc.sty
+// The texvc package contains macros available in mediawiki pages.
+// We omit the functions deprecated at
+// https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax
+// We also omit texvc's \O, which conflicts with \text{\O}
+
+defineMacro("\\darr", "\\downarrow");
+defineMacro("\\dArr", "\\Downarrow");
+defineMacro("\\Darr", "\\Downarrow");
+defineMacro("\\lang", "\\langle");
+defineMacro("\\rang", "\\rangle");
+defineMacro("\\uarr", "\\uparrow");
+defineMacro("\\uArr", "\\Uparrow");
+defineMacro("\\Uarr", "\\Uparrow");
+defineMacro("\\N", "\\mathbb{N}");
+defineMacro("\\R", "\\mathbb{R}");
+defineMacro("\\Z", "\\mathbb{Z}");
+defineMacro("\\alef", "\\aleph");
+defineMacro("\\alefsym", "\\aleph");
+defineMacro("\\Alpha", "\\mathrm{A}");
+defineMacro("\\Beta", "\\mathrm{B}");
+defineMacro("\\bull", "\\bullet");
+defineMacro("\\Chi", "\\mathrm{X}");
+defineMacro("\\clubs", "\\clubsuit");
+defineMacro("\\cnums", "\\mathbb{C}");
+defineMacro("\\Complex", "\\mathbb{C}");
+defineMacro("\\Dagger", "\\ddagger");
+defineMacro("\\diamonds", "\\diamondsuit");
+defineMacro("\\empty", "\\emptyset");
+defineMacro("\\Epsilon", "\\mathrm{E}");
+defineMacro("\\Eta", "\\mathrm{H}");
+defineMacro("\\exist", "\\exists");
+defineMacro("\\harr", "\\leftrightarrow");
+defineMacro("\\hArr", "\\Leftrightarrow");
+defineMacro("\\Harr", "\\Leftrightarrow");
+defineMacro("\\hearts", "\\heartsuit");
+defineMacro("\\image", "\\Im");
+defineMacro("\\infin", "\\infty");
+defineMacro("\\Iota", "\\mathrm{I}");
+defineMacro("\\isin", "\\in");
+defineMacro("\\Kappa", "\\mathrm{K}");
+defineMacro("\\larr", "\\leftarrow");
+defineMacro("\\lArr", "\\Leftarrow");
+defineMacro("\\Larr", "\\Leftarrow");
+defineMacro("\\lrarr", "\\leftrightarrow");
+defineMacro("\\lrArr", "\\Leftrightarrow");
+defineMacro("\\Lrarr", "\\Leftrightarrow");
+defineMacro("\\Mu", "\\mathrm{M}");
+defineMacro("\\natnums", "\\mathbb{N}");
+defineMacro("\\Nu", "\\mathrm{N}");
+defineMacro("\\Omicron", "\\mathrm{O}");
+defineMacro("\\plusmn", "\\pm");
+defineMacro("\\rarr", "\\rightarrow");
+defineMacro("\\rArr", "\\Rightarrow");
+defineMacro("\\Rarr", "\\Rightarrow");
+defineMacro("\\real", "\\Re");
+defineMacro("\\reals", "\\mathbb{R}");
+defineMacro("\\Reals", "\\mathbb{R}");
+defineMacro("\\Rho", "\\mathrm{P}");
+defineMacro("\\sdot", "\\cdot");
+defineMacro("\\sect", "\\S");
+defineMacro("\\spades", "\\spadesuit");
+defineMacro("\\sub", "\\subset");
+defineMacro("\\sube", "\\subseteq");
+defineMacro("\\supe", "\\supseteq");
+defineMacro("\\Tau", "\\mathrm{T}");
+defineMacro("\\thetasym", "\\vartheta"); // TODO: defineMacro("\\varcoppa", "\\\mbox{\\coppa}");
+
+defineMacro("\\weierp", "\\wp");
+defineMacro("\\Zeta", "\\mathrm{Z}"); //////////////////////////////////////////////////////////////////////
+// statmath.sty
+// https://ctan.math.illinois.edu/macros/latex/contrib/statmath/statmath.pdf
+
+defineMacro("\\argmin", "\\DOTSB\\operatorname*{arg\\,min}");
+defineMacro("\\argmax", "\\DOTSB\\operatorname*{arg\\,max}");
+defineMacro("\\plim", "\\DOTSB\\mathop{\\operatorname{plim}}\\limits"); //////////////////////////////////////////////////////////////////////
+// braket.sty
+// http://ctan.math.washington.edu/tex-archive/macros/latex/contrib/braket/braket.pdf
+
+defineMacro("\\bra", "\\mathinner{\\langle{#1}|}");
+defineMacro("\\ket", "\\mathinner{|{#1}\\rangle}");
+defineMacro("\\braket", "\\mathinner{\\langle{#1}\\rangle}");
+defineMacro("\\Bra", "\\left\\langle#1\\right|");
+defineMacro("\\Ket", "\\left|#1\\right\\rangle"); // Custom Khan Academy colors, should be moved to an optional package
+
+defineMacro("\\blue", "\\textcolor{##6495ed}{#1}");
+defineMacro("\\orange", "\\textcolor{##ffa500}{#1}");
+defineMacro("\\pink", "\\textcolor{##ff00af}{#1}");
+defineMacro("\\red", "\\textcolor{##df0030}{#1}");
+defineMacro("\\green", "\\textcolor{##28ae7b}{#1}");
+defineMacro("\\gray", "\\textcolor{gray}{#1}");
+defineMacro("\\purple", "\\textcolor{##9d38bd}{#1}");
+defineMacro("\\blueA", "\\textcolor{##ccfaff}{#1}");
+defineMacro("\\blueB", "\\textcolor{##80f6ff}{#1}");
+defineMacro("\\blueC", "\\textcolor{##63d9ea}{#1}");
+defineMacro("\\blueD", "\\textcolor{##11accd}{#1}");
+defineMacro("\\blueE", "\\textcolor{##0c7f99}{#1}");
+defineMacro("\\tealA", "\\textcolor{##94fff5}{#1}");
+defineMacro("\\tealB", "\\textcolor{##26edd5}{#1}");
+defineMacro("\\tealC", "\\textcolor{##01d1c1}{#1}");
+defineMacro("\\tealD", "\\textcolor{##01a995}{#1}");
+defineMacro("\\tealE", "\\textcolor{##208170}{#1}");
+defineMacro("\\greenA", "\\textcolor{##b6ffb0}{#1}");
+defineMacro("\\greenB", "\\textcolor{##8af281}{#1}");
+defineMacro("\\greenC", "\\textcolor{##74cf70}{#1}");
+defineMacro("\\greenD", "\\textcolor{##1fab54}{#1}");
+defineMacro("\\greenE", "\\textcolor{##0d923f}{#1}");
+defineMacro("\\goldA", "\\textcolor{##ffd0a9}{#1}");
+defineMacro("\\goldB", "\\textcolor{##ffbb71}{#1}");
+defineMacro("\\goldC", "\\textcolor{##ff9c39}{#1}");
+defineMacro("\\goldD", "\\textcolor{##e07d10}{#1}");
+defineMacro("\\goldE", "\\textcolor{##a75a05}{#1}");
+defineMacro("\\redA", "\\textcolor{##fca9a9}{#1}");
+defineMacro("\\redB", "\\textcolor{##ff8482}{#1}");
+defineMacro("\\redC", "\\textcolor{##f9685d}{#1}");
+defineMacro("\\redD", "\\textcolor{##e84d39}{#1}");
+defineMacro("\\redE", "\\textcolor{##bc2612}{#1}");
+defineMacro("\\maroonA", "\\textcolor{##ffbde0}{#1}");
+defineMacro("\\maroonB", "\\textcolor{##ff92c6}{#1}");
+defineMacro("\\maroonC", "\\textcolor{##ed5fa6}{#1}");
+defineMacro("\\maroonD", "\\textcolor{##ca337c}{#1}");
+defineMacro("\\maroonE", "\\textcolor{##9e034e}{#1}");
+defineMacro("\\purpleA", "\\textcolor{##ddd7ff}{#1}");
+defineMacro("\\purpleB", "\\textcolor{##c6b9fc}{#1}");
+defineMacro("\\purpleC", "\\textcolor{##aa87ff}{#1}");
+defineMacro("\\purpleD", "\\textcolor{##7854ab}{#1}");
+defineMacro("\\purpleE", "\\textcolor{##543b78}{#1}");
+defineMacro("\\mintA", "\\textcolor{##f5f9e8}{#1}");
+defineMacro("\\mintB", "\\textcolor{##edf2df}{#1}");
+defineMacro("\\mintC", "\\textcolor{##e0e5cc}{#1}");
+defineMacro("\\grayA", "\\textcolor{##f6f7f7}{#1}");
+defineMacro("\\grayB", "\\textcolor{##f0f1f2}{#1}");
+defineMacro("\\grayC", "\\textcolor{##e3e5e6}{#1}");
+defineMacro("\\grayD", "\\textcolor{##d6d8da}{#1}");
+defineMacro("\\grayE", "\\textcolor{##babec2}{#1}");
+defineMacro("\\grayF", "\\textcolor{##888d93}{#1}");
+defineMacro("\\grayG", "\\textcolor{##626569}{#1}");
+defineMacro("\\grayH", "\\textcolor{##3b3e40}{#1}");
+defineMacro("\\grayI", "\\textcolor{##21242c}{#1}");
+defineMacro("\\kaBlue", "\\textcolor{##314453}{#1}");
+defineMacro("\\kaGreen", "\\textcolor{##71B307}{#1}");
+
+/**
+ * This file contains the “gullet” where macros are expanded
+ * until only non-macro tokens remain.
+ */
+// List of commands that act like macros but aren't defined as a macro,
+// function, or symbol.  Used in `isDefined`.
+const implicitCommands = {
+  "\\relax": true,
+  // MacroExpander.js
+  "^": true,
+  // Parser.js
+  "_": true,
+  // Parser.js
+  "\\limits": true,
+  // Parser.js
+  "\\nolimits": true // Parser.js
+
+};
+class MacroExpander {
+  constructor(input, settings, mode) {
+    this.settings = void 0;
+    this.expansionCount = void 0;
+    this.lexer = void 0;
+    this.macros = void 0;
+    this.stack = void 0;
+    this.mode = void 0;
+    this.settings = settings;
+    this.expansionCount = 0;
+    this.feed(input); // Make new global namespace
+
+    this.macros = new Namespace(builtinMacros, settings.macros);
+    this.mode = mode;
+    this.stack = []; // contains tokens in REVERSE order
+  }
+  /**
+   * Feed a new input string to the same MacroExpander
+   * (with existing macros etc.).
+   */
+
+
+  feed(input) {
+    this.lexer = new Lexer(input, this.settings);
+  }
+  /**
+   * Switches between "text" and "math" modes.
+   */
+
+
+  switchMode(newMode) {
+    this.mode = newMode;
+  }
+  /**
+   * Start a new group nesting within all namespaces.
+   */
+
+
+  beginGroup() {
+    this.macros.beginGroup();
+  }
+  /**
+   * End current group nesting within all namespaces.
+   */
+
+
+  endGroup() {
+    this.macros.endGroup();
+  }
+  /**
+   * Returns the topmost token on the stack, without expanding it.
+   * Similar in behavior to TeX's `\futurelet`.
+   */
+
+
+  future() {
+    if (this.stack.length === 0) {
+      this.pushToken(this.lexer.lex());
+    }
+
+    return this.stack[this.stack.length - 1];
+  }
+  /**
+   * Remove and return the next unexpanded token.
+   */
+
+
+  popToken() {
+    this.future(); // ensure non-empty stack
+
+    return this.stack.pop();
+  }
+  /**
+   * Add a given token to the token stack.  In particular, this get be used
+   * to put back a token returned from one of the other methods.
+   */
+
+
+  pushToken(token) {
+    this.stack.push(token);
+  }
+  /**
+   * Append an array of tokens to the token stack.
+   */
+
+
+  pushTokens(tokens) {
+    this.stack.push(...tokens);
+  }
+  /**
+   * Consume all following space tokens, without expansion.
+   */
+
+
+  consumeSpaces() {
+    for (;;) {
+      const token = this.future();
+
+      if (token.text === " ") {
+        this.stack.pop();
+      } else {
+        break;
+      }
+    }
+  }
+  /**
+   * Consume the specified number of arguments from the token stream,
+   * and return the resulting array of arguments.
+   */
+
+
+  consumeArgs(numArgs) {
+    const args = []; // obtain arguments, either single token or balanced {…} group
+
+    for (let i = 0; i < numArgs; ++i) {
+      this.consumeSpaces(); // ignore spaces before each argument
+
+      const startOfArg = this.popToken();
+
+      if (startOfArg.text === "{") {
+        const arg = [];
+        let depth = 1;
+
+        while (depth !== 0) {
+          const tok = this.popToken();
+          arg.push(tok);
+
+          if (tok.text === "{") {
+            ++depth;
+          } else if (tok.text === "}") {
+            --depth;
+          } else if (tok.text === "EOF") {
+            throw new ParseError("End of input in macro argument", startOfArg);
+          }
+        }
+
+        arg.pop(); // remove last }
+
+        arg.reverse(); // like above, to fit in with stack order
+
+        args[i] = arg;
+      } else if (startOfArg.text === "EOF") {
+        throw new ParseError("End of input expecting macro argument");
+      } else {
+        args[i] = [startOfArg];
+      }
+    }
+
+    return args;
+  }
+  /**
+   * Expand the next token only once if possible.
+   *
+   * If the token is expanded, the resulting tokens will be pushed onto
+   * the stack in reverse order and will be returned as an array,
+   * also in reverse order.
+   *
+   * If not, the next token will be returned without removing it
+   * from the stack.  This case can be detected by a `Token` return value
+   * instead of an `Array` return value.
+   *
+   * In either case, the next token will be on the top of the stack,
+   * or the stack will be empty.
+   *
+   * Used to implement `expandAfterFuture` and `expandNextToken`.
+   *
+   * At the moment, macro expansion doesn't handle delimited macros,
+   * i.e. things like those defined by \def\foo#1\end{…}.
+   * See the TeX book page 202ff. for details on how those should behave.
+   *
+   * If expandableOnly, only expandable tokens are expanded and
+   * an undefined control sequence results in an error.
+   */
+
+
+  expandOnce(expandableOnly) {
+    const topToken = this.popToken();
+    const name = topToken.text;
+    const expansion = !topToken.noexpand ? this._getExpansion(name) : null;
+
+    if (expansion == null || expandableOnly && expansion.unexpandable) {
+      if (expandableOnly && expansion == null && name[0] === "\\" && !this.isDefined(name)) {
+        throw new ParseError("Undefined control sequence: " + name);
+      }
+
+      this.pushToken(topToken);
+      return topToken;
+    }
+
+    this.expansionCount++;
+
+    if (this.expansionCount > this.settings.maxExpand) {
+      throw new ParseError("Too many expansions: infinite loop or " + "need to increase maxExpand setting");
+    }
+
+    let tokens = expansion.tokens;
+
+    if (expansion.numArgs) {
+      const args = this.consumeArgs(expansion.numArgs); // paste arguments in place of the placeholders
+
+      tokens = tokens.slice(); // make a shallow copy
+
+      for (let i = tokens.length - 1; i >= 0; --i) {
+        let tok = tokens[i];
+
+        if (tok.text === "#") {
+          if (i === 0) {
+            throw new ParseError("Incomplete placeholder at end of macro body", tok);
+          }
+
+          tok = tokens[--i]; // next token on stack
+
+          if (tok.text === "#") {
+            // ## → #
+            tokens.splice(i + 1, 1); // drop first #
+          } else if (/^[1-9]$/.test(tok.text)) {
+            // replace the placeholder with the indicated argument
+            tokens.splice(i, 2, ...args[+tok.text - 1]);
+          } else {
+            throw new ParseError("Not a valid argument number", tok);
+          }
+        }
+      }
+    } // Concatenate expansion onto top of stack.
+
+
+    this.pushTokens(tokens);
+    return tokens;
+  }
+  /**
+   * Expand the next token only once (if possible), and return the resulting
+   * top token on the stack (without removing anything from the stack).
+   * Similar in behavior to TeX's `\expandafter\futurelet`.
+   * Equivalent to expandOnce() followed by future().
+   */
+
+
+  expandAfterFuture() {
+    this.expandOnce();
+    return this.future();
+  }
+  /**
+   * Recursively expand first token, then return first non-expandable token.
+   */
+
+
+  expandNextToken() {
+    for (;;) {
+      const expanded = this.expandOnce(); // expandOnce returns Token if and only if it's fully expanded.
+
+      if (expanded instanceof Token) {
+        // \relax stops the expansion, but shouldn't get returned (a
+        // null return value couldn't get implemented as a function).
+        // the token after \noexpand is interpreted as if its meaning
+        // were ‘\relax’
+        if (expanded.text === "\\relax" || expanded.treatAsRelax) {
+          this.stack.pop();
+        } else {
+          return this.stack.pop(); // === expanded
+        }
+      }
+    } // Flow unable to figure out that this pathway is impossible.
+    // https://github.com/facebook/flow/issues/4808
+
+
+    throw new Error(); // eslint-disable-line no-unreachable
+  }
+  /**
+   * Fully expand the given macro name and return the resulting list of
+   * tokens, or return `undefined` if no such macro is defined.
+   */
+
+
+  expandMacro(name) {
+    return this.macros.has(name) ? this.expandTokens([new Token(name)]) : undefined;
+  }
+  /**
+   * Fully expand the given token stream and return the resulting list of tokens
+   */
+
+
+  expandTokens(tokens) {
+    const output = [];
+    const oldStackLength = this.stack.length;
+    this.pushTokens(tokens);
+
+    while (this.stack.length > oldStackLength) {
+      const expanded = this.expandOnce(true); // expand only expandable tokens
+      // expandOnce returns Token if and only if it's fully expanded.
+
+      if (expanded instanceof Token) {
+        if (expanded.treatAsRelax) {
+          // the expansion of \noexpand is the token itself
+          expanded.noexpand = false;
+          expanded.treatAsRelax = false;
+        }
+
+        output.push(this.stack.pop());
+      }
+    }
+
+    return output;
+  }
+  /**
+   * Fully expand the given macro name and return the result as a string,
+   * or return `undefined` if no such macro is defined.
+   */
+
+
+  expandMacroAsText(name) {
+    const tokens = this.expandMacro(name);
+
+    if (tokens) {
+      return tokens.map(token => token.text).join("");
+    } else {
+      return tokens;
+    }
+  }
+  /**
+   * Returns the expanded macro as a reversed array of tokens and a macro
+   * argument count.  Or returns `null` if no such macro.
+   */
+
+
+  _getExpansion(name) {
+    const definition = this.macros.get(name);
+
+    if (definition == null) {
+      // mainly checking for undefined here
+      return definition;
+    }
+
+    const expansion = typeof definition === "function" ? definition(this) : definition;
+
+    if (typeof expansion === "string") {
+      let numArgs = 0;
+
+      if (expansion.indexOf("#") !== -1) {
+        const stripped = expansion.replace(/##/g, "");
+
+        while (stripped.indexOf("#" + (numArgs + 1)) !== -1) {
+          ++numArgs;
+        }
+      }
+
+      const bodyLexer = new Lexer(expansion, this.settings);
+      const tokens = [];
+      let tok = bodyLexer.lex();
+
+      while (tok.text !== "EOF") {
+        tokens.push(tok);
+        tok = bodyLexer.lex();
+      }
+
+      tokens.reverse(); // to fit in with stack using push and pop
+
+      const expanded = {
+        tokens,
+        numArgs
+      };
+      return expanded;
+    }
+
+    return expansion;
+  }
+  /**
+   * Determine whether a command is currently "defined" (has some
+   * functionality), meaning that it's a macro (in the current group),
+   * a function, a symbol, or one of the special commands listed in
+   * `implicitCommands`.
+   */
+
+
+  isDefined(name) {
+    return this.macros.has(name) || functions.hasOwnProperty(name) || symbols.math.hasOwnProperty(name) || symbols.text.hasOwnProperty(name) || implicitCommands.hasOwnProperty(name);
+  }
+  /**
+   * Determine whether a command is expandable.
+   */
+
+
+  isExpandable(name) {
+    const macro = this.macros.get(name);
+    return macro != null ? typeof macro === "string" || typeof macro === "function" || !macro.unexpandable // TODO(ylem): #2085
+    : functions.hasOwnProperty(name)
+    /* && !functions[name].primitive*/
+    ;
+  }
+
+}
+
+/* eslint no-constant-condition:0 */
+
+const unicodeAccents = {
+  "́": {
+    "text": "\\'",
+    "math": "\\acute"
+  },
+  "̀": {
+    "text": "\\`",
+    "math": "\\grave"
+  },
+  "̈": {
+    "text": "\\\"",
+    "math": "\\ddot"
+  },
+  "̃": {
+    "text": "\\~",
+    "math": "\\tilde"
+  },
+  "̄": {
+    "text": "\\=",
+    "math": "\\bar"
+  },
+  "̆": {
+    "text": "\\u",
+    "math": "\\breve"
+  },
+  "̌": {
+    "text": "\\v",
+    "math": "\\check"
+  },
+  "̂": {
+    "text": "\\^",
+    "math": "\\hat"
+  },
+  "̇": {
+    "text": "\\.",
+    "math": "\\dot"
+  },
+  "̊": {
+    "text": "\\r",
+    "math": "\\mathring"
+  },
+  "̋": {
+    "text": "\\H"
+  }
+};
+const unicodeSymbols = {
+  "á": "á",
+  "à": "à",
+  "ä": "ä",
+  "ǟ": "ǟ",
+  "ã": "ã",
+  "ā": "ā",
+  "ă": "ă",
+  "ắ": "ắ",
+  "ằ": "ằ",
+  "ẵ": "ẵ",
+  "ǎ": "ǎ",
+  "â": "â",
+  "ấ": "ấ",
+  "ầ": "ầ",
+  "ẫ": "ẫ",
+  "ȧ": "ȧ",
+  "ǡ": "ǡ",
+  "å": "å",
+  "ǻ": "ǻ",
+  "ḃ": "ḃ",
+  "ć": "ć",
+  "č": "č",
+  "ĉ": "ĉ",
+  "ċ": "ċ",
+  "ď": "ď",
+  "ḋ": "ḋ",
+  "é": "é",
+  "è": "è",
+  "ë": "ë",
+  "ẽ": "ẽ",
+  "ē": "ē",
+  "ḗ": "ḗ",
+  "ḕ": "ḕ",
+  "ĕ": "ĕ",
+  "ě": "ě",
+  "ê": "ê",
+  "ế": "ế",
+  "ề": "ề",
+  "ễ": "ễ",
+  "ė": "ė",
+  "ḟ": "ḟ",
+  "ǵ": "ǵ",
+  "ḡ": "ḡ",
+  "ğ": "ğ",
+  "ǧ": "ǧ",
+  "ĝ": "ĝ",
+  "ġ": "ġ",
+  "ḧ": "ḧ",
+  "ȟ": "ȟ",
+  "ĥ": "ĥ",
+  "ḣ": "ḣ",
+  "í": "í",
+  "ì": "ì",
+  "ï": "ï",
+  "ḯ": "ḯ",
+  "ĩ": "ĩ",
+  "ī": "ī",
+  "ĭ": "ĭ",
+  "ǐ": "ǐ",
+  "î": "î",
+  "ǰ": "ǰ",
+  "ĵ": "ĵ",
+  "ḱ": "ḱ",
+  "ǩ": "ǩ",
+  "ĺ": "ĺ",
+  "ľ": "ľ",
+  "ḿ": "ḿ",
+  "ṁ": "ṁ",
+  "ń": "ń",
+  "ǹ": "ǹ",
+  "ñ": "ñ",
+  "ň": "ň",
+  "ṅ": "ṅ",
+  "ó": "ó",
+  "ò": "ò",
+  "ö": "ö",
+  "ȫ": "ȫ",
+  "õ": "õ",
+  "ṍ": "ṍ",
+  "ṏ": "ṏ",
+  "ȭ": "ȭ",
+  "ō": "ō",
+  "ṓ": "ṓ",
+  "ṑ": "ṑ",
+  "ŏ": "ŏ",
+  "ǒ": "ǒ",
+  "ô": "ô",
+  "ố": "ố",
+  "ồ": "ồ",
+  "ỗ": "ỗ",
+  "ȯ": "ȯ",
+  "ȱ": "ȱ",
+  "ő": "ő",
+  "ṕ": "ṕ",
+  "ṗ": "ṗ",
+  "ŕ": "ŕ",
+  "ř": "ř",
+  "ṙ": "ṙ",
+  "ś": "ś",
+  "ṥ": "ṥ",
+  "š": "š",
+  "ṧ": "ṧ",
+  "ŝ": "ŝ",
+  "ṡ": "ṡ",
+  "ẗ": "ẗ",
+  "ť": "ť",
+  "ṫ": "ṫ",
+  "ú": "ú",
+  "ù": "ù",
+  "ü": "ü",
+  "ǘ": "ǘ",
+  "ǜ": "ǜ",
+  "ǖ": "ǖ",
+  "ǚ": "ǚ",
+  "ũ": "ũ",
+  "ṹ": "ṹ",
+  "ū": "ū",
+  "ṻ": "ṻ",
+  "ŭ": "ŭ",
+  "ǔ": "ǔ",
+  "û": "û",
+  "ů": "ů",
+  "ű": "ű",
+  "ṽ": "ṽ",
+  "ẃ": "ẃ",
+  "ẁ": "ẁ",
+  "ẅ": "ẅ",
+  "ŵ": "ŵ",
+  "ẇ": "ẇ",
+  "ẘ": "ẘ",
+  "ẍ": "ẍ",
+  "ẋ": "ẋ",
+  "ý": "ý",
+  "ỳ": "ỳ",
+  "ÿ": "ÿ",
+  "ỹ": "ỹ",
+  "ȳ": "ȳ",
+  "ŷ": "ŷ",
+  "ẏ": "ẏ",
+  "ẙ": "ẙ",
+  "ź": "ź",
+  "ž": "ž",
+  "ẑ": "ẑ",
+  "ż": "ż",
+  "Á": "Á",
+  "À": "À",
+  "Ä": "Ä",
+  "Ǟ": "Ǟ",
+  "Ã": "Ã",
+  "Ā": "Ā",
+  "Ă": "Ă",
+  "Ắ": "Ắ",
+  "Ằ": "Ằ",
+  "Ẵ": "Ẵ",
+  "Ǎ": "Ǎ",
+  "Â": "Â",
+  "Ấ": "Ấ",
+  "Ầ": "Ầ",
+  "Ẫ": "Ẫ",
+  "Ȧ": "Ȧ",
+  "Ǡ": "Ǡ",
+  "Å": "Å",
+  "Ǻ": "Ǻ",
+  "Ḃ": "Ḃ",
+  "Ć": "Ć",
+  "Č": "Č",
+  "Ĉ": "Ĉ",
+  "Ċ": "Ċ",
+  "Ď": "Ď",
+  "Ḋ": "Ḋ",
+  "É": "É",
+  "È": "È",
+  "Ë": "Ë",
+  "Ẽ": "Ẽ",
+  "Ē": "Ē",
+  "Ḗ": "Ḗ",
+  "Ḕ": "Ḕ",
+  "Ĕ": "Ĕ",
+  "Ě": "Ě",
+  "Ê": "Ê",
+  "Ế": "Ế",
+  "Ề": "Ề",
+  "Ễ": "Ễ",
+  "Ė": "Ė",
+  "Ḟ": "Ḟ",
+  "Ǵ": "Ǵ",
+  "Ḡ": "Ḡ",
+  "Ğ": "Ğ",
+  "Ǧ": "Ǧ",
+  "Ĝ": "Ĝ",
+  "Ġ": "Ġ",
+  "Ḧ": "Ḧ",
+  "Ȟ": "Ȟ",
+  "Ĥ": "Ĥ",
+  "Ḣ": "Ḣ",
+  "Í": "Í",
+  "Ì": "Ì",
+  "Ï": "Ï",
+  "Ḯ": "Ḯ",
+  "Ĩ": "Ĩ",
+  "Ī": "Ī",
+  "Ĭ": "Ĭ",
+  "Ǐ": "Ǐ",
+  "Î": "Î",
+  "İ": "İ",
+  "Ĵ": "Ĵ",
+  "Ḱ": "Ḱ",
+  "Ǩ": "Ǩ",
+  "Ĺ": "Ĺ",
+  "Ľ": "Ľ",
+  "Ḿ": "Ḿ",
+  "Ṁ": "Ṁ",
+  "Ń": "Ń",
+  "Ǹ": "Ǹ",
+  "Ñ": "Ñ",
+  "Ň": "Ň",
+  "Ṅ": "Ṅ",
+  "Ó": "Ó",
+  "Ò": "Ò",
+  "Ö": "Ö",
+  "Ȫ": "Ȫ",
+  "Õ": "Õ",
+  "Ṍ": "Ṍ",
+  "Ṏ": "Ṏ",
+  "Ȭ": "Ȭ",
+  "Ō": "Ō",
+  "Ṓ": "Ṓ",
+  "Ṑ": "Ṑ",
+  "Ŏ": "Ŏ",
+  "Ǒ": "Ǒ",
+  "Ô": "Ô",
+  "Ố": "Ố",
+  "Ồ": "Ồ",
+  "Ỗ": "Ỗ",
+  "Ȯ": "Ȯ",
+  "Ȱ": "Ȱ",
+  "Ő": "Ő",
+  "Ṕ": "Ṕ",
+  "Ṗ": "Ṗ",
+  "Ŕ": "Ŕ",
+  "Ř": "Ř",
+  "Ṙ": "Ṙ",
+  "Ś": "Ś",
+  "Ṥ": "Ṥ",
+  "Š": "Š",
+  "Ṧ": "Ṧ",
+  "Ŝ": "Ŝ",
+  "Ṡ": "Ṡ",
+  "Ť": "Ť",
+  "Ṫ": "Ṫ",
+  "Ú": "Ú",
+  "Ù": "Ù",
+  "Ü": "Ü",
+  "Ǘ": "Ǘ",
+  "Ǜ": "Ǜ",
+  "Ǖ": "Ǖ",
+  "Ǚ": "Ǚ",
+  "Ũ": "Ũ",
+  "Ṹ": "Ṹ",
+  "Ū": "Ū",
+  "Ṻ": "Ṻ",
+  "Ŭ": "Ŭ",
+  "Ǔ": "Ǔ",
+  "Û": "Û",
+  "Ů": "Ů",
+  "Ű": "Ű",
+  "Ṽ": "Ṽ",
+  "Ẃ": "Ẃ",
+  "Ẁ": "Ẁ",
+  "Ẅ": "Ẅ",
+  "Ŵ": "Ŵ",
+  "Ẇ": "Ẇ",
+  "Ẍ": "Ẍ",
+  "Ẋ": "Ẋ",
+  "Ý": "Ý",
+  "Ỳ": "Ỳ",
+  "Ÿ": "Ÿ",
+  "Ỹ": "Ỹ",
+  "Ȳ": "Ȳ",
+  "Ŷ": "Ŷ",
+  "Ẏ": "Ẏ",
+  "Ź": "Ź",
+  "Ž": "Ž",
+  "Ẑ": "Ẑ",
+  "Ż": "Ż",
+  "ά": "ά",
+  "ὰ": "ὰ",
+  "ᾱ": "ᾱ",
+  "ᾰ": "ᾰ",
+  "έ": "έ",
+  "ὲ": "ὲ",
+  "ή": "ή",
+  "ὴ": "ὴ",
+  "ί": "ί",
+  "ὶ": "ὶ",
+  "ϊ": "ϊ",
+  "ΐ": "ΐ",
+  "ῒ": "ῒ",
+  "ῑ": "ῑ",
+  "ῐ": "ῐ",
+  "ό": "ό",
+  "ὸ": "ὸ",
+  "ύ": "ύ",
+  "ὺ": "ὺ",
+  "ϋ": "ϋ",
+  "ΰ": "ΰ",
+  "ῢ": "ῢ",
+  "ῡ": "ῡ",
+  "ῠ": "ῠ",
+  "ώ": "ώ",
+  "ὼ": "ὼ",
+  "Ύ": "Ύ",
+  "Ὺ": "Ὺ",
+  "Ϋ": "Ϋ",
+  "Ῡ": "Ῡ",
+  "Ῠ": "Ῠ",
+  "Ώ": "Ώ",
+  "Ὼ": "Ὼ"
+};
+
+/**
+ * This file contains the parser used to parse out a TeX expression from the
+ * input. Since TeX isn't context-free, standard parsers don't work particularly
+ * well.
+ *
+ * The strategy of this parser is as such:
+ *
+ * The main functions (the `.parse...` ones) take a position in the current
+ * parse string to parse tokens from. The lexer (found in Lexer.js, stored at
+ * this.gullet.lexer) also supports pulling out tokens at arbitrary places. When
+ * individual tokens are needed at a position, the lexer is called to pull out a
+ * token, which is then used.
+ *
+ * The parser has a property called "mode" indicating the mode that
+ * the parser is currently in. Currently it has to be one of "math" or
+ * "text", which denotes whether the current environment is a math-y
+ * one or a text-y one (e.g. inside \text). Currently, this serves to
+ * limit the functions which can be used in text mode.
+ *
+ * The main functions then return an object which contains the useful data that
+ * was parsed at its given point, and a new position at the end of the parsed
+ * data. The main functions can call each other and continue the parsing by
+ * using the returned position as a new starting point.
+ *
+ * There are also extra `.handle...` functions, which pull out some reused
+ * functionality into self-contained functions.
+ *
+ * The functions return ParseNodes.
+ */
+class Parser {
+  constructor(input, settings) {
+    this.mode = void 0;
+    this.gullet = void 0;
+    this.settings = void 0;
+    this.leftrightDepth = void 0;
+    this.nextToken = void 0;
+    // Start in math mode
+    this.mode = "math"; // Create a new macro expander (gullet) and (indirectly via that) also a
+    // new lexer (mouth) for this parser (stomach, in the language of TeX)
+
+    this.gullet = new MacroExpander(input, settings, this.mode); // Store the settings for use in parsing
+
+    this.settings = settings; // Count leftright depth (for \middle errors)
+
+    this.leftrightDepth = 0;
+  }
+  /**
+   * Checks a result to make sure it has the right type, and throws an
+   * appropriate error otherwise.
+   */
+
+
+  expect(text, consume) {
+    if (consume === void 0) {
+      consume = true;
+    }
+
+    if (this.fetch().text !== text) {
+      throw new ParseError(`Expected '${text}', got '${this.fetch().text}'`, this.fetch());
+    }
+
+    if (consume) {
+      this.consume();
+    }
+  }
+  /**
+   * Discards the current lookahead token, considering it consumed.
+   */
+
+
+  consume() {
+    this.nextToken = null;
+  }
+  /**
+   * Return the current lookahead token, or if there isn't one (at the
+   * beginning, or if the previous lookahead token was consume()d),
+   * fetch the next token as the new lookahead token and return it.
+   */
+
+
+  fetch() {
+    if (this.nextToken == null) {
+      this.nextToken = this.gullet.expandNextToken();
+    }
+
+    return this.nextToken;
+  }
+  /**
+   * Switches between "text" and "math" modes.
+   */
+
+
+  switchMode(newMode) {
+    this.mode = newMode;
+    this.gullet.switchMode(newMode);
+  }
+  /**
+   * Main parsing function, which parses an entire input.
+   */
+
+
+  parse() {
+    if (!this.settings.globalGroup) {
+      // Create a group namespace for the math expression.
+      // (LaTeX creates a new group for every $...$, $$...$$, \[...\].)
+      this.gullet.beginGroup();
+    } // Use old \color behavior (same as LaTeX's \textcolor) if requested.
+    // We do this within the group for the math expression, so it doesn't
+    // pollute settings.macros.
+
+
+    if (this.settings.colorIsTextColor) {
+      this.gullet.macros.set("\\color", "\\textcolor");
+    } // Try to parse the input
+
+
+    const parse = this.parseExpression(false); // If we succeeded, make sure there's an EOF at the end
+
+    this.expect("EOF"); // End the group namespace for the expression
+
+    if (!this.settings.globalGroup) {
+      this.gullet.endGroup();
+    }
+
+    return parse;
+  }
+
+  parseExpression(breakOnInfix, breakOnTokenText) {
+    const body = []; // Keep adding atoms to the body until we can't parse any more atoms (either
+    // we reached the end, a }, or a \right)
+
+    while (true) {
+      // Ignore spaces in math mode
+      if (this.mode === "math") {
+        this.consumeSpaces();
+      }
+
+      const lex = this.fetch();
+
+      if (Parser.endOfExpression.indexOf(lex.text) !== -1) {
+        break;
+      }
+
+      if (breakOnTokenText && lex.text === breakOnTokenText) {
+        break;
+      }
+
+      if (breakOnInfix && functions[lex.text] && functions[lex.text].infix) {
+        break;
+      }
+
+      const atom = this.parseAtom(breakOnTokenText);
+
+      if (!atom) {
+        break;
+      } else if (atom.type === "internal") {
+        continue;
+      }
+
+      body.push(atom);
+    }
+
+    if (this.mode === "text") {
+      this.formLigatures(body);
+    }
+
+    return this.handleInfixNodes(body);
+  }
+  /**
+   * Rewrites infix operators such as \over with corresponding commands such
+   * as \frac.
+   *
+   * There can only be one infix operator per group.  If there's more than one
+   * then the expression is ambiguous.  This can be resolved by adding {}.
+   */
+
+
+  handleInfixNodes(body) {
+    let overIndex = -1;
+    let funcName;
+
+    for (let i = 0; i < body.length; i++) {
+      if (body[i].type === "infix") {
+        if (overIndex !== -1) {
+          throw new ParseError("only one infix operator per group", body[i].token);
+        }
+
+        overIndex = i;
+        funcName = body[i].replaceWith;
+      }
+    }
+
+    if (overIndex !== -1 && funcName) {
+      let numerNode;
+      let denomNode;
+      const numerBody = body.slice(0, overIndex);
+      const denomBody = body.slice(overIndex + 1);
+
+      if (numerBody.length === 1 && numerBody[0].type === "ordgroup") {
+        numerNode = numerBody[0];
+      } else {
+        numerNode = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: numerBody
+        };
+      }
+
+      if (denomBody.length === 1 && denomBody[0].type === "ordgroup") {
+        denomNode = denomBody[0];
+      } else {
+        denomNode = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: denomBody
+        };
+      }
+
+      let node;
+
+      if (funcName === "\\\\abovefrac") {
+        node = this.callFunction(funcName, [numerNode, body[overIndex], denomNode], []);
+      } else {
+        node = this.callFunction(funcName, [numerNode, denomNode], []);
+      }
+
+      return [node];
+    } else {
+      return body;
+    }
+  } // The greediness of a superscript or subscript
+
+
+  /**
+   * Handle a subscript or superscript with nice errors.
+   */
+  handleSupSubscript(name) {
+    const symbolToken = this.fetch();
+    const symbol = symbolToken.text;
+    this.consume();
+    const group = this.parseGroup(name, false, Parser.SUPSUB_GREEDINESS, undefined, undefined, true); // ignore spaces before sup/subscript argument
+
+    if (!group) {
+      throw new ParseError("Expected group after '" + symbol + "'", symbolToken);
+    }
+
+    return group;
+  }
+  /**
+   * Converts the textual input of an unsupported command into a text node
+   * contained within a color node whose color is determined by errorColor
+   */
+
+
+  formatUnsupportedCmd(text) {
+    const textordArray = [];
+
+    for (let i = 0; i < text.length; i++) {
+      textordArray.push({
+        type: "textord",
+        mode: "text",
+        text: text[i]
+      });
+    }
+
+    const textNode = {
+      type: "text",
+      mode: this.mode,
+      body: textordArray
+    };
+    const colorNode = {
+      type: "color",
+      mode: this.mode,
+      color: this.settings.errorColor,
+      body: [textNode]
+    };
+    return colorNode;
+  }
+  /**
+   * Parses a group with optional super/subscripts.
+   */
+
+
+  parseAtom(breakOnTokenText) {
+    // The body of an atom is an implicit group, so that things like
+    // \left(x\right)^2 work correctly.
+    const base = this.parseGroup("atom", false, null, breakOnTokenText); // In text mode, we don't have superscripts or subscripts
+
+    if (this.mode === "text") {
+      return base;
+    } // Note that base may be empty (i.e. null) at this point.
+
+
+    let superscript;
+    let subscript;
+
+    while (true) {
+      // Guaranteed in math mode, so eat any spaces first.
+      this.consumeSpaces(); // Lex the first token
+
+      const lex = this.fetch();
+
+      if (lex.text === "\\limits" || lex.text === "\\nolimits") {
+        // We got a limit control
+        if (base && base.type === "op") {
+          const limits = lex.text === "\\limits";
+          base.limits = limits;
+          base.alwaysHandleSupSub = true;
+        } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub) {
+          const limits = lex.text === "\\limits";
+          base.limits = limits;
+        } else {
+          throw new ParseError("Limit controls must follow a math operator", lex);
+        }
+
+        this.consume();
+      } else if (lex.text === "^") {
+        // We got a superscript start
+        if (superscript) {
+          throw new ParseError("Double superscript", lex);
+        }
+
+        superscript = this.handleSupSubscript("superscript");
+      } else if (lex.text === "_") {
+        // We got a subscript start
+        if (subscript) {
+          throw new ParseError("Double subscript", lex);
+        }
+
+        subscript = this.handleSupSubscript("subscript");
+      } else if (lex.text === "'") {
+        // We got a prime
+        if (superscript) {
+          throw new ParseError("Double superscript", lex);
+        }
+
+        const prime = {
+          type: "textord",
+          mode: this.mode,
+          text: "\\prime"
+        }; // Many primes can be grouped together, so we handle this here
+
+        const primes = [prime];
+        this.consume(); // Keep lexing tokens until we get something that's not a prime
+
+        while (this.fetch().text === "'") {
+          // For each one, add another prime to the list
+          primes.push(prime);
+          this.consume();
+        } // If there's a superscript following the primes, combine that
+        // superscript in with the primes.
+
+
+        if (this.fetch().text === "^") {
+          primes.push(this.handleSupSubscript("superscript"));
+        } // Put everything into an ordgroup as the superscript
+
+
+        superscript = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: primes
+        };
+      } else {
+        // If it wasn't ^, _, or ', stop parsing super/subscripts
+        break;
+      }
+    } // Base must be set if superscript or subscript are set per logic above,
+    // but need to check here for type check to pass.
+
+
+    if (superscript || subscript) {
+      // If we got either a superscript or subscript, create a supsub
+      return {
+        type: "supsub",
+        mode: this.mode,
+        base: base,
+        sup: superscript,
+        sub: subscript
+      };
+    } else {
+      // Otherwise return the original body
+      return base;
+    }
+  }
+  /**
+   * Parses an entire function, including its base and all of its arguments.
+   */
+
+
+  parseFunction(breakOnTokenText, name, // For error reporting.
+  greediness) {
+    const token = this.fetch();
+    const func = token.text;
+    const funcData = functions[func];
+
+    if (!funcData) {
+      return null;
+    }
+
+    this.consume(); // consume command token
+
+    if (greediness != null && funcData.greediness <= greediness) {
+      throw new ParseError("Got function '" + func + "' with no arguments" + (name ? " as " + name : ""), token);
+    } else if (this.mode === "text" && !funcData.allowedInText) {
+      throw new ParseError("Can't use function '" + func + "' in text mode", token);
+    } else if (this.mode === "math" && funcData.allowedInMath === false) {
+      throw new ParseError("Can't use function '" + func + "' in math mode", token);
+    }
+
+    const _this$parseArguments = this.parseArguments(func, funcData),
+          args = _this$parseArguments.args,
+          optArgs = _this$parseArguments.optArgs;
+
+    return this.callFunction(func, args, optArgs, token, breakOnTokenText);
+  }
+  /**
+   * Call a function handler with a suitable context and arguments.
+   */
+
+
+  callFunction(name, args, optArgs, token, breakOnTokenText) {
+    const context = {
+      funcName: name,
+      parser: this,
+      token,
+      breakOnTokenText
+    };
+    const func = functions[name];
+
+    if (func && func.handler) {
+      return func.handler(context, args, optArgs);
+    } else {
+      throw new ParseError(`No function handler for ${name}`);
+    }
+  }
+  /**
+   * Parses the arguments of a function or environment
+   */
+
+
+  parseArguments(func, // Should look like "\name" or "\begin{name}".
+  funcData) {
+    const totalArgs = funcData.numArgs + funcData.numOptionalArgs;
+
+    if (totalArgs === 0) {
+      return {
+        args: [],
+        optArgs: []
+      };
+    }
+
+    const baseGreediness = funcData.greediness;
+    const args = [];
+    const optArgs = [];
+
+    for (let i = 0; i < totalArgs; i++) {
+      const argType = funcData.argTypes && funcData.argTypes[i];
+      const isOptional = i < funcData.numOptionalArgs; // Ignore spaces between arguments.  As the TeXbook says:
+      // "After you have said ‘\def\row#1#2{...}’, you are allowed to
+      //  put spaces between the arguments (e.g., ‘\row x n’), because
+      //  TeX doesn’t use single spaces as undelimited arguments."
+
+      const consumeSpaces = i > 0 && !isOptional || // Also consume leading spaces in math mode, as parseSymbol
+      // won't know what to do with them.  This can only happen with
+      // macros, e.g. \frac\foo\foo where \foo expands to a space symbol.
+      // In LaTeX, the \foo's get treated as (blank) arguments.
+      // In KaTeX, for now, both spaces will get consumed.
+      // TODO(edemaine)
+      i === 0 && !isOptional && this.mode === "math";
+      const arg = this.parseGroupOfType(`argument to '${func}'`, argType, isOptional, baseGreediness, consumeSpaces);
+
+      if (!arg) {
+        if (isOptional) {
+          optArgs.push(null);
+          continue;
+        }
+
+        throw new ParseError(`Expected group after '${func}'`, this.fetch());
+      }
+
+      (isOptional ? optArgs : args).push(arg);
+    }
+
+    return {
+      args,
+      optArgs
+    };
+  }
+  /**
+   * Parses a group when the mode is changing.
+   */
+
+
+  parseGroupOfType(name, type, optional, greediness, consumeSpaces) {
+    switch (type) {
+      case "color":
+        if (consumeSpaces) {
+          this.consumeSpaces();
+        }
+
+        return this.parseColorGroup(optional);
+
+      case "size":
+        if (consumeSpaces) {
+          this.consumeSpaces();
+        }
+
+        return this.parseSizeGroup(optional);
+
+      case "url":
+        return this.parseUrlGroup(optional, consumeSpaces);
+
+      case "math":
+      case "text":
+        return this.parseGroup(name, optional, greediness, undefined, type, consumeSpaces);
+
+      case "hbox":
+        {
+          // hbox argument type wraps the argument in the equivalent of
+          // \hbox, which is like \text but switching to \textstyle size.
+          const group = this.parseGroup(name, optional, greediness, undefined, "text", consumeSpaces);
+
+          if (!group) {
+            return group;
+          }
+
+          const styledGroup = {
+            type: "styling",
+            mode: group.mode,
+            body: [group],
+            style: "text" // simulate \textstyle
+
+          };
+          return styledGroup;
+        }
+
+      case "raw":
+        {
+          if (consumeSpaces) {
+            this.consumeSpaces();
+          }
+
+          if (optional && this.fetch().text === "{") {
+            return null;
+          }
+
+          const token = this.parseStringGroup("raw", optional, true);
+
+          if (token) {
+            return {
+              type: "raw",
+              mode: "text",
+              string: token.text
+            };
+          } else {
+            throw new ParseError("Expected raw group", this.fetch());
+          }
+        }
+
+      case "original":
+      case null:
+      case undefined:
+        return this.parseGroup(name, optional, greediness, undefined, undefined, consumeSpaces);
+
+      default:
+        throw new ParseError("Unknown group type as " + name, this.fetch());
+    }
+  }
+  /**
+   * Discard any space tokens, fetching the next non-space token.
+   */
+
+
+  consumeSpaces() {
+    while (this.fetch().text === " ") {
+      this.consume();
+    }
+  }
+  /**
+   * Parses a group, essentially returning the string formed by the
+   * brace-enclosed tokens plus some position information.
+   */
+
+
+  parseStringGroup(modeName, // Used to describe the mode in error messages.
+  optional, raw) {
+    const groupBegin = optional ? "[" : "{";
+    const groupEnd = optional ? "]" : "}";
+    const beginToken = this.fetch();
+
+    if (beginToken.text !== groupBegin) {
+      if (optional) {
+        return null;
+      } else if (raw && beginToken.text !== "EOF" && /[^{}[\]]/.test(beginToken.text)) {
+        this.consume();
+        return beginToken;
+      }
+    }
+
+    const outerMode = this.mode;
+    this.mode = "text";
+    this.expect(groupBegin);
+    let str = "";
+    const firstToken = this.fetch();
+    let nested = 0; // allow nested braces in raw string group
+
+    let lastToken = firstToken;
+    let nextToken;
+
+    while ((nextToken = this.fetch()).text !== groupEnd || raw && nested > 0) {
+      switch (nextToken.text) {
+        case "EOF":
+          throw new ParseError("Unexpected end of input in " + modeName, firstToken.range(lastToken, str));
+
+        case groupBegin:
+          nested++;
+          break;
+
+        case groupEnd:
+          nested--;
+          break;
+      }
+
+      lastToken = nextToken;
+      str += lastToken.text;
+      this.consume();
+    }
+
+    this.expect(groupEnd);
+    this.mode = outerMode;
+    return firstToken.range(lastToken, str);
+  }
+  /**
+   * Parses a regex-delimited group: the largest sequence of tokens
+   * whose concatenated strings match `regex`. Returns the string
+   * formed by the tokens plus some position information.
+   */
+
+
+  parseRegexGroup(regex, modeName) {
+    const outerMode = this.mode;
+    this.mode = "text";
+    const firstToken = this.fetch();
+    let lastToken = firstToken;
+    let str = "";
+    let nextToken;
+
+    while ((nextToken = this.fetch()).text !== "EOF" && regex.test(str + nextToken.text)) {
+      lastToken = nextToken;
+      str += lastToken.text;
+      this.consume();
+    }
+
+    if (str === "") {
+      throw new ParseError("Invalid " + modeName + ": '" + firstToken.text + "'", firstToken);
+    }
+
+    this.mode = outerMode;
+    return firstToken.range(lastToken, str);
+  }
+  /**
+   * Parses a color description.
+   */
+
+
+  parseColorGroup(optional) {
+    const res = this.parseStringGroup("color", optional);
+
+    if (!res) {
+      return null;
+    }
+
+    const match = /^(#[a-f0-9]{3}|#?[a-f0-9]{6}|[a-z]+)$/i.exec(res.text);
+
+    if (!match) {
+      throw new ParseError("Invalid color: '" + res.text + "'", res);
+    }
+
+    let color = match[0];
+
+    if (/^[0-9a-f]{6}$/i.test(color)) {
+      // We allow a 6-digit HTML color spec without a leading "#".
+      // This follows the xcolor package's HTML color model.
+      // Predefined color names are all missed by this RegEx pattern.
+      color = "#" + color;
+    }
+
+    return {
+      type: "color-token",
+      mode: this.mode,
+      color
+    };
+  }
+  /**
+   * Parses a size specification, consisting of magnitude and unit.
+   */
+
+
+  parseSizeGroup(optional) {
+    let res;
+    let isBlank = false;
+
+    if (!optional && this.fetch().text !== "{") {
+      res = this.parseRegexGroup(/^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2} *$/, "size");
+    } else {
+      res = this.parseStringGroup("size", optional);
+    }
+
+    if (!res) {
+      return null;
+    }
+
+    if (!optional && res.text.length === 0) {
+      // Because we've tested for what is !optional, this block won't
+      // affect \kern, \hspace, etc. It will capture the mandatory arguments
+      // to \genfrac and \above.
+      res.text = "0pt"; // Enable \above{}
+
+      isBlank = true; // This is here specifically for \genfrac
+    }
+
+    const match = /([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(res.text);
+
+    if (!match) {
+      throw new ParseError("Invalid size: '" + res.text + "'", res);
+    }
+
+    const data = {
+      number: +(match[1] + match[2]),
+      // sign + magnitude, cast to number
+      unit: match[3]
+    };
+
+    if (!validUnit(data)) {
+      throw new ParseError("Invalid unit: '" + data.unit + "'", res);
+    }
+
+    return {
+      type: "size",
+      mode: this.mode,
+      value: data,
+      isBlank
+    };
+  }
+  /**
+   * Parses an URL, checking escaped letters and allowed protocols,
+   * and setting the catcode of % as an active character (as in \hyperref).
+   */
+
+
+  parseUrlGroup(optional, consumeSpaces) {
+    this.gullet.lexer.setCatcode("%", 13); // active character
+
+    const res = this.parseStringGroup("url", optional, true); // get raw string
+
+    this.gullet.lexer.setCatcode("%", 14); // comment character
+
+    if (!res) {
+      return null;
+    } // hyperref package allows backslashes alone in href, but doesn't
+    // generate valid links in such cases; we interpret this as
+    // "undefined" behaviour, and keep them as-is. Some browser will
+    // replace backslashes with forward slashes.
+
+
+    const url = res.text.replace(/\\([#$%&~_^{}])/g, '$1');
+    return {
+      type: "url",
+      mode: this.mode,
+      url
+    };
+  }
+  /**
+   * If `optional` is false or absent, this parses an ordinary group,
+   * which is either a single nucleus (like "x") or an expression
+   * in braces (like "{x+y}") or an implicit group, a group that starts
+   * at the current position, and ends right before a higher explicit
+   * group ends, or at EOF.
+   * If `optional` is true, it parses either a bracket-delimited expression
+   * (like "[x+y]") or returns null to indicate the absence of a
+   * bracket-enclosed group.
+   * If `mode` is present, switches to that mode while parsing the group,
+   * and switches back after.
+   */
+
+
+  parseGroup(name, // For error reporting.
+  optional, greediness, breakOnTokenText, mode, consumeSpaces) {
+    // Switch to specified mode
+    const outerMode = this.mode;
+
+    if (mode) {
+      this.switchMode(mode);
+    } // Consume spaces if requested, crucially *after* we switch modes,
+    // so that the next non-space token is parsed in the correct mode.
+
+
+    if (consumeSpaces) {
+      this.consumeSpaces();
+    } // Get first token
+
+
+    const firstToken = this.fetch();
+    const text = firstToken.text;
+    let result; // Try to parse an open brace or \begingroup
+
+    if (optional ? text === "[" : text === "{" || text === "\\begingroup") {
+      this.consume();
+      const groupEnd = Parser.endOfGroup[text]; // Start a new group namespace
+
+      this.gullet.beginGroup(); // If we get a brace, parse an expression
+
+      const expression = this.parseExpression(false, groupEnd);
+      const lastToken = this.fetch(); // Check that we got a matching closing brace
+
+      this.expect(groupEnd); // End group namespace
+
+      this.gullet.endGroup();
+      result = {
+        type: "ordgroup",
+        mode: this.mode,
+        loc: SourceLocation.range(firstToken, lastToken),
+        body: expression,
+        // A group formed by \begingroup...\endgroup is a semi-simple group
+        // which doesn't affect spacing in math mode, i.e., is transparent.
+        // https://tex.stackexchange.com/questions/1930/when-should-one-
+        // use-begingroup-instead-of-bgroup
+        semisimple: text === "\\begingroup" || undefined
+      };
+    } else if (optional) {
+      // Return nothing for an optional group
+      result = null;
+    } else {
+      // If there exists a function with this name, parse the function.
+      // Otherwise, just return a nucleus
+      result = this.parseFunction(breakOnTokenText, name, greediness) || this.parseSymbol();
+
+      if (result == null && text[0] === "\\" && !implicitCommands.hasOwnProperty(text)) {
+        if (this.settings.throwOnError) {
+          throw new ParseError("Undefined control sequence: " + text, firstToken);
+        }
+
+        result = this.formatUnsupportedCmd(text);
+        this.consume();
+      }
+    } // Switch mode back
+
+
+    if (mode) {
+      this.switchMode(outerMode);
+    }
+
+    return result;
+  }
+  /**
+   * Form ligature-like combinations of characters for text mode.
+   * This includes inputs like "--", "---", "``" and "''".
+   * The result will simply replace multiple textord nodes with a single
+   * character in each value by a single textord node having multiple
+   * characters in its value.  The representation is still ASCII source.
+   * The group will be modified in place.
+   */
+
+
+  formLigatures(group) {
+    let n = group.length - 1;
+
+    for (let i = 0; i < n; ++i) {
+      const a = group[i]; // $FlowFixMe: Not every node type has a `text` property.
+
+      const v = a.text;
+
+      if (v === "-" && group[i + 1].text === "-") {
+        if (i + 1 < n && group[i + 2].text === "-") {
+          group.splice(i, 3, {
+            type: "textord",
+            mode: "text",
+            loc: SourceLocation.range(a, group[i + 2]),
+            text: "---"
+          });
+          n -= 2;
+        } else {
+          group.splice(i, 2, {
+            type: "textord",
+            mode: "text",
+            loc: SourceLocation.range(a, group[i + 1]),
+            text: "--"
+          });
+          n -= 1;
+        }
+      }
+
+      if ((v === "'" || v === "`") && group[i + 1].text === v) {
+        group.splice(i, 2, {
+          type: "textord",
+          mode: "text",
+          loc: SourceLocation.range(a, group[i + 1]),
+          text: v + v
+        });
+        n -= 1;
+      }
+    }
+  }
+  /**
+   * Parse a single symbol out of the string. Here, we handle single character
+   * symbols and special functions like \verb.
+   */
+
+
+  parseSymbol() {
+    const nucleus = this.fetch();
+    let text = nucleus.text;
+
+    if (/^\\verb[^a-zA-Z]/.test(text)) {
+      this.consume();
+      let arg = text.slice(5);
+      const star = arg.charAt(0) === "*";
+
+      if (star) {
+        arg = arg.slice(1);
+      } // Lexer's tokenRegex is constructed to always have matching
+      // first/last characters.
+
+
+      if (arg.length < 2 || arg.charAt(0) !== arg.slice(-1)) {
+        throw new ParseError(`\\verb assertion failed --
+                    please report what input caused this bug`);
+      }
+
+      arg = arg.slice(1, -1); // remove first and last char
+
+      return {
+        type: "verb",
+        mode: "text",
+        body: arg,
+        star
+      };
+    } // At this point, we should have a symbol, possibly with accents.
+    // First expand any accented base symbol according to unicodeSymbols.
+
+
+    if (unicodeSymbols.hasOwnProperty(text[0]) && !symbols[this.mode][text[0]]) {
+      // This behavior is not strict (XeTeX-compatible) in math mode.
+      if (this.settings.strict && this.mode === "math") {
+        this.settings.reportNonstrict("unicodeTextInMathMode", `Accented Unicode text character "${text[0]}" used in ` + `math mode`, nucleus);
+      }
+
+      text = unicodeSymbols[text[0]] + text.substr(1);
+    } // Strip off any combining characters
+
+
+    const match = combiningDiacriticalMarksEndRegex.exec(text);
+
+    if (match) {
+      text = text.substring(0, match.index);
+
+      if (text === 'i') {
+        text = '\u0131'; // dotless i, in math and text mode
+      } else if (text === 'j') {
+        text = '\u0237'; // dotless j, in math and text mode
+      }
+    } // Recognize base symbol
+
+
+    let symbol;
+
+    if (symbols[this.mode][text]) {
+      if (this.settings.strict && this.mode === 'math' && extraLatin.indexOf(text) >= 0) {
+        this.settings.reportNonstrict("unicodeTextInMathMode", `Latin-1/Unicode text character "${text[0]}" used in ` + `math mode`, nucleus);
+      }
+
+      const group = symbols[this.mode][text].group;
+      const loc = SourceLocation.range(nucleus);
+      let s;
+
+      if (ATOMS.hasOwnProperty(group)) {
+        // $FlowFixMe
+        const family = group;
+        s = {
+          type: "atom",
+          mode: this.mode,
+          family,
+          loc,
+          text
+        };
+      } else {
+        // $FlowFixMe
+        s = {
+          type: group,
+          mode: this.mode,
+          loc,
+          text
+        };
+      }
+
+      symbol = s;
+    } else if (text.charCodeAt(0) >= 0x80) {
+      // no symbol for e.g. ^
+      if (this.settings.strict) {
+        if (!supportedCodepoint(text.charCodeAt(0))) {
+          this.settings.reportNonstrict("unknownSymbol", `Unrecognized Unicode character "${text[0]}"` + ` (${text.charCodeAt(0)})`, nucleus);
+        } else if (this.mode === "math") {
+          this.settings.reportNonstrict("unicodeTextInMathMode", `Unicode text character "${text[0]}" used in math mode`, nucleus);
+        }
+      } // All nonmathematical Unicode characters are rendered as if they
+      // are in text mode (wrapped in \text) because that's what it
+      // takes to render them in LaTeX.  Setting `mode: this.mode` is
+      // another natural choice (the user requested math mode), but
+      // this makes it more difficult for getCharacterMetrics() to
+      // distinguish Unicode characters without metrics and those for
+      // which we want to simulate the letter M.
+
+
+      symbol = {
+        type: "textord",
+        mode: "text",
+        loc: SourceLocation.range(nucleus),
+        text
+      };
+    } else {
+      return null; // EOF, ^, _, {, }, etc.
+    }
+
+    this.consume(); // Transform combining characters into accents
+
+    if (match) {
+      for (let i = 0; i < match[0].length; i++) {
+        const accent = match[0][i];
+
+        if (!unicodeAccents[accent]) {
+          throw new ParseError(`Unknown accent ' ${accent}'`, nucleus);
+        }
+
+        const command = unicodeAccents[accent][this.mode];
+
+        if (!command) {
+          throw new ParseError(`Accent ${accent} unsupported in ${this.mode} mode`, nucleus);
+        }
+
+        symbol = {
+          type: "accent",
+          mode: this.mode,
+          loc: SourceLocation.range(nucleus),
+          label: command,
+          isStretchy: false,
+          isShifty: true,
+          base: symbol
+        };
+      }
+    }
+
+    return symbol;
+  }
+
+}
+Parser.endOfExpression = ["}", "\\endgroup", "\\end", "\\right", "&"];
+Parser.endOfGroup = {
+  "[": "]",
+  "{": "}",
+  "\\begingroup": "\\endgroup"
+  /**
+   * Parses an "expression", which is a list of atoms.
+   *
+   * `breakOnInfix`: Should the parsing stop when we hit infix nodes? This
+   *                 happens when functions have higher precendence han infix
+   *                 nodes in implicit parses.
+   *
+   * `breakOnTokenText`: The text of the token that the expression should end
+   *                     with, or `null` if something else should end the
+   *                     expression.
+   */
+
+};
+Parser.SUPSUB_GREEDINESS = 1;
+
+/**
+ * Provides a single function for parsing an expression using a Parser
+ * TODO(emily): Remove this
+ */
+
+/**
+ * Parses an expression using a Parser, then returns the parsed result.
+ */
+const parseTree = function parseTree(toParse, settings) {
+  if (!(typeof toParse === 'string' || toParse instanceof String)) {
+    throw new TypeError('KaTeX can only parse string typed expression');
+  }
+
+  const parser = new Parser(toParse, settings); // Blank out any \df@tag to avoid spurious "Duplicate \tag" errors
+
+  delete parser.gullet.macros.current["\\df@tag"];
+  let tree = parser.parse(); // If the input used \tag, it will set the \df@tag macro to the tag.
+  // In this case, we separately parse the tag and wrap the tree.
+
+  if (parser.gullet.macros.get("\\df@tag")) {
+    if (!settings.displayMode) {
+      throw new ParseError("\\tag works only in display equations");
+    }
+
+    parser.gullet.feed("\\df@tag");
+    tree = [{
+      type: "tag",
+      mode: "text",
+      body: tree,
+      tag: parser.parse()
+    }];
+  }
+
+  return tree;
+};
+
+/* eslint no-console:0 */
+
+/**
+ * Parse and build an expression, and place that expression in the DOM node
+ * given.
+ */
+let render = function render(expression, baseNode, options) {
+  baseNode.textContent = "";
+  const node = renderToDomTree(expression, options).toNode();
+  baseNode.appendChild(node);
+}; // KaTeX's styles don't work properly in quirks mode. Print out an error, and
+// disable rendering.
+
+
+if (typeof document !== "undefined") {
+  if (document.compatMode !== "CSS1Compat") {
+    typeof console !== "undefined" && console.warn("Warning: KaTeX doesn't work in quirks mode. Make sure your " + "website has a suitable doctype.");
+
+    render = function render() {
+      throw new ParseError("KaTeX doesn't work in quirks mode.");
+    };
+  }
+}
+/**
+ * Parse and build an expression, and return the markup for that.
+ */
+
+
+const renderToString = function renderToString(expression, options) {
+  const markup = renderToDomTree(expression, options).toMarkup();
+  return markup;
+};
+/**
+ * Parse an expression and return the parse tree.
+ */
+
+
+const generateParseTree = function generateParseTree(expression, options) {
+  const settings = new Settings(options);
+  return parseTree(expression, settings);
+};
+/**
+ * If the given error is a KaTeX ParseError and options.throwOnError is false,
+ * renders the invalid LaTeX as a span with hover title giving the KaTeX
+ * error message.  Otherwise, simply throws the error.
+ */
+
+
+const renderError = function renderError(error, expression, options) {
+  if (options.throwOnError || !(error instanceof ParseError)) {
+    throw error;
+  }
+
+  const node = buildCommon.makeSpan(["katex-error"], [new SymbolNode(expression)]);
+  node.setAttribute("title", error.toString());
+  node.setAttribute("style", `color:${options.errorColor}`);
+  return node;
+};
+/**
+ * Generates and returns the katex build tree. This is used for advanced
+ * use cases (like rendering to custom output).
+ */
+
+
+const renderToDomTree = function renderToDomTree(expression, options) {
+  const settings = new Settings(options);
+
+  try {
+    const tree = parseTree(expression, settings);
+    return buildTree(tree, expression, settings);
+  } catch (error) {
+    return renderError(error, expression, settings);
+  }
+};
+/**
+ * Generates and returns the katex build tree, with just HTML (no MathML).
+ * This is used for advanced use cases (like rendering to custom output).
+ */
+
+
+const renderToHTMLTree = function renderToHTMLTree(expression, options) {
+  const settings = new Settings(options);
+
+  try {
+    const tree = parseTree(expression, settings);
+    return buildHTMLTree(tree, expression, settings);
+  } catch (error) {
+    return renderError(error, expression, settings);
+  }
+};
+
+var katex = {
+  /**
+   * Current KaTeX version
+   */
+  version: "0.12.0",
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML combination, and adds
+   * it as a child to the specified DOM node.
+   */
+  render,
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML combination string,
+   * for sending to the client.
+   */
+  renderToString,
+
+  /**
+   * KaTeX error, usually during parsing.
+   */
+  ParseError,
+
+  /**
+   * Parses the given LaTeX into KaTeX's internal parse tree structure,
+   * without rendering to HTML or MathML.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __parse: generateParseTree,
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML internal DOM tree
+   * representation, without flattening that representation to a string.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __renderToDomTree: renderToDomTree,
+
+  /**
+   * Renders the given LaTeX into an HTML internal DOM tree representation,
+   * without MathML and without flattening that representation to a string.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __renderToHTMLTree: renderToHTMLTree,
+
+  /**
+   * extends internal font metrics object with a new object
+   * each key in the new object represents a font name
+  */
+  __setFontMetrics: setFontMetrics,
+
+  /**
+   * adds a new symbol to builtin symbols table
+   */
+  __defineSymbol: defineSymbol,
+
+  /**
+   * adds a new macro to builtin macro list
+   */
+  __defineMacro: defineMacro,
+
+  /**
+   * Expose the dom tree node types, which can be useful for type checking nodes.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __domTree: {
+    Span,
+    Anchor,
+    SymbolNode,
+    SvgNode,
+    PathNode,
+    LineNode
+  }
+};
+
+export default katex;
diff --git a/docs/pt/opensearch.xml b/docs/pt/opensearch.xml
new file mode 100644
index 0000000..2b090b4
--- /dev/null
+++ b/docs/pt/opensearch.xml
@@ -0,0 +1,11 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<OpenSearchDescription xmlns="http://a9.com/-/spec/opensearch/1.1/"
+                       xmlns:moz="http://www.mozilla.org/2006/browser/search/">
+  <ShortName>PSP-UFU</ShortName>
+  <Description>Search PSP-UFU</Description>
+  <InputEncoding>UTF-8</InputEncoding>
+  <Image width="16" height="16" type="image/x-icon">https://thales1330.github.io/PSP/pt/img/favicon.ico</Image>
+  <Url type="text/html" method="get" template="https://thales1330.github.io/PSP/pt/search?q={searchTerms}"/>
+  <Url type="application/opensearchdescription+xml" rel="self" template="https://thales1330.github.io/PSP/pt/opensearch.xml" />
+  <moz:SearchForm>https://thales1330.github.io/PSP/pt/</moz:SearchForm>
+</OpenSearchDescription>
\ No newline at end of file
diff --git a/docs/pt/search/index.html b/docs/pt/search/index.html
new file mode 100644
index 0000000..767d238
--- /dev/null
+++ b/docs/pt/search/index.html
@@ -0,0 +1,27 @@
+<!doctype html>
+<html lang="pt" dir="ltr">
+<head>
+<meta charset="UTF-8">
+<meta name="viewport" content="width=device-width,initial-scale=1">
+<meta name="generator" content="Docusaurus v2.0.0-alpha.71">
+<link rel="alternate" type="application/rss+xml" href="/PSP/pt/blog/rss.xml" title="PSP-UFU Blog RSS Feed">
+<link rel="alternate" type="application/atom+xml" href="/PSP/pt/blog/atom.xml" title="PSP-UFU Blog Atom Feed">
+<link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/pt/opensearch.xml"><title data-react-helmet="true">Search the documentation | PSP-UFU</title><meta data-react-helmet="true" property="og:url" content="https://thales1330.github.io/PSP/pt/search"><meta data-react-helmet="true" name="docsearch:language" content="pt"><meta data-react-helmet="true" name="docsearch:docusaurus_tag" content="default"><meta data-react-helmet="true" property="robots" content="noindex, follow"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/pt/img/favicon.ico"><link data-react-helmet="true" rel="canonical" href="https://thales1330.github.io/PSP/pt/search"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/search" hreflang="en"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/pt/search" hreflang="pt"><link data-react-helmet="true" rel="alternate" href="https://thales1330.github.io/PSP/search" hreflang="x-default"><link data-react-helmet="true" rel="preconnect" href="https://BH4D9OD16A-dsn.algolia.net" crossorigin="anonymous"><link rel="stylesheet" href="/PSP/pt/assets/css/styles.73f64b8b.css">
+<link rel="preload" href="/PSP/pt/assets/js/styles.cc4b2cfe.js" as="script">
+<link rel="preload" href="/PSP/pt/assets/js/runtime~main.b7ffaf86.js" as="script">
+<link rel="preload" href="/PSP/pt/assets/js/main.168dd2d0.js" as="script">
+<link rel="preload" href="/PSP/pt/assets/js/1.c1697df3.js" as="script">
+<link rel="preload" href="/PSP/pt/assets/js/2.6b1d437b.js" as="script">
+<link rel="preload" href="/PSP/pt/assets/js/63.d045ac5e.js" as="script">
+</head>
+<body>
+<script>!function(){function e(e){document.documentElement.setAttribute("data-theme",e)}var t=function(){var e=null;try{e=localStorage.getItem("theme")}catch(e){}return e}();null!==t?e(t):window.matchMedia("(prefers-color-scheme: dark)").matches?e("dark"):window.matchMedia("(prefers-color-scheme: light)").matches?e("light"):e("dark")}()</script><div id="__docusaurus">
+<div><a href="#main" class="skipToContent_1oUP">Skip to main content</a></div><nav class="navbar navbar--fixed-top"><div class="navbar__inner"><div class="navbar__items"><button aria-label="Navigation bar toggle" class="navbar__toggle" type="button" tabindex="0"><svg aria-label="Menu" width="30" height="30" viewBox="0 0 30 30" role="img" focusable="false"><title>Menu</title><path stroke="currentColor" stroke-linecap="round" stroke-miterlimit="10" stroke-width="2" d="M4 7h22M4 15h22M4 23h22"></path></svg></button><a class="navbar__brand" href="/PSP/pt/"><img src="/PSP/pt/img/logo.svg" alt="PSP-UFU logo" class="themedImage_1VuW themedImage--light_3UqQ navbar__logo"><img src="/PSP/pt/img/logo.svg" alt="PSP-UFU logo" class="themedImage_1VuW themedImage--dark_hz6m navbar__logo"><strong class="navbar__title">PSP-UFU</strong></a><a class="navbar__item navbar__link" href="/PSP/pt/docs/">Manual do Usuário</a><a class="navbar__item navbar__link" href="/PSP/pt/blog">Registro de 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class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item">Documentação do Código</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Comunidade</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Faça uma pergunta</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Crédito das Imagens do Site</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Imagens criadas por upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="footer__bottom text--center"><div class="footer__copyright">Copyright © 2021 Thales Lima Oliveira. Construído com Docusaurus.</div></div></div></footer></div>
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+  <property name="kdenlive:docproperties.position">51</property>
+  <property name="kdenlive:docproperties.previewextension"/>
+  <property name="kdenlive:docproperties.previewparameters"/>
+  <property name="kdenlive:docproperties.profile">C:/Users/thale/AppData/Roaming/kdenlive/profiles/customprofile0</property>
+  <property name="kdenlive:docproperties.proxyextension"/>
+  <property name="kdenlive:docproperties.proxyimageminsize">2000</property>
+  <property name="kdenlive:docproperties.proxyimagesize">800</property>
+  <property name="kdenlive:docproperties.proxyminsize">1000</property>
+  <property name="kdenlive:docproperties.proxyparams"/>
+  <property name="kdenlive:docproperties.renderaudioquality">6</property>
+  <property name="kdenlive:docproperties.rendercategory">Generic (HD for web, mobile devices...)</property>
+  <property name="kdenlive:docproperties.renderendguide">-1</property>
+  <property name="kdenlive:docproperties.renderexportaudio">0</property>
+  <property name="kdenlive:docproperties.renderfield">0</property>
+  <property name="kdenlive:docproperties.renderguide">0</property>
+  <property name="kdenlive:docproperties.renderplay">0</property>
+  <property name="kdenlive:docproperties.renderprofile">WebM - the widespread free format (VP8/Vorbis)</property>
+  <property name="kdenlive:docproperties.renderquality">23</property>
+  <property name="kdenlive:docproperties.renderratio">1</property>
+  <property name="kdenlive:docproperties.renderrescale">0</property>
+  <property name="kdenlive:docproperties.renderrescaleheight">540</property>
+  <property name="kdenlive:docproperties.renderrescalewidth">960</property>
+  <property name="kdenlive:docproperties.renderscanning">0</property>
+  <property name="kdenlive:docproperties.renderspeed">2</property>
+  <property name="kdenlive:docproperties.renderstartguide">-1</property>
+  <property name="kdenlive:docproperties.rendertcoverlay">0</property>
+  <property name="kdenlive:docproperties.rendertctype">0</property>
+  <property name="kdenlive:docproperties.rendertwopass">0</property>
+  <property name="kdenlive:docproperties.renderurl">delete.webm</property>
+  <property name="kdenlive:docproperties.renderzone">0</property>
+  <property name="kdenlive:docproperties.scrollPos">0</property>
+  <property name="kdenlive:docproperties.seekOffset">30000</property>
+  <property name="kdenlive:docproperties.version">0.99</property>
+  <property name="kdenlive:docproperties.verticalzoom">1</property>
+  <property name="kdenlive:docproperties.videoTarget">-1</property>
+  <property name="kdenlive:docproperties.zonein">0</property>
+  <property name="kdenlive:docproperties.zoneout">75</property>
+  <property name="kdenlive:docproperties.zoom">4</property>
+  <property name="kdenlive:documentnotes"/>
+  <property name="xml_retain">1</property>
+  <entry producer="producer0" in="00:00:00.000" out="00:00:11.000"/>
+ </playlist>
+ <producer id="black_track" in="00:00:00.000" out="00:16:46.033">
+  <property name="length">2147483647</property>
+  <property name="eof">continue</property>
+  <property name="resource">black</property>
+  <property name="aspect_ratio">1</property>
+  <property name="mlt_service">color</property>
+  <property name="mlt_image_format">rgb24a</property>
+  <property name="set.test_audio">0</property>
+ </producer>
+ <playlist id="playlist0">
+  <property name="kdenlive:audio_track">1</property>
+  <filter id="filter0">
+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter1">
+   <property name="channel">-1</property>
+   <property name="mlt_service">panner</property>
+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
+   <property name="disable">1</property>
+  </filter>
+ </playlist>
+ <playlist id="playlist1">
+  <property name="kdenlive:audio_track">1</property>
+ </playlist>
+ <tractor id="tractor0" in="00:00:00.000">
+  <property name="kdenlive:audio_track">1</property>
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="video" producer="playlist0"/>
+  <track hide="video" producer="playlist1"/>
+  <filter id="filter2">
+   <property name="iec_scale">0</property>
+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
+  </filter>
+ </tractor>
+ <producer id="producer1" in="00:00:00.000" out="00:00:11.000">
+  <property name="length">340</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-07-17 20-22-43.mkv</property>
+  <property name="audio_index">1</property>
+  <property name="video_index">0</property>
+  <property name="mute_on_pause">0</property>
+  <property name="mlt_service">avformat-novalidate</property>
+  <property name="seekable">1</property>
+  <property name="aspect_ratio">1</property>
+  <property name="kdenlive:clipname"/>
+  <property name="kdenlive:folderid">-1</property>
+  <property name="kdenlive:id">2</property>
+  <property name="kdenlive:file_size">3540690</property>
+  <property name="kdenlive:file_hash">da785c97e6b5e7e07e5c7b52335cfb3c</property>
+  <property name="global_feed">1</property>
+  <property name="xml">was here</property>
+  <property name="meta.media.nb_streams">2</property>
+  <property name="meta.media.0.stream.type">video</property>
+  <property name="meta.media.0.stream.frame_rate">30</property>
+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
+  <property name="meta.media.0.codec.width">1920</property>
+  <property name="meta.media.0.codec.height">1080</property>
+  <property name="meta.media.0.codec.rotate">0</property>
+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
+  <property name="meta.media.0.codec.bit_rate">0</property>
+  <property name="meta.attr.0.stream.DURATION.markup">00:00:11.333000000</property>
+  <property name="meta.media.1.stream.type">audio</property>
+  <property name="meta.media.1.codec.sample_fmt">fltp</property>
+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:11.262000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="meta.media.frame_rate_num">30</property>
+  <property name="meta.media.frame_rate_den">1</property>
+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+  <property name="set.test_audio">0</property>
+  <property name="set.test_image">1</property>
+ </producer>
+ <playlist id="playlist2">
+  <property name="kdenlive:audio_track">1</property>
+  <entry producer="producer1" in="00:00:00.000" out="00:00:06.000">
+   <property name="kdenlive:id">2</property>
+  </entry>
+  <filter id="filter3">
+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter4">
+   <property name="channel">-1</property>
+   <property name="mlt_service">panner</property>
+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
+   <property name="disable">1</property>
+  </filter>
+ </playlist>
+ <playlist id="playlist3">
+  <property name="kdenlive:audio_track">1</property>
+ </playlist>
+ <tractor id="tractor1" in="00:00:00.000" out="00:00:06.000">
+  <property name="kdenlive:audio_track">1</property>
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="video" producer="playlist2"/>
+  <track hide="video" producer="playlist3"/>
+  <filter id="filter5">
+   <property name="iec_scale">0</property>
+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
+  </filter>
+ </tractor>
+ <producer id="producer2" in="00:00:00.000" out="00:00:11.000">
+  <property name="length">340</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-07-17 20-22-43.mkv</property>
+  <property name="audio_index">1</property>
+  <property name="video_index">0</property>
+  <property name="mute_on_pause">0</property>
+  <property name="mlt_service">avformat-novalidate</property>
+  <property name="seekable">1</property>
+  <property name="aspect_ratio">1</property>
+  <property name="kdenlive:clipname"/>
+  <property name="kdenlive:folderid">-1</property>
+  <property name="kdenlive:id">2</property>
+  <property name="kdenlive:file_size">3540690</property>
+  <property name="kdenlive:file_hash">da785c97e6b5e7e07e5c7b52335cfb3c</property>
+  <property name="global_feed">1</property>
+  <property name="xml">was here</property>
+  <property name="meta.media.nb_streams">2</property>
+  <property name="meta.media.0.stream.type">video</property>
+  <property name="meta.media.0.stream.frame_rate">30</property>
+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
+  <property name="meta.media.0.codec.width">1920</property>
+  <property name="meta.media.0.codec.height">1080</property>
+  <property name="meta.media.0.codec.rotate">0</property>
+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
+  <property name="meta.media.0.codec.bit_rate">0</property>
+  <property name="meta.attr.0.stream.DURATION.markup">00:00:11.333000000</property>
+  <property name="meta.media.1.stream.type">audio</property>
+  <property name="meta.media.1.codec.sample_fmt">fltp</property>
+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:11.262000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="meta.media.frame_rate_num">30</property>
+  <property name="meta.media.frame_rate_den">1</property>
+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+  <property name="set.test_audio">1</property>
+  <property name="set.test_image">0</property>
+ </producer>
+ <playlist id="playlist4">
+  <entry producer="producer2" in="00:00:00.000" out="00:00:06.000">
+   <property name="kdenlive:id">2</property>
+  </entry>
+ </playlist>
+ <playlist id="playlist5"/>
+ <tractor id="tractor2" in="00:00:00.000" out="00:00:06.000">
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="audio" producer="playlist4"/>
+  <track hide="audio" producer="playlist5"/>
+ </tractor>
+ <playlist id="playlist6"/>
+ <playlist id="playlist7"/>
+ <tractor id="tractor3" in="00:00:00.000">
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="audio" producer="playlist6"/>
+  <track hide="audio" producer="playlist7"/>
+ </tractor>
+ <tractor id="tractor4" global_feed="1" in="00:00:00.000" out="00:16:46.033">
+  <track producer="black_track"/>
+  <track producer="tractor0"/>
+  <track producer="tractor1"/>
+  <track producer="tractor2"/>
+  <track producer="tractor3"/>
+  <transition id="transition0">
+   <property name="a_track">0</property>
+   <property name="b_track">1</property>
+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
+  <transition id="transition1">
+   <property name="a_track">0</property>
+   <property name="b_track">2</property>
+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
+  <transition id="transition2">
+   <property name="a_track">0</property>
+   <property name="b_track">3</property>
+   <property name="compositing">0</property>
+   <property name="distort">0</property>
+   <property name="rotate_center">0</property>
+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+  </transition>
+  <transition id="transition3">
+   <property name="a_track">0</property>
+   <property name="b_track">4</property>
+   <property name="compositing">0</property>
+   <property name="distort">0</property>
+   <property name="rotate_center">0</property>
+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+  </transition>
+  <filter id="filter6">
+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter7">
+   <property name="channel">-1</property>
+   <property name="mlt_service">panner</property>
+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter8">
+   <property name="iec_scale">0</property>
+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
+  </filter>
+ </tractor>
+</mlt>
diff --git a/docs/pt/videos/delete.mp4 b/docs/pt/videos/delete.mp4
new file mode 100644
index 0000000..7395c9d
--- /dev/null
+++ b/docs/pt/videos/delete.mp4
diff --git a/docs/pt/videos/delete.webm b/docs/pt/videos/delete.webm
new file mode 100644
index 0000000..4298c0a
--- /dev/null
+++ b/docs/pt/videos/delete.webm
diff --git a/docs/pt/videos/disconnect.mp4 b/docs/pt/videos/disconnect.mp4
new file mode 100644
index 0000000..e06275a
--- /dev/null
+++ b/docs/pt/videos/disconnect.mp4
diff --git a/docs/pt/videos/disconnect.webm b/docs/pt/videos/disconnect.webm
new file mode 100644
index 0000000..13c2875
--- /dev/null
+++ b/docs/pt/videos/disconnect.webm
diff --git a/docs/pt/videos/drag.kdenlive b/docs/pt/videos/drag.kdenlive
new file mode 100644
index 0000000..57f6ed5
--- /dev/null
+++ b/docs/pt/videos/drag.kdenlive
@@ -0,0 +1,416 @@
+<?xml version='1.0' encoding='utf-8'?>
+<mlt LC_NUMERIC="pt_BR" producer="main_bin" version="6.21.0" root="C:/Users/thale/Documents/GitHub/PSP/docusaurus/static/videos">
+ <profile frame_rate_num="30" sample_aspect_num="1" display_aspect_den="1080" colorspace="601" progressive="1" description="1920x1080 30.00fps" display_aspect_num="1920" frame_rate_den="1" width="1920" height="1080" sample_aspect_den="1"/>
+ <producer id="producer0" in="00:00:00.000" out="00:00:11.000">
+  <property name="length">337</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-07-17 18-55-55.mkv</property>
+  <property name="audio_index">1</property>
+  <property name="video_index">0</property>
+  <property name="mute_on_pause">0</property>
+  <property name="mlt_service">avformat-novalidate</property>
+  <property name="seekable">1</property>
+  <property name="aspect_ratio">1</property>
+  <property name="kdenlive:clipname"/>
+  <property name="kdenlive:folderid">-1</property>
+  <property name="kdenlive:id">2</property>
+  <property name="kdenlive:file_size">3509274</property>
+  <property name="kdenlive:file_hash">09b0d81a59a3e265e609c23bf9e153a1</property>
+  <property name="global_feed">1</property>
+  <property name="meta.media.nb_streams">2</property>
+  <property name="meta.media.0.stream.type">video</property>
+  <property name="meta.media.0.stream.frame_rate">30</property>
+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
+  <property name="meta.media.0.codec.width">1920</property>
+  <property name="meta.media.0.codec.height">1080</property>
+  <property name="meta.media.0.codec.rotate">0</property>
+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
+  <property name="meta.media.0.codec.bit_rate">0</property>
+  <property name="meta.attr.0.stream.DURATION.markup">00:00:11.233000000</property>
+  <property name="meta.media.1.stream.type">audio</property>
+  <property name="meta.media.1.codec.sample_fmt">fltp</property>
+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:11.146000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="meta.media.frame_rate_num">30</property>
+  <property name="meta.media.frame_rate_den">1</property>
+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+ </producer>
+ <playlist id="main_bin">
+  <property name="kdenlive:docproperties.activeTrack">2</property>
+  <property name="kdenlive:docproperties.audioTarget">-1</property>
+  <property name="kdenlive:docproperties.decimalPoint">.</property>
+  <property name="kdenlive:docproperties.disablepreview">0</property>
+  <property name="kdenlive:docproperties.documentid">1595022975488</property>
+  <property name="kdenlive:docproperties.enableTimelineZone">0</property>
+  <property name="kdenlive:docproperties.enableexternalproxy">0</property>
+  <property name="kdenlive:docproperties.enableproxy">0</property>
+  <property name="kdenlive:docproperties.externalproxyparams">../Sub;;S03.MP4;../Clip;;.MXF</property>
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+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
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+  <track hide="audio" producer="playlist4"/>
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+ </tractor>
+ <tractor id="tractor4" global_feed="1" in="00:00:00.000" out="00:16:46.033">
+  <track producer="black_track"/>
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+  <track producer="tractor1"/>
+  <track producer="tractor2"/>
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+  <transition id="transition0">
+   <property name="a_track">0</property>
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+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
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+   <property name="always_active">1</property>
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+   <property name="always_active">1</property>
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+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
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+   <property name="channel">-1</property>
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+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
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+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
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+ </tractor>
+</mlt>
diff --git a/docs/pt/videos/drag.mp4 b/docs/pt/videos/drag.mp4
new file mode 100644
index 0000000..c4086a8
--- /dev/null
+++ b/docs/pt/videos/drag.mp4
diff --git a/docs/pt/videos/drag.webm b/docs/pt/videos/drag.webm
new file mode 100644
index 0000000..a152955
--- /dev/null
+++ b/docs/pt/videos/drag.webm
diff --git a/docs/pt/videos/fit.kdenlive b/docs/pt/videos/fit.kdenlive
new file mode 100644
index 0000000..d6be013
--- /dev/null
+++ b/docs/pt/videos/fit.kdenlive
@@ -0,0 +1,453 @@
+<?xml version='1.0' encoding='utf-8'?>
+<mlt LC_NUMERIC="pt_BR" producer="main_bin" version="6.21.0" root="C:/Users/thale/Documents/GitHub/PSP/docusaurus/static/videos">
+ <profile frame_rate_num="30" sample_aspect_num="1" display_aspect_den="1080" colorspace="601" progressive="1" description="1920x1080 30.00fps" display_aspect_num="1920" frame_rate_den="1" width="1920" height="1080" sample_aspect_den="1"/>
+ <producer id="producer0" in="00:00:00.000" out="00:00:13.000">
+  <property name="length">394</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-07-17 19-01-12.mkv</property>
+  <property name="audio_index">1</property>
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+  <property name="mlt_service">avformat-novalidate</property>
+  <property name="seekable">1</property>
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+  <property name="kdenlive:folderid">-1</property>
+  <property name="kdenlive:id">2</property>
+  <property name="kdenlive:file_size">4094126</property>
+  <property name="kdenlive:file_hash">867a75a8e98fb344c2437301b7610bb1</property>
+  <property name="global_feed">1</property>
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+  <property name="meta.media.0.stream.frame_rate">30</property>
+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
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+  <property name="meta.media.0.codec.height">1080</property>
+  <property name="meta.media.0.codec.rotate">0</property>
+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
+  <property name="meta.media.0.codec.bit_rate">0</property>
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+  <property name="meta.media.1.stream.type">audio</property>
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+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:13.026000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
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+  <property name="meta.media.frame_rate_num">30</property>
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+  <property name="meta.media.color_range">mpeg</property>
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+ </producer>
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+  <property name="kdenlive:docproperties.decimalPoint">.</property>
+  <property name="kdenlive:docproperties.disablepreview">0</property>
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diff --git a/docs/pt/videos/fit.mp4 b/docs/pt/videos/fit.mp4
new file mode 100644
index 0000000..9ec3098
--- /dev/null
+++ b/docs/pt/videos/fit.mp4
diff --git a/docs/pt/videos/fit.webm b/docs/pt/videos/fit.webm
new file mode 100644
index 0000000..6f4b3ba
--- /dev/null
+++ b/docs/pt/videos/fit.webm
diff --git a/docs/pt/videos/graphicProps.kdenlive b/docs/pt/videos/graphicProps.kdenlive
new file mode 100644
index 0000000..a74ab25
--- /dev/null
+++ b/docs/pt/videos/graphicProps.kdenlive
@@ -0,0 +1,539 @@
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+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+ </producer>
+ <playlist id="playlist4">
+  <entry producer="producer3" in="00:00:00,000" out="00:00:04,267">
+   <property name="kdenlive:id">2</property>
+  </entry>
+  <entry producer="producer4" in="00:00:07,500" out="00:00:23,433">
+   <property name="kdenlive:id">2</property>
+  </entry>
+ </playlist>
+ <playlist id="playlist5"/>
+ <tractor id="tractor2" in="00:00:00,000" out="00:00:20,233">
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="audio" producer="playlist4"/>
+  <track hide="audio" producer="playlist5"/>
+ </tractor>
+ <playlist id="playlist6"/>
+ <playlist id="playlist7"/>
+ <tractor id="tractor3" in="00:00:00,000">
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="audio" producer="playlist6"/>
+  <track hide="audio" producer="playlist7"/>
+ </tractor>
+ <tractor id="tractor4" global_feed="1" in="00:00:00,000" out="00:17:00,267">
+  <track producer="black_track"/>
+  <track producer="tractor0"/>
+  <track producer="tractor1"/>
+  <track producer="tractor2"/>
+  <track producer="tractor3"/>
+  <transition id="transition0">
+   <property name="a_track">0</property>
+   <property name="b_track">1</property>
+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
+  <transition id="transition1">
+   <property name="a_track">0</property>
+   <property name="b_track">2</property>
+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
+  <transition id="transition2">
+   <property name="a_track">0</property>
+   <property name="b_track">3</property>
+   <property name="compositing">0</property>
+   <property name="distort">0</property>
+   <property name="rotate_center">0</property>
+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+  </transition>
+  <transition id="transition3">
+   <property name="a_track">0</property>
+   <property name="b_track">4</property>
+   <property name="compositing">0</property>
+   <property name="distort">0</property>
+   <property name="rotate_center">0</property>
+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+  </transition>
+  <filter id="filter6">
+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter7">
+   <property name="channel">-1</property>
+   <property name="mlt_service">panner</property>
+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter8">
+   <property name="iec_scale">0</property>
+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
+  </filter>
+ </tractor>
+</mlt>
diff --git a/docs/pt/videos/graphicPropsChange.mp4 b/docs/pt/videos/graphicPropsChange.mp4
new file mode 100644
index 0000000..c549ca2
--- /dev/null
+++ b/docs/pt/videos/graphicPropsChange.mp4
diff --git a/docs/pt/videos/graphicPropsChange.webm b/docs/pt/videos/graphicPropsChange.webm
new file mode 100644
index 0000000..8dfd3e5
--- /dev/null
+++ b/docs/pt/videos/graphicPropsChange.webm
diff --git a/docs/pt/videos/heatmap.kdenlive b/docs/pt/videos/heatmap.kdenlive
new file mode 100644
index 0000000..79fe2ca
--- /dev/null
+++ b/docs/pt/videos/heatmap.kdenlive
@@ -0,0 +1,406 @@
+<?xml version='1.0' encoding='utf-8'?>
+<mlt LC_NUMERIC="pt_BR" producer="main_bin" version="6.21.0" root="C:/Users/thale/Documents/GitHub/PSP/docusaurus/static/videos">
+ <profile frame_rate_num="30" sample_aspect_num="1" display_aspect_den="1080" colorspace="601" progressive="1" description="1920x1080 30.00fps" display_aspect_num="1920" frame_rate_den="1" width="1920" height="1080" sample_aspect_den="1"/>
+ <producer id="producer0" in="00:00:00,000" out="00:00:22,000">
+  <property name="length">661</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-08-01 17-10-55.mkv</property>
+  <property name="audio_index">1</property>
+  <property name="video_index">0</property>
+  <property name="mute_on_pause">0</property>
+  <property name="mlt_service">avformat-novalidate</property>
+  <property name="seekable">1</property>
+  <property name="aspect_ratio">1</property>
+  <property name="kdenlive:clipname"/>
+  <property name="kdenlive:folderid">-1</property>
+  <property name="kdenlive:id">2</property>
+  <property name="kdenlive:file_size">6998585</property>
+  <property name="kdenlive:file_hash">ae822e60d4e2248265daaa3d21841b38</property>
+  <property name="global_feed">1</property>
+  <property name="meta.media.nb_streams">2</property>
+  <property name="meta.media.0.stream.type">video</property>
+  <property name="meta.media.0.stream.frame_rate">30</property>
+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
+  <property name="meta.media.0.codec.width">1920</property>
+  <property name="meta.media.0.codec.height">1080</property>
+  <property name="meta.media.0.codec.rotate">0</property>
+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
+  <property name="meta.media.0.codec.bit_rate">0</property>
+  <property name="meta.attr.0.stream.DURATION.markup">00:00:22.033000000</property>
+  <property name="meta.media.1.stream.type">audio</property>
+  <property name="meta.media.1.codec.sample_fmt">fltp</property>
+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:21.989000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="meta.media.frame_rate_num">30</property>
+  <property name="meta.media.frame_rate_den">1</property>
+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+ </producer>
+ <playlist id="main_bin">
+  <property name="kdenlive:docproperties.activeTrack">2</property>
+  <property name="kdenlive:docproperties.audioTarget">1</property>
+  <property name="kdenlive:docproperties.decimalPoint">,</property>
+  <property name="kdenlive:docproperties.disablepreview">0</property>
+  <property name="kdenlive:docproperties.documentid">1596312714884</property>
+  <property name="kdenlive:docproperties.enableTimelineZone">0</property>
+  <property name="kdenlive:docproperties.enableexternalproxy">0</property>
+  <property name="kdenlive:docproperties.enableproxy">0</property>
+  <property name="kdenlive:docproperties.externalproxyparams">../Sub;;S03.MP4;../Clip;;.MXF</property>
+  <property name="kdenlive:docproperties.generateimageproxy">0</property>
+  <property name="kdenlive:docproperties.generateproxy">0</property>
+  <property name="kdenlive:docproperties.kdenliveversion">20.04.1</property>
+  <property name="kdenlive:docproperties.position">321</property>
+  <property name="kdenlive:docproperties.previewextension"/>
+  <property name="kdenlive:docproperties.previewparameters"/>
+  <property name="kdenlive:docproperties.profile">C:/Users/thale/AppData/Roaming/kdenlive/profiles/customprofile0</property>
+  <property name="kdenlive:docproperties.proxyextension"/>
+  <property name="kdenlive:docproperties.proxyimageminsize">2000</property>
+  <property name="kdenlive:docproperties.proxyimagesize">800</property>
+  <property name="kdenlive:docproperties.proxyminsize">1000</property>
+  <property name="kdenlive:docproperties.proxyparams"/>
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+  <property name="kdenlive:docproperties.seekOffset">30000</property>
+  <property name="kdenlive:docproperties.version">0.99</property>
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+  <property name="kdenlive:docproperties.videoTarget">2</property>
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+  <property name="kdenlive:docproperties.zoneout">75</property>
+  <property name="kdenlive:docproperties.zoom">8</property>
+  <property name="kdenlive:documentnotes"/>
+  <property name="kdenlive:docproperties.groups">[
+    {
+        "children": [
+            {
+                "data": "2:0",
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+]
+</property>
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+ </playlist>
+ <producer id="black_track" in="00:00:00,000" out="00:16:50,700">
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+  <property name="eof">continue</property>
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+ <playlist id="playlist1">
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+ </playlist>
+ <tractor id="tractor0" in="00:00:00,000">
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+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="video" producer="playlist0"/>
+  <track hide="video" producer="playlist1"/>
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+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
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+  <filter id="filter1">
+   <property name="channel">-1</property>
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+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
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+  </filter>
+  <filter id="filter2">
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+   <property name="mlt_service">audiolevel</property>
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+  <property name="aspect_ratio">1</property>
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+  <property name="meta.media.0.stream.type">video</property>
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+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
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+  <property name="meta.media.0.codec.height">1080</property>
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+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
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+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
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+  <property name="meta.media.1.stream.type">audio</property>
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+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:21.989000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
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+  <property name="mute_on_pause">1</property>
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+  <property name="warp_resource">C:/Users/thale/Videos/2020-08-01 17-10-55.mkv</property>
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+  <property name="meta.media.colorspace">601</property>
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+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+ </producer>
+ <playlist id="playlist2">
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+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="video" producer="playlist2"/>
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+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
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+  <filter id="filter4">
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+  <property name="meta.media.0.stream.frame_rate">30</property>
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+  <property name="meta.media.0.codec.width">1920</property>
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+  <property name="meta.media.0.codec.frame_rate">30</property>
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+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
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+  <property name="meta.media.1.codec.sample_rate">44100</property>
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+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
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diff --git a/docs/pt/videos/heatmap.mp4 b/docs/pt/videos/heatmap.mp4
new file mode 100644
index 0000000..b5dd309
--- /dev/null
+++ b/docs/pt/videos/heatmap.mp4
diff --git a/docs/pt/videos/heatmap.webm b/docs/pt/videos/heatmap.webm
new file mode 100644
index 0000000..e3186c5
--- /dev/null
+++ b/docs/pt/videos/heatmap.webm
diff --git a/docs/pt/videos/move.kdenlive b/docs/pt/videos/move.kdenlive
new file mode 100644
index 0000000..a96c25b
--- /dev/null
+++ b/docs/pt/videos/move.kdenlive
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+  <track hide="audio" producer="playlist4"/>
+  <track hide="audio" producer="playlist5"/>
+ </tractor>
+ <playlist id="playlist6"/>
+ <playlist id="playlist7"/>
+ <tractor id="tractor3" in="00:00:00,000">
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="audio" producer="playlist6"/>
+  <track hide="audio" producer="playlist7"/>
+ </tractor>
+ <tractor id="tractor4" global_feed="1" in="00:00:00,000" out="00:16:57,100">
+  <track producer="black_track"/>
+  <track producer="tractor0"/>
+  <track producer="tractor1"/>
+  <track producer="tractor2"/>
+  <track producer="tractor3"/>
+  <transition id="transition0">
+   <property name="a_track">0</property>
+   <property name="b_track">1</property>
+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
+  <transition id="transition1">
+   <property name="a_track">0</property>
+   <property name="b_track">2</property>
+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
+  <transition id="transition2">
+   <property name="a_track">0</property>
+   <property name="b_track">3</property>
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+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
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+   <property name="always_active">1</property>
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+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+  </transition>
+  <filter id="filter7">
+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter8">
+   <property name="channel">-1</property>
+   <property name="mlt_service">panner</property>
+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter9">
+   <property name="iec_scale">0</property>
+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
+  </filter>
+ </tractor>
+</mlt>
diff --git a/docs/pt/videos/move.mp4 b/docs/pt/videos/move.mp4
new file mode 100644
index 0000000..f2d95c9
--- /dev/null
+++ b/docs/pt/videos/move.mp4
diff --git a/docs/pt/videos/move.webm b/docs/pt/videos/move.webm
new file mode 100644
index 0000000..871578a
--- /dev/null
+++ b/docs/pt/videos/move.webm
diff --git a/docs/pt/videos/rotate.kdenlive b/docs/pt/videos/rotate.kdenlive
new file mode 100644
index 0000000..f69d2f4
--- /dev/null
+++ b/docs/pt/videos/rotate.kdenlive
@@ -0,0 +1,443 @@
+<?xml version='1.0' encoding='utf-8'?>
+<mlt LC_NUMERIC="pt_BR" producer="main_bin" version="6.21.0" root="C:/Users/thale/Documents/GitHub/PSP/docusaurus/static/videos">
+ <profile frame_rate_num="30" sample_aspect_num="1" display_aspect_den="1080" colorspace="601" progressive="1" description="1920x1080 30.00fps" display_aspect_num="1920" frame_rate_den="1" width="1920" height="1080" sample_aspect_den="1"/>
+ <producer id="producer0" in="00:00:00.000" out="00:00:22.000">
+  <property name="length">688</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-07-17 15-40-29.mkv</property>
+  <property name="audio_index">1</property>
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+  <property name="mlt_service">avformat-novalidate</property>
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+  <property name="kdenlive:folderid">-1</property>
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+  <property name="kdenlive:file_size">7174637</property>
+  <property name="kdenlive:file_hash">e7265dd54408e9012b865d66707237c6</property>
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+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
+  <property name="meta.media.0.codec.width">1920</property>
+  <property name="meta.media.0.codec.height">1080</property>
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+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
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+  <property name="meta.attr.0.stream.DURATION.markup">00:00:22.933000000</property>
+  <property name="meta.media.1.stream.type">audio</property>
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+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:22.848000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="meta.media.frame_rate_num">30</property>
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+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+ </producer>
+ <producer id="producer1" in="00:00:00.000" out="00:00:13.000">
+  <property name="length">416</property>
+  <property name="eof">pause</property>
+  <property name="resource">0x000000ff</property>
+  <property name="aspect_ratio">1</property>
+  <property name="mlt_service">color</property>
+  <property name="kdenlive:clipname">Clipe de Cor</property>
+  <property name="kdenlive:duration">00:00:05,000</property>
+  <property name="kdenlive:folderid">-1</property>
+  <property name="kdenlive:id">3</property>
+  <property name="kdenlive:file_hash">9196813120dc5c505072516b15baf414</property>
+ </producer>
+ <playlist id="main_bin">
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+  <property name="kdenlive:docproperties.enableproxy">0</property>
+  <property name="kdenlive:docproperties.externalproxyparams">../Sub;;S03.MP4;../Clip;;.MXF</property>
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+  <property name="kdenlive:docproperties.generateproxy">0</property>
+  <property name="kdenlive:docproperties.groups">[
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+   <property name="disable">1</property>
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+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="meta.media.frame_rate_num">30</property>
+  <property name="meta.media.frame_rate_den">1</property>
+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+  <property name="set.test_audio">1</property>
+  <property name="set.test_image">0</property>
+ </producer>
+ <playlist id="playlist4">
+  <entry producer="producer3" in="00:00:00.000" out="00:00:14.000">
+   <property name="kdenlive:id">2</property>
+  </entry>
+ </playlist>
+ <playlist id="playlist5"/>
+ <tractor id="tractor2" in="00:00:00.000" out="00:00:14.000">
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="audio" producer="playlist4"/>
+  <track hide="audio" producer="playlist5"/>
+ </tractor>
+ <playlist id="playlist6">
+  <entry producer="producer1" in="00:00:00.000" out="00:00:13.000">
+   <property name="kdenlive:id">3</property>
+   <filter id="filter6">
+    <property name="rotate_center">1</property>
+    <property name="mlt_service">qtblend</property>
+    <property name="kdenlive_id">qtblend</property>
+    <property name="rect">00:00:00.000=0 1027 1920 1080 1</property>
+    <property name="rotation">00:00:00.000=0</property>
+    <property name="compositing">0</property>
+    <property name="distort">0</property>
+    <property name="kdenlive:collapsed">0</property>
+   </filter>
+  </entry>
+ </playlist>
+ <playlist id="playlist7"/>
+ <tractor id="tractor3" in="00:00:00.000" out="00:00:13.000">
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="audio" producer="playlist6"/>
+  <track hide="audio" producer="playlist7"/>
+ </tractor>
+ <tractor id="tractor4" global_feed="1" in="00:00:00.000" out="00:16:54.033">
+  <track producer="black_track"/>
+  <track producer="tractor0"/>
+  <track producer="tractor1"/>
+  <track producer="tractor2"/>
+  <track producer="tractor3"/>
+  <transition id="transition0">
+   <property name="a_track">0</property>
+   <property name="b_track">1</property>
+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
+  <transition id="transition1">
+   <property name="a_track">0</property>
+   <property name="b_track">2</property>
+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
+  <transition id="transition2">
+   <property name="a_track">0</property>
+   <property name="b_track">3</property>
+   <property name="compositing">0</property>
+   <property name="distort">0</property>
+   <property name="rotate_center">0</property>
+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+  </transition>
+  <transition id="transition3">
+   <property name="a_track">0</property>
+   <property name="b_track">4</property>
+   <property name="compositing">0</property>
+   <property name="distort">0</property>
+   <property name="rotate_center">0</property>
+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+  </transition>
+  <filter id="filter7">
+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter8">
+   <property name="channel">-1</property>
+   <property name="mlt_service">panner</property>
+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter9">
+   <property name="iec_scale">0</property>
+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
+  </filter>
+ </tractor>
+</mlt>
diff --git a/docs/pt/videos/rotate.mp4 b/docs/pt/videos/rotate.mp4
new file mode 100644
index 0000000..5c67343
--- /dev/null
+++ b/docs/pt/videos/rotate.mp4
diff --git a/docs/pt/videos/rotate.webm b/docs/pt/videos/rotate.webm
new file mode 100644
index 0000000..fb94f7e
--- /dev/null
+++ b/docs/pt/videos/rotate.webm
diff --git a/docs/pt/videos/select.kdenlive b/docs/pt/videos/select.kdenlive
new file mode 100644
index 0000000..db5f352
--- /dev/null
+++ b/docs/pt/videos/select.kdenlive
@@ -0,0 +1,416 @@
+<?xml version='1.0' encoding='utf-8'?>
+<mlt LC_NUMERIC="pt_BR" producer="main_bin" version="6.21.0" root="C:/Users/thale/Documents/GitHub/PSP/docusaurus/static/videos">
+ <profile frame_rate_num="30" sample_aspect_num="1" display_aspect_den="1080" colorspace="601" progressive="1" description="1920x1080 30.00fps" display_aspect_num="1920" frame_rate_den="1" width="1920" height="1080" sample_aspect_den="1"/>
+ <producer id="producer0" in="00:00:00.000" out="00:00:17.000">
+  <property name="length">520</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-07-17 15-23-56.mkv</property>
+  <property name="audio_index">1</property>
+  <property name="video_index">0</property>
+  <property name="mute_on_pause">0</property>
+  <property name="mlt_service">avformat-novalidate</property>
+  <property name="meta.media.nb_streams">2</property>
+  <property name="meta.media.0.stream.type">video</property>
+  <property name="meta.media.0.stream.frame_rate">30</property>
+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
+  <property name="meta.media.0.codec.width">1920</property>
+  <property name="meta.media.0.codec.height">1080</property>
+  <property name="meta.media.0.codec.rotate">0</property>
+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
+  <property name="meta.media.0.codec.bit_rate">0</property>
+  <property name="meta.attr.0.stream.DURATION.markup">00:00:17.333000000</property>
+  <property name="meta.media.1.stream.type">audio</property>
+  <property name="meta.media.1.codec.sample_fmt">fltp</property>
+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:17.299000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="seekable">1</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="aspect_ratio">1</property>
+  <property name="kdenlive:clipname"/>
+  <property name="meta.media.frame_rate_num">30</property>
+  <property name="meta.media.frame_rate_den">1</property>
+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+  <property name="kdenlive:folderid">-1</property>
+  <property name="kdenlive:id">2</property>
+  <property name="kdenlive:file_size">5430759</property>
+  <property name="kdenlive:file_hash">13b4c117affe7445bb51baa73623f7ab</property>
+  <property name="global_feed">1</property>
+ </producer>
+ <playlist id="main_bin">
+  <property name="kdenlive:docproperties.activeTrack">2</property>
+  <property name="kdenlive:docproperties.audioTarget">-1</property>
+  <property name="kdenlive:docproperties.decimalPoint">.</property>
+  <property name="kdenlive:docproperties.disablepreview">0</property>
+  <property name="kdenlive:docproperties.documentid">1595009192967</property>
+  <property name="kdenlive:docproperties.enableTimelineZone">0</property>
+  <property name="kdenlive:docproperties.enableexternalproxy">0</property>
+  <property name="kdenlive:docproperties.enableproxy">0</property>
+  <property name="kdenlive:docproperties.externalproxyparams">../Sub;;S03.MP4;../Clip;;.MXF</property>
+  <property name="kdenlive:docproperties.generateimageproxy">0</property>
+  <property name="kdenlive:docproperties.generateproxy">0</property>
+  <property name="kdenlive:docproperties.kdenliveversion">20.04.1</property>
+  <property name="kdenlive:docproperties.position">420</property>
+  <property name="kdenlive:docproperties.previewextension"/>
+  <property name="kdenlive:docproperties.previewparameters"/>
+  <property name="kdenlive:docproperties.profile">C:/Users/thale/AppData/Roaming/kdenlive/profiles/customprofile0</property>
+  <property name="kdenlive:docproperties.proxyextension"/>
+  <property name="kdenlive:docproperties.proxyimageminsize">2000</property>
+  <property name="kdenlive:docproperties.proxyimagesize">800</property>
+  <property name="kdenlive:docproperties.proxyminsize">1000</property>
+  <property name="kdenlive:docproperties.proxyparams"/>
+  <property name="kdenlive:docproperties.scrollPos">39</property>
+  <property name="kdenlive:docproperties.seekOffset">30000</property>
+  <property name="kdenlive:docproperties.version">0.99</property>
+  <property name="kdenlive:docproperties.verticalzoom">1</property>
+  <property name="kdenlive:docproperties.videoTarget">-1</property>
+  <property name="kdenlive:docproperties.zonein">0</property>
+  <property name="kdenlive:docproperties.zoneout">75</property>
+  <property name="kdenlive:docproperties.zoom">5</property>
+  <property name="kdenlive:documentnotes"/>
+  <property name="kdenlive:docproperties.groups">[
+    {
+        "children": [
+            {
+                "data": "2:0",
+                "leaf": "clip",
+                "type": "Leaf"
+            },
+            {
+                "data": "1:0",
+                "leaf": "clip",
+                "type": "Leaf"
+            }
+        ],
+        "type": "AVSplit"
+    }
+]
+</property>
+  <property name="kdenlive:docproperties.renderaudioquality">6</property>
+  <property name="kdenlive:docproperties.rendercategory">Generic (HD for web, mobile devices...)</property>
+  <property name="kdenlive:docproperties.renderendguide">-1</property>
+  <property name="kdenlive:docproperties.renderexportaudio">0</property>
+  <property name="kdenlive:docproperties.renderfield">0</property>
+  <property name="kdenlive:docproperties.renderguide">0</property>
+  <property name="kdenlive:docproperties.renderplay">0</property>
+  <property name="kdenlive:docproperties.renderprofile">WebM - the widespread free format (VP8/Vorbis)</property>
+  <property name="kdenlive:docproperties.renderquality">23</property>
+  <property name="kdenlive:docproperties.renderratio">1</property>
+  <property name="kdenlive:docproperties.renderrescale">0</property>
+  <property name="kdenlive:docproperties.renderrescaleheight">540</property>
+  <property name="kdenlive:docproperties.renderrescalewidth">960</property>
+  <property name="kdenlive:docproperties.renderscanning">0</property>
+  <property name="kdenlive:docproperties.renderspeed">2</property>
+  <property name="kdenlive:docproperties.renderstartguide">-1</property>
+  <property name="kdenlive:docproperties.rendertcoverlay">0</property>
+  <property name="kdenlive:docproperties.rendertctype">0</property>
+  <property name="kdenlive:docproperties.rendertwopass">0</property>
+  <property name="kdenlive:docproperties.renderurl">select.webm</property>
+  <property name="kdenlive:docproperties.renderzone">0</property>
+  <property name="xml_retain">1</property>
+  <entry producer="producer0" in="00:00:00.000" out="00:00:17.000"/>
+ </playlist>
+ <producer id="black_track" in="00:00:00.000" out="00:16:54.033">
+  <property name="length">2147483647</property>
+  <property name="eof">continue</property>
+  <property name="resource">black</property>
+  <property name="aspect_ratio">1</property>
+  <property name="mlt_service">color</property>
+  <property name="mlt_image_format">rgb24a</property>
+  <property name="set.test_audio">0</property>
+ </producer>
+ <playlist id="playlist0">
+  <property name="kdenlive:audio_track">1</property>
+  <filter id="filter0">
+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter1">
+   <property name="channel">-1</property>
+   <property name="mlt_service">panner</property>
+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
+   <property name="disable">1</property>
+  </filter>
+ </playlist>
+ <playlist id="playlist1">
+  <property name="kdenlive:audio_track">1</property>
+ </playlist>
+ <tractor id="tractor0" in="00:00:00.000">
+  <property name="kdenlive:audio_track">1</property>
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="video" producer="playlist0"/>
+  <track hide="video" producer="playlist1"/>
+  <filter id="filter2">
+   <property name="iec_scale">0</property>
+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
+  </filter>
+ </tractor>
+ <producer id="producer1" in="00:00:00.000" out="00:00:17.000">
+  <property name="length">520</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-07-17 15-23-56.mkv</property>
+  <property name="audio_index">1</property>
+  <property name="video_index">0</property>
+  <property name="mute_on_pause">0</property>
+  <property name="mlt_service">avformat-novalidate</property>
+  <property name="seekable">1</property>
+  <property name="aspect_ratio">1</property>
+  <property name="kdenlive:clipname"/>
+  <property name="kdenlive:folderid">-1</property>
+  <property name="kdenlive:id">2</property>
+  <property name="kdenlive:file_size">5430759</property>
+  <property name="kdenlive:file_hash">13b4c117affe7445bb51baa73623f7ab</property>
+  <property name="global_feed">1</property>
+  <property name="xml">was here</property>
+  <property name="meta.media.nb_streams">2</property>
+  <property name="meta.media.0.stream.type">video</property>
+  <property name="meta.media.0.stream.frame_rate">30</property>
+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
+  <property name="meta.media.0.codec.width">1920</property>
+  <property name="meta.media.0.codec.height">1080</property>
+  <property name="meta.media.0.codec.rotate">0</property>
+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
+  <property name="meta.media.0.codec.bit_rate">0</property>
+  <property name="meta.attr.0.stream.DURATION.markup">00:00:17.333000000</property>
+  <property name="meta.media.1.stream.type">audio</property>
+  <property name="meta.media.1.codec.sample_fmt">fltp</property>
+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:17.299000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="meta.media.frame_rate_num">30</property>
+  <property name="meta.media.frame_rate_den">1</property>
+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+  <property name="set.test_audio">0</property>
+  <property name="set.test_image">1</property>
+ </producer>
+ <playlist id="playlist2">
+  <property name="kdenlive:audio_track">1</property>
+  <entry producer="producer1" in="00:00:01.000" out="00:00:15.000">
+   <property name="kdenlive:id">2</property>
+  </entry>
+  <filter id="filter3">
+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter4">
+   <property name="channel">-1</property>
+   <property name="mlt_service">panner</property>
+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
+   <property name="disable">1</property>
+  </filter>
+ </playlist>
+ <playlist id="playlist3">
+  <property name="kdenlive:audio_track">1</property>
+ </playlist>
+ <tractor id="tractor1" in="00:00:00.000" out="00:00:14.000">
+  <property name="kdenlive:audio_track">1</property>
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="video" producer="playlist2"/>
+  <track hide="video" producer="playlist3"/>
+  <filter id="filter5">
+   <property name="iec_scale">0</property>
+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
+  </filter>
+ </tractor>
+ <producer id="producer2" in="00:00:00.000" out="00:00:17.000">
+  <property name="length">520</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-07-17 15-23-56.mkv</property>
+  <property name="audio_index">1</property>
+  <property name="video_index">0</property>
+  <property name="mute_on_pause">0</property>
+  <property name="mlt_service">avformat-novalidate</property>
+  <property name="seekable">1</property>
+  <property name="aspect_ratio">1</property>
+  <property name="kdenlive:clipname"/>
+  <property name="kdenlive:folderid">-1</property>
+  <property name="kdenlive:id">2</property>
+  <property name="kdenlive:file_size">5430759</property>
+  <property name="kdenlive:file_hash">13b4c117affe7445bb51baa73623f7ab</property>
+  <property name="global_feed">1</property>
+  <property name="xml">was here</property>
+  <property name="meta.media.nb_streams">2</property>
+  <property name="meta.media.0.stream.type">video</property>
+  <property name="meta.media.0.stream.frame_rate">30</property>
+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
+  <property name="meta.media.0.codec.width">1920</property>
+  <property name="meta.media.0.codec.height">1080</property>
+  <property name="meta.media.0.codec.rotate">0</property>
+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
+  <property name="meta.media.0.codec.bit_rate">0</property>
+  <property name="meta.attr.0.stream.DURATION.markup">00:00:17.333000000</property>
+  <property name="meta.media.1.stream.type">audio</property>
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+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:00:17.299000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="meta.media.frame_rate_num">30</property>
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+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+  <property name="set.test_audio">1</property>
+  <property name="set.test_image">0</property>
+ </producer>
+ <playlist id="playlist4">
+  <entry producer="producer2" in="00:00:01.000" out="00:00:15.000">
+   <property name="kdenlive:id">2</property>
+  </entry>
+ </playlist>
+ <playlist id="playlist5"/>
+ <tractor id="tractor2" in="00:00:00.000" out="00:00:14.000">
+  <property name="kdenlive:trackheight">67</property>
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+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="audio" producer="playlist4"/>
+  <track hide="audio" producer="playlist5"/>
+ </tractor>
+ <playlist id="playlist6"/>
+ <playlist id="playlist7"/>
+ <tractor id="tractor3" in="00:00:00.000">
+  <property name="kdenlive:trackheight">67</property>
+  <property name="kdenlive:collapsed">0</property>
+  <property name="kdenlive:thumbs_format"/>
+  <property name="kdenlive:audio_rec"/>
+  <property name="kdenlive:timeline_active">1</property>
+  <track hide="audio" producer="playlist6"/>
+  <track hide="audio" producer="playlist7"/>
+ </tractor>
+ <tractor id="tractor4" global_feed="1" in="00:00:00.000" out="00:16:54.033">
+  <track producer="black_track"/>
+  <track producer="tractor0"/>
+  <track producer="tractor1"/>
+  <track producer="tractor2"/>
+  <track producer="tractor3"/>
+  <transition id="transition0">
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+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
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+   <property name="a_track">0</property>
+   <property name="b_track">2</property>
+   <property name="mlt_service">mix</property>
+   <property name="kdenlive_id">mix</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+   <property name="sum">1</property>
+  </transition>
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+   <property name="a_track">0</property>
+   <property name="b_track">3</property>
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+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+  </transition>
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+   <property name="b_track">4</property>
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+   <property name="rotate_center">0</property>
+   <property name="mlt_service">qtblend</property>
+   <property name="kdenlive_id">qtblend</property>
+   <property name="internal_added">237</property>
+   <property name="always_active">1</property>
+  </transition>
+  <filter id="filter6">
+   <property name="window">75</property>
+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
+   <property name="internal_added">237</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter7">
+   <property name="channel">-1</property>
+   <property name="mlt_service">panner</property>
+   <property name="internal_added">237</property>
+   <property name="start">0,5</property>
+   <property name="disable">1</property>
+  </filter>
+  <filter id="filter8">
+   <property name="iec_scale">0</property>
+   <property name="mlt_service">audiolevel</property>
+   <property name="disable">1</property>
+  </filter>
+ </tractor>
+</mlt>
diff --git a/docs/pt/videos/select.mp4 b/docs/pt/videos/select.mp4
new file mode 100644
index 0000000..c53e049
--- /dev/null
+++ b/docs/pt/videos/select.mp4
diff --git a/docs/pt/videos/select.webm b/docs/pt/videos/select.webm
new file mode 100644
index 0000000..0aa1e98
--- /dev/null
+++ b/docs/pt/videos/select.webm
diff --git a/docs/pt/videos/timelapseBuild.kdenlive b/docs/pt/videos/timelapseBuild.kdenlive
new file mode 100644
index 0000000..1383738
--- /dev/null
+++ b/docs/pt/videos/timelapseBuild.kdenlive
@@ -0,0 +1,406 @@
+<?xml version='1.0' encoding='utf-8'?>
+<mlt LC_NUMERIC="pt_BR" producer="main_bin" version="6.21.0" root="C:/Users/thale/Documents/GitHub/PSP/docusaurus/static/videos">
+ <profile frame_rate_num="30" sample_aspect_num="1" display_aspect_den="1080" colorspace="601" progressive="1" description="1920x1080 30.00fps" display_aspect_num="1920" frame_rate_den="1" width="1920" height="1080" sample_aspect_den="1"/>
+ <producer id="producer0" in="00:00:00,000" out="00:03:32,500">
+  <property name="length">6376</property>
+  <property name="eof">pause</property>
+  <property name="resource">C:/Users/thale/Videos/2020-07-18 18-35-35.mkv</property>
+  <property name="meta.media.nb_streams">2</property>
+  <property name="meta.media.0.stream.type">video</property>
+  <property name="meta.media.0.stream.frame_rate">30</property>
+  <property name="meta.media.0.stream.sample_aspect_ratio">0</property>
+  <property name="meta.media.0.codec.width">1920</property>
+  <property name="meta.media.0.codec.height">1080</property>
+  <property name="meta.media.0.codec.rotate">0</property>
+  <property name="meta.media.0.codec.frame_rate">30</property>
+  <property name="meta.media.0.codec.pix_fmt">yuv420p</property>
+  <property name="meta.media.0.codec.sample_aspect_ratio">1</property>
+  <property name="meta.media.0.codec.colorspace">601</property>
+  <property name="meta.media.0.codec.name">h264</property>
+  <property name="meta.media.0.codec.long_name">H.264 / AVC / MPEG-4 AVC / MPEG-4 part 10</property>
+  <property name="meta.media.0.codec.bit_rate">0</property>
+  <property name="meta.attr.0.stream.DURATION.markup">00:03:32.533000000</property>
+  <property name="meta.media.1.stream.type">audio</property>
+  <property name="meta.media.1.codec.sample_fmt">fltp</property>
+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
+  <property name="meta.media.1.codec.name">aac</property>
+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
+  <property name="meta.attr.1.stream.DURATION.markup">00:03:32.486000000</property>
+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="seekable">1</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
+  <property name="aspect_ratio">1</property>
+  <property name="audio_index">1</property>
+  <property name="video_index">0</property>
+  <property name="mute_on_pause">0</property>
+  <property name="mlt_service">avformat-novalidate</property>
+  <property name="kdenlive:clipname"/>
+  <property name="meta.media.frame_rate_num">30</property>
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+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1920</property>
+  <property name="meta.media.height">1080</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
+  <property name="meta.media.progressive">1</property>
+  <property name="kdenlive:folderid">-1</property>
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+  <property name="kdenlive:file_hash">2382d1f43ecb21dc1cb5db6244d9eea3</property>
+  <property name="global_feed">1</property>
+ </producer>
+ <playlist id="main_bin">
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+  <property name="kdenlive:docproperties.audioTarget">1</property>
+  <property name="kdenlive:docproperties.decimalPoint">,</property>
+  <property name="kdenlive:docproperties.disablepreview">0</property>
+  <property name="kdenlive:docproperties.documentid">1595107997711</property>
+  <property name="kdenlive:docproperties.enableTimelineZone">0</property>
+  <property name="kdenlive:docproperties.enableexternalproxy">0</property>
+  <property name="kdenlive:docproperties.enableproxy">0</property>
+  <property name="kdenlive:docproperties.externalproxyparams">../Sub;;S03.MP4;../Clip;;.MXF</property>
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+  <property name="kdenlive:docproperties.generateproxy">0</property>
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+  <property name="kdenlive:docproperties.position">1058</property>
+  <property name="kdenlive:docproperties.previewextension"/>
+  <property name="kdenlive:docproperties.previewparameters"/>
+  <property name="kdenlive:docproperties.profile">C:/Users/thale/AppData/Roaming/kdenlive/profiles/customprofile0</property>
+  <property name="kdenlive:docproperties.proxyextension"/>
+  <property name="kdenlive:docproperties.proxyimageminsize">2000</property>
+  <property name="kdenlive:docproperties.proxyimagesize">800</property>
+  <property name="kdenlive:docproperties.proxyminsize">1000</property>
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+  <property name="kdenlive:documentnotes"/>
+  <property name="kdenlive:docproperties.groups">[
+    {
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+</property>
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+  <property name="kdenlive:thumbs_format"/>
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+  <track hide="video" producer="playlist0"/>
+  <track hide="video" producer="playlist1"/>
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+   <property name="max_gain">20dB</property>
+   <property name="mlt_service">volume</property>
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+  <property name="meta.media.0.codec.colorspace">601</property>
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+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
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+  <property name="warp_resource">C:/Users/thale/Videos/2020-07-18 18-35-35.mkv</property>
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+  <property name="meta.media.width">1920</property>
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+  <property name="meta.media.color_range">mpeg</property>
+ </producer>
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+ </playlist>
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+  <track producer="black_track"/>
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+  </filter>
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+ </tractor>
+</mlt>
diff --git a/docs/pt/videos/timelapseBuild.mp4 b/docs/pt/videos/timelapseBuild.mp4
new file mode 100644
index 0000000..0c83e56
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diff --git a/docs/pt/videos/timelapseBuild.webm b/docs/pt/videos/timelapseBuild.webm
new file mode 100644
index 0000000..ef07f3d
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diff --git a/docs/pt/videos/timelapseControlEditor.mp4 b/docs/pt/videos/timelapseControlEditor.mp4
new file mode 100644
index 0000000..14dd64b
--- /dev/null
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diff --git a/docs/pt/videos/timelapseControlEditor.webm b/docs/pt/videos/timelapseControlEditor.webm
new file mode 100644
index 0000000..7d9853a
--- /dev/null
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diff --git a/docs/pt/videos/zoom.kdenlive b/docs/pt/videos/zoom.kdenlive
new file mode 100644
index 0000000..b1e2a0e
--- /dev/null
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+<?xml version='1.0' encoding='utf-8'?>
+<mlt LC_NUMERIC="pt_BR" producer="main_bin" version="6.21.0" root="C:/Users/thale/Documents/GitHub/PSP/docusaurus/static/videos">
+ <profile frame_rate_num="25" sample_aspect_num="1" display_aspect_den="9" colorspace="709" progressive="1" description="HD 1080p 25 fps" display_aspect_num="16" frame_rate_den="1" width="1920" height="1080" sample_aspect_den="1"/>
+ <producer id="producer0" in="00:00:00,000" out="00:00:11,520">
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+  <property name="meta.media.1.codec.sample_rate">44100</property>
+  <property name="meta.media.1.codec.channels">2</property>
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+  <property name="meta.media.1.codec.long_name">AAC (Advanced Audio Coding)</property>
+  <property name="meta.media.1.codec.bit_rate">0</property>
+  <property name="meta.attr.1.stream.title.markup">simple_aac</property>
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+  <property name="meta.attr.ENCODER.markup">Lavf58.29.100</property>
+  <property name="seekable">1</property>
+  <property name="meta.media.sample_aspect_num">1</property>
+  <property name="meta.media.sample_aspect_den">1</property>
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+  <property name="kdenlive:clipname"/>
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+  <property name="meta.media.colorspace">601</property>
+  <property name="meta.media.color_trc">2</property>
+  <property name="meta.media.width">1280</property>
+  <property name="meta.media.height">720</property>
+  <property name="meta.media.color_range">mpeg</property>
+  <property name="meta.media.top_field_first">0</property>
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+  <property name="kdenlive:folderid">-1</property>
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+  <property name="kdenlive:file_hash">7064adfdf64bab829e988d3aa2c1cb8d</property>
+  <property name="global_feed">1</property>
+ </producer>
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+  <property name="mlt_service">kdenlivetitle</property>
+  <property name="kdenlive:duration">125</property>
+  <property name="kdenlive:clipname">Scroll cima ...</property>
+  <property name="xmldata">&lt;kdenlivetitle duration="125" LC_NUMERIC="Portuguese_Brazil.1252" width="1920" height="1080" out="124">&#xd;&#xd;
+ &lt;item type="QGraphicsTextItem" z-index="0">&#xd;&#xd;
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+  <property name="kdenlive:clipname">Scroll baixo</property>
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diff --git a/docs/styles.1ab9c66d.css b/docs/styles.1ab9c66d.css
deleted file mode 100644
index 30bd8d9..0000000
--- a/docs/styles.1ab9c66d.css
+++ /dev/null
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diff --git a/docs/styles.f1364c44.js b/docs/styles.f1364c44.js
deleted file mode 100644
index e2c3470..0000000
--- a/docs/styles.f1364c44.js
+++ /dev/null
@@ -1 +0,0 @@
-(window.webpackJsonp=window.webpackJsonp||[]).push([[0],{113:function(e,o,n){e.exports={heroBanner:"heroBanner_2Ftp",buttons:"buttons_1Wc3",features:"features_P2SU",featureImage:"featureImage_3Xqx"}},122:function(e,o,n){},15:function(e,o,n){},39:function(e,o,n){},43:function(e,o,n){},44:function(e,o,n){},47:function(e,o,n){e.exports={tabItem:"tabItem_1w39"}},48:function(e,o,n){e.exports={codeBlockContent:"codeBlockContent_1u-d",codeBlockTitle:"codeBlockTitle_3nn1",codeBlock:"codeBlock_3iAC",codeBlockWithTitle:"codeBlockWithTitle_3QsD",copyButton:"copyButton_10dd",codeBlockLines:"codeBlockLines_b7E3"}},49:function(e,o,n){},50:function(e,o,n){e.exports={enhancedAnchor:"enhancedAnchor_2cZh"}},51:function(e,o,n){e.exports={mdxCodeBlock:"mdxCodeBlock_1XEh"}},52:function(e,o,n){},53:function(e,o,n){e.exports={announcementBar:"announcementBar_1l0Z",announcementBarClose:"announcementBarClose_21wD",announcementBarContent:"announcementBarContent_1xni"}},54:function(e,o,n){e.exports={toggle:"toggle_BsTx"}},55:function(e,o,n){e.exports={displayOnlyInLargeViewport:"displayOnlyInLargeViewport_2aTZ",hideLogoText:"hideLogoText_GP-Q",navbarHideable:"navbarHideable_3046",navbarHidden:"navbarHidden_1Uo_"}},56:function(e,o,n){e.exports={footerLogoLink:"footerLogoLink_1zJy"}},57:function(e,o,n){},62:function(e,o,n){e.exports={searchQueryInput:"searchQueryInput_2wzZ",searchVersionInput:"searchVersionInput_onPS",searchResultsColumn:"searchResultsColumn_gCu4",searchLogoColumn:"searchLogoColumn_3CX6",algoliaLogo:"algoliaLogo_2w8o",algoliaLogoPathFill:"algoliaLogoPathFill_2lQN",searchResultItem:"searchResultItem_2Ox1",searchResultItemHeading:"searchResultItemHeading_E89F",searchResultItemPath:"searchResultItemPath_W0N5",searchResultItemSummary:"searchResultItemSummary_2EZC",searchQueryColumn:"searchQueryColumn_Sxqm",searchVersionColumn:"searchVersionColumn_2cau",loadingSpinner:"loadingSpinner_My1l",loadingspin:"loadingspin_XSpe",loader:"loader_15BX"}},65:function(e,o,n){e.exports={docTitle:"docTitle_1Lrw",docItemContainer:"docItemContainer_3QWW",docItemCol:"docItemCol_2ASc",docItemWrapper:"docItemWrapper_1EkI",tableOfContents:"tableOfContents_3klQ",docLastUpdatedAt:"docLastUpdatedAt_217_"}},66:function(e,o,n){e.exports={sidebar:"sidebar_2urC",sidebarWithHideableNavbar:"sidebarWithHideableNavbar_2IoJ",sidebarLogo:"sidebarLogo_Snse",menu:"menu_5FrY",menuLinkText:"menuLinkText_2zSB",menuWithAnnouncementBar:"menuWithAnnouncementBar_18ic",sidebarMenuIcon:"sidebarMenuIcon_Dm3K",sidebarMenuCloseIcon:"sidebarMenuCloseIcon_hir9"}},67:function(e,o,n){e.exports={docPage:"docPage_2gpo",docSidebarContainer:"docSidebarContainer_3_JD",docMainContainer:"docMainContainer_3EyW"}},85:function(e,o,n){e.exports={blogPostTitle:"blogPostTitle_1mse",blogPostDate:"blogPostDate_3bQP"}}}]);
\ No newline at end of file
diff --git a/docusaurus/blog/2019-10-13-2019w42a-beta.md b/docusaurus/blog/2019-10-13-2019w42a-beta.md
index 6fd385b..472d5c6 100644
--- a/docusaurus/blog/2019-10-13-2019w42a-beta.md
+++ b/docusaurus/blog/2019-10-13-2019w42a-beta.md
@@ -1,5 +1,4 @@
 ---
-id: 2019w42a-beta
 title: 2019w42a-beta
 author: Thales
 author_title: PhD
diff --git a/docusaurus/blog/2019-11-01-2019w44a-beta.md b/docusaurus/blog/2019-11-01-2019w44a-beta.md
index 5e38a3d..785872c 100644
--- a/docusaurus/blog/2019-11-01-2019w44a-beta.md
+++ b/docusaurus/blog/2019-11-01-2019w44a-beta.md
@@ -1,5 +1,4 @@
 ---
-id: 2019w44a-beta
 title: 22019w44a-beta
 author: Thales
 author_title: PhD
diff --git a/docusaurus/blog/2020-06-13-2020w24a-beta.md b/docusaurus/blog/2020-06-13-2020w24a-beta.md
index 4511f20..f4f4c12 100644
--- a/docusaurus/blog/2020-06-13-2020w24a-beta.md
+++ b/docusaurus/blog/2020-06-13-2020w24a-beta.md
@@ -1,5 +1,4 @@
 ---
-id: 2020w24a-beta
 title: 2020w24a-beta
 author: Thales
 author_title: PhD
diff --git a/docusaurus/blog/2020-07-09-2020w28a-beta.md b/docusaurus/blog/2020-07-09-2020w28a-beta.md
index 2440de4..d55bca2 100644
--- a/docusaurus/blog/2020-07-09-2020w28a-beta.md
+++ b/docusaurus/blog/2020-07-09-2020w28a-beta.md
@@ -1,5 +1,4 @@
 ---
-id: 2020w28a-beta
 title: 2020w28a-beta
 author: Thales
 author_title: PhD
diff --git a/docusaurus/blog/2020-08-01-2020w31a-beta.md b/docusaurus/blog/2020-08-01-2020w31a-beta.md
index 1f10f66..de93c74 100644
--- a/docusaurus/blog/2020-08-01-2020w31a-beta.md
+++ b/docusaurus/blog/2020-08-01-2020w31a-beta.md
@@ -1,5 +1,4 @@
 ---
-id: 2020w31a-beta
 title: 2020w31a-beta
 author: Thales
 author_title: PhD
diff --git a/docusaurus/docs/aboutPSP.md b/docusaurus/docs/aboutPSP.md
index 36de9a5..34d20ad 100644
--- a/docusaurus/docs/aboutPSP.md
+++ b/docusaurus/docs/aboutPSP.md
@@ -1,40 +1,41 @@
 ---
 id: aboutPSP
+slug: /
 title: PSP-UFU
 sidebar_label: PSP-UFU
 ---
 
 :::important
-This user guide was written in Brazilian Portuguese. If you want to help me translate this guide, contact-me on [GitHub](https://github.com/Thales1330/PSP/issues) or [Twitter](https://twitter.com/PspUfu).
+Most of this user guide was written in Brazilian Portuguese. If you want to help me translate this guide, contact-me on [GitHub](https://github.com/Thales1330/PSP/issues) or [Twitter](https://twitter.com/PspUfu).
 :::
 
-## Sobre o PSP-UFU
+## About PSP-UFU
 
-O PSP-UFU (Plataforma de Sistemas de Potência da Universidade Federal de Uberlândia) é um software **multi-plataforma**, **multilíngue**, **livre e de código aberto (FOSS)** com **recursos avançados de GUI (*Graphical User Interface*)** e **ferramentas CAD (*Computer Aided Design*)** para estudos de sistemas elétricos de potência.
+PSP-UFU (Power Systems Platform of Federal University of Uberlândia) is a **multi-platform**, **multilingual**, **free and open source (FOSS)** software with **advanced GUI (Graphical User Interface) features** and **CAD (Computer Aided Design) tools** for studies of electrical power systems.
 
-O software permite a construção de qualquer rede de transmissão elétrica e sistemas de controle através da inserção de elementos visuais. Para a visualização dos resultados, o programa oferece **elementos de texto vinculados** na tela principal e também **editores de tabelas e gráficos**.
+The software allows the construction of any electrical transmission network and control systems through the deployment of visual elements. For viewing the results, the program offers **linked text elements** on the main screen and also **table and graph editors**.
 
-O PSP-UFU tem como objetivo fornecer ferramentas eficientes de simulação para fins de **pesquisa** e **educação**, além de **aplicações industriais** em sistemas elétricos de potência.
+The PSP-UFU aims to provide efficient simulation tools for **research** and **education** purposes, in addition to **industrial applications** in electrical power systems.
 
-Resumidamente, o *software* pode executar os seguintes estudos:
+Briefly, software can perform the following studies:
 
-- Fluxo de potência
-- Cálculo de curto-circuito
-- Harmônicos
-- Estabilidade transitória e dinâmica
+- Power flow
+- Short circuit calculation
+- Harmonics
+- Transient and dynamic stability
 
-### Publicações
-Os artigos abaixo possuem maiores detalhes a respeito do PSP-UFU:
+### Papers
+The articles below have more details about the PSP-UFU:
 >Oliveira, T. L., Guimarães, G. C., & Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. *International Transactions on Electrical Energy Systems*, e12185. doi: [10.1002/2050-7038.12185](https://doi.org/10.1002/2050-7038.12185)
 
 >Oliveira, T. L., Guimarães, G. C., Silva, L. R., & Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. *The International Journal of Electrical Engineering & Education*, 0020720919879058. doi: [10.1177/0020720919879058](https://doi.org/10.1177/0020720919879058)
 
-A Tese de Doutorado abaixo detalha em profundidade os cálculos envolvidos, arquitetura de software e ferramentas implementadas no PSP-UFU:
+The PhD. Thesis below details in depth the calculations involved, software architecture and tools implemented in the PSP-UFU:
 >Oliveira, T. L. (2019). Desenvolvimento de um programa computacional livre, gráfico, e multiplataforma para analisar sistemas elétricos de potência em regime permanente e dinâmico. doi: [10.14393/ufu.te.2019.2444](https://dx.doi.org/10.14393/ufu.te.2019.2444)
 
-### Contribuindo com o projeto
-Se você está interessado em contribuir com o PSP-UFU de alguma forma (desenvolvimento, críticas, sugestões, etc.), entre em contato pelo [GitHub](https://github.com/Thales1330/PSP/issues) ou [Twitter](https://twitter.com/PspUfu).
+### Contributing to the project
+If you are interested in contributing to PSP-UFU in any way (development, criticism, suggestions, etc.), contact us at [GitHub](https://github.com/Thales1330/PSP/issues) or [Twitter](https://twitter.com/PspUfu).
 
-### Desenvolvedor~~es~~
+### Developer~~s~~
 [Thales Lima Oliveira](https://github.com/Thales1330)
 
diff --git a/docusaurus/docs/installation.md b/docusaurus/docs/installation.md
index 4b937e9..534861c 100644
--- a/docusaurus/docs/installation.md
+++ b/docusaurus/docs/installation.md
@@ -1,24 +1,24 @@
 ---
 id: installation
-title: Instalação
-sidebar_label: Instalação
+title: Installation
+sidebar_label: Installation
 ---
 import useBaseUrl from "@docusaurus/useBaseUrl";
 
 ## Download
 
-O download do PSP-UFU pode ser realizado pelo ["*last release*" do repositório do PSP-UFU no GitHub](https://github.com/Thales1330/PSP/releases/latest).
+The PSP-UFU can be downloaded from the [*last release* of the PSP-UFU repository on GitHub](https://github.com/Thales1330/PSP/releases/latest).
 
-Ao acessar ao link, role a página até encontrar os *Assets*:
+When accessing the link, scroll down the page until you find the * Assets *:
 
 <div><center><img src={useBaseUrl("images/download.png")} alt="Assets download" title="Informações da Figura" /></center></div>
 
-Como o PSP-UFU é um *software* multiplataforma, você deve baixar o arquivo correto:
-- Para o sistema operacional [Windows](installation#windows) baixe os arquivos *.exe* (recomendado) ou *.zip*.
-- Para os sistemas [Linux](installation#linux) baseados no Debian (por exemplo, Ubuntu, Mint) *64 bits* baixe o pacote de instalação *.deb*.
+As the PSP-UFU is a multiplatform software, you must download the correct file:
+- For the operating system [Windows](installation#windows) download the files *.exe* (recommended) or *.zip*.
+- For [Linux](installation#linux) systems based on Debian (for example, Ubuntu, Mint) *64 bits* download the installation package *.deb*.
 
-:::tip Dica
-Se você está interessado em contribuir com o PSP-UFU construindo-o para outros sistemas operacionais (principalmente macOS!) entre em contato pelo [GitHub](https://github.com/Thales1330/PSP/issues) ou [Twitter](https://twitter.com/PspUfu).
+:::tip Tip
+If you are interested in contributing to PSP-UFU by building it for other operating systems (mainly macOS!) Contact us at [GitHub](https://github.com/Thales1330/PSP/issues) or [Twitter](https://twitter.com/PspUfu).
 :::
 
 Na sequência são apresentados o passo-a-passo para instalação dos SOs [Windows](installation#windows) e [Linux](installation#linux). Também são apresentadas as soluções para os erros mais recorrentes.
diff --git a/docusaurus/docusaurus.config.js b/docusaurus/docusaurus.config.js
index 32ecd64..36a4c4f 100644
--- a/docusaurus/docusaurus.config.js
+++ b/docusaurus/docusaurus.config.js
@@ -21,7 +21,7 @@ module.exports = {
           // For more information about styling options visit: https://reactjs.org/docs/dom-elements.html#style
           marginLeft: '2px',
         },
-        lightIcon: '\u2600',
+        lightIcon: '☀',
         lightIconStyle: {
           marginLeft: '1px',
         },
@@ -42,8 +42,12 @@ module.exports = {
         },
         {to: 'blog', label: 'Changelog', position: 'left'},
         {
+          type: 'localeDropdown',
+          position: 'right',
+        },
+        {
           href: 'https://github.com/Thales1330/PSP',
-          label: 'PSP-UFU Source Code',
+          label: 'GitHub',
           position: 'right',
         },
       ],
@@ -101,7 +105,7 @@ module.exports = {
       {
         docs: {
           // It is recommended to set document id as docs home page (`docs/` path).
-          homePageId: 'aboutPSP',
+          // homePageId: 'aboutPSP',
           sidebarPath: require.resolve('./sidebars.js'),
           // Please change this to your repo.
           editUrl:
@@ -119,11 +123,23 @@ module.exports = {
           customCss: require.resolve('./src/css/custom.css'),
         },
         sitemap: {
-            cacheTime: 600 * 1000, // 600 sec - cache purge period
+            //cacheTime: 600 * 1000, // 600 sec - cache purge period
             changefreq: 'weekly',
             priority: 0.5,
 		},
       },
     ],
   ],
+  i18n: {
+    defaultLocale: 'en',
+    locales: ['en', 'pt'],
+    localeConfigs: {
+      en: {
+        label: 'English',
+      },
+      pt: {
+        label: 'Português',
+      },
+    },
+  },
 };
diff --git a/docusaurus/i18n/pt/code.json b/docusaurus/i18n/pt/code.json
new file mode 100644
index 0000000..a5e60e9
--- /dev/null
+++ b/docusaurus/i18n/pt/code.json
@@ -0,0 +1,338 @@
+{
+  "Complete Platform": {
+    "message": "Plataforma Completa"
+  },
+  "theme.NotFound.title": {
+    "message": "Página não encontrada",
+    "description": "The title of the 404 page"
+  },
+  "theme.NotFound.p1": {
+    "message": "Não pudemos encontrar o que você estava procurando.",
+    "description": "The first paragraph of the 404 page"
+  },
+  "theme.NotFound.p2": {
+    "message": "Entre em contato com o proprietário do site que o vinculou ao URL original e informe-o que o link está quebrado.",
+    "description": "The 2nd paragraph of the 404 page"
+  },
+  "theme.AnnouncementBar.closeButtonAriaLabel": {
+    "message": "Fechar",
+    "description": "The ARIA label for close button of announcement bar"
+  },
+  "theme.blog.paginator.navAriaLabel": {
+    "message": "Blog list page navigation",
+    "description": "The ARIA label for the blog pagination"
+  },
+  "theme.blog.paginator.newerEntries": {
+    "message": "Newer Entries",
+    "description": "The label used to navigate to the newer blog posts page (previous page)"
+  },
+  "theme.blog.paginator.olderEntries": {
+    "message": "Older Entries",
+    "description": "The label used to navigate to the older blog posts page (next page)"
+  },
+  "theme.blog.post.readingTime.plurals": {
+    "message": "One min read|{readingTime} min read",
+    "description": "Pluralized label for \"{readingTime} min read\". Use as much plural forms (separated by \"|\") as your language support (see https://www.unicode.org/cldr/cldr-aux/charts/34/supplemental/language_plural_rules.html)"
+  },
+  "theme.tags.tagsListLabel": {
+    "message": "Tags:",
+    "description": "The label alongside a tag list"
+  },
+  "theme.blog.post.readMore": {
+    "message": "Read More",
+    "description": "The label used in blog post item excerpts to link to full blog posts"
+  },
+  "theme.blog.post.paginator.navAriaLabel": {
+    "message": "Blog post page navigation",
+    "description": "The ARIA label for the blog posts pagination"
+  },
+  "theme.blog.post.paginator.newerPost": {
+    "message": "Newer Post",
+    "description": "The blog post button label to navigate to the newer/previous post"
+  },
+  "theme.blog.post.paginator.olderPost": {
+    "message": "Older Post",
+    "description": "The blog post button label to navigate to the older/next post"
+  },
+  "theme.tags.tagsPageTitle": {
+    "message": "Tags",
+    "description": "The title of the tag list page"
+  },
+  "theme.blog.post.plurals": {
+    "message": "One post|{count} posts",
+    "description": "Pluralized label for \"{count} posts\". Use as much plural forms (separated by \"|\") as your language support (see https://www.unicode.org/cldr/cldr-aux/charts/34/supplemental/language_plural_rules.html)"
+  },
+  "theme.blog.tagTitle": {
+    "message": "{nPosts} tagged with \"{tagName}\"",
+    "description": "The title of the page for a blog tag"
+  },
+  "theme.tags.tagsPageLink": {
+    "message": "View All Tags",
+    "description": "The label of the link targeting the tag list page"
+  },
+  "theme.CodeBlock.copyButtonAriaLabel": {
+    "message": "Copy code to clipboard",
+    "description": "The ARIA label for copy code blocks button"
+  },
+  "theme.CodeBlock.copied": {
+    "message": "Copied",
+    "description": "The copied button label on code blocks"
+  },
+  "theme.CodeBlock.copy": {
+    "message": "Copy",
+    "description": "The copy button label on code blocks"
+  },
+  "theme.docs.sidebar.expandButtonTitle": {
+    "message": "Expand sidebar",
+    "description": "The ARIA label and title attribute for expand button of doc sidebar"
+  },
+  "theme.docs.sidebar.expandButtonAriaLabel": {
+    "message": "Expand sidebar",
+    "description": "The ARIA label and title attribute for expand button of doc sidebar"
+  },
+  "theme.docs.paginator.navAriaLabel": {
+    "message": "Docs pages navigation",
+    "description": "The ARIA label for the docs pagination"
+  },
+  "theme.docs.paginator.previous": {
+    "message": "Previous",
+    "description": "The label used to navigate to the previous doc"
+  },
+  "theme.docs.paginator.next": {
+    "message": "Next",
+    "description": "The label used to navigate to the next doc"
+  },
+  "theme.docs.sidebar.responsiveCloseButtonLabel": {
+    "message": "Close menu",
+    "description": "The ARIA label for close button of mobile doc sidebar"
+  },
+  "theme.docs.sidebar.responsiveOpenButtonLabel": {
+    "message": "Open menu",
+    "description": "The ARIA label for open button of mobile doc sidebar"
+  },
+  "theme.docs.sidebar.collapseButtonTitle": {
+    "message": "Collapse sidebar",
+    "description": "The title attribute for collapse button of doc sidebar"
+  },
+  "theme.docs.sidebar.collapseButtonAriaLabel": {
+    "message": "Collapse sidebar",
+    "description": "The title attribute for collapse button of doc sidebar"
+  },
+  "theme.docs.versions.unreleasedVersionLabel": {
+    "message": "This is unreleased documentation for {siteTitle} {versionLabel} version.",
+    "description": "The label used to tell the user that he's browsing an unreleased doc version"
+  },
+  "theme.docs.versions.unmaintainedVersionLabel": {
+    "message": "This is documentation for {siteTitle} {versionLabel}, which is no longer actively maintained.",
+    "description": "The label used to tell the user that he's browsing an unmaintained doc version"
+  },
+  "theme.docs.versions.latestVersionSuggestionLabel": {
+    "message": "For up-to-date documentation, see the {latestVersionLink} ({versionLabel}).",
+    "description": "The label userd to tell the user that he's browsing an unmaintained doc version"
+  },
+  "theme.docs.versions.latestVersionLinkLabel": {
+    "message": "latest version",
+    "description": "The label used for the latest version suggestion link label"
+  },
+  "theme.common.editThisPage": {
+    "message": "Edit this page",
+    "description": "The link label to edit the current page"
+  },
+  "theme.common.headingLinkTitle": {
+    "message": "Direct link to heading",
+    "description": "Title for link to heading"
+  },
+  "theme.lastUpdated.atDate": {
+    "message": " on {date}",
+    "description": "The words used to describe on which date a page has been last updated"
+  },
+  "theme.lastUpdated.byUser": {
+    "message": " by {user}",
+    "description": "The words used to describe by who the page has been last updated"
+  },
+  "theme.lastUpdated.lastUpdatedAtBy": {
+    "message": "Last updated{atDate}{byUser}",
+    "description": "The sentence used to display when a page has been last updated, and by who"
+  },
+  "theme.common.skipToMainContent": {
+    "message": "Skip to main content",
+    "description": "The skip to content label used for accessibility, allowing to rapidly navigate to main content with keyboard tab/enter navigation"
+  },
+  "theme.SearchPage.documentsFound.plurals": {
+    "message": "One document found|{count} documents found",
+    "description": "Pluralized label for \"{count} documents found\". Use as much plural forms (separated by \"|\") as your language support (see https://www.unicode.org/cldr/cldr-aux/charts/34/supplemental/language_plural_rules.html)"
+  },
+  "theme.SearchPage.existingResultsTitle": {
+    "message": "Search results for \"{query}\"",
+    "description": "The search page title for non-empty query"
+  },
+  "theme.SearchPage.emptyResultsTitle": {
+    "message": "Search the documentation",
+    "description": "The search page title for empty query"
+  },
+  "theme.SearchPage.inputPlaceholder": {
+    "message": "Type your search here",
+    "description": "The placeholder for search page input"
+  },
+  "theme.SearchPage.inputLabel": {
+    "message": "Search",
+    "description": "The ARIA label for search page input"
+  },
+  "theme.SearchPage.algoliaLabel": {
+    "message": "Search by Algolia",
+    "description": "The ARIA label for Algolia mention"
+  },
+  "theme.SearchPage.noResultsText": {
+    "message": "No results were found",
+    "description": "The paragraph for empty search result"
+  },
+  "theme.SearchPage.fetchingNewResults": {
+    "message": "Fetching new results...",
+    "description": "The paragraph for fetching new search results"
+  },
+  "theme.SearchBar.label": {
+    "message": "Search",
+    "description": "The ARIA label and placeholder for search button"
+  },
+  "Advanced CAD Features": {
+    "message": "Recursos CAD avançados"
+  },
+  "Easy Visualization": {
+    "message": "Fácil Vizualização"
+  },
+  "For the visualization of results, the program offers linked text elements in the main screen, voltage heatmap display, and also table and graph editors.": {
+    "message": "Para a visualização dos resultados, o programa oferece elementos de texto vinculados na tela principal, mapa de calor de tensão e também editores de tabelas e gráficos."
+  },
+  "Application": {
+    "message": "Aplicação"
+  },
+  "Power Flow": {
+    "message": "Fluxo de carga"
+  },
+  "Hybrid Newton-Gauss": {
+    "message": "Newton-Gauss Híbrido"
+  },
+  "Three-phase induction motors included": {
+    "message": "Motores de indução trifásicos inclusos"
+  },
+  "PSP-UFU is a": {
+    "message": "O PSP-UFU é um"
+  },
+  "cross-platform, multilingual, Free and Open-Source Software (FOSS)": {
+    "message": "Software Livre e de Código Aberto (FOSS), multiplataforma e multi-idiomas"
+  },
+  "with": {
+    "message": "com"
+  },
+  "advanced GUI (Graphical User Interface) features": {
+    "message": "recursos de GUI (Interface Gráfica do Usuário) avançados"
+  },
+  "CAD (Computer-Aided Design) tools": {
+    "message": "ferramentas CAD (Desenho Assistido por Computador)"
+  },
+  "for electrical power system studies": {
+    "message": "para estudos de sistemas elétricos de potência."
+  },
+  "The software allows the construction of any": {
+    "message": "O software permite a construção de qualquer"
+  },
+  "electric transmission network": {
+    "message": "rede de transmissão elétrica"
+  },
+  "and": {
+    "message": "e"
+  },
+  "control systems": {
+    "message": "sistemas de controle"
+  },
+  "through the deployment of visual elements": {
+    "message": "por meio da inserção de elementos visuais"
+  },
+  "The PSP-UFU aims to provide efficient computer simulation tools for": {
+    "message": "O PSP-UFU tem como objetivo fornecer ferramentas eficientes de simulação computacional para"
+  },
+  "research and education purposes": {
+    "message": "fins de pesquisa e educação"
+  },
+  "in addition to": {
+    "message": "além de"
+  },
+  "industrial applications": {
+    "message": "aplicações industriais"
+  },
+  "in electrical power systems": {
+    "message": "no sistema elétrico de potência"
+  },
+  "Short-Circuit calculation": {
+    "message": "Cálculod e curto-circuito"
+  },
+  "Balanced": {
+    "message": "Balanceado"
+  },
+  "Unbalanced": {
+    "message": "Desbalanceado"
+  },
+  "Short-Circuit power in all system buses": {
+    "message": "Potência de curto-circuito em todas as barras"
+  },
+  "Harmonics": {
+    "message": "Harmônicos"
+  },
+  "Harmonic voltages and THD (Total Harmonic Distortion) calculation": {
+    "message": "Cálculo das tensões harmônicas e DHT (Distorção Harmônica de Tensão)"
+  },
+  "Frequency scan": {
+    "message": "Varredura de frequências"
+  },
+  "Transient and Dynamic Stability": {
+    "message": "Estabilidade transitória e dinâmica"
+  },
+  "Several synchronous machine models automatically selected": {
+    "message": "Vários modelos de máquinas síncronas selecionados automaticamente"
+  },
+  "Three-phase induction motors": {
+    "message": "Motores de indução trifásicos"
+  },
+  "User-defined machine controls, exciters and prime moves created using block diagrams (Exciters, AVR, PSS, Hydro and Thermal turbines, Speed Governor, etc.)": {
+    "message": "Controles de máquina definidos pelo usuário, excitatrizes e reguladores de velocidade criados usando diagramas de blocos (Excitatrizes, AVR, PSS, turbinas hidrelétricas e térmicas, regulador de velocidade, etc.)"
+  },
+  "Download": {
+    "message": "Download"
+  },
+  "User Guide": {
+    "message": "Manual do Usuário"
+  },
+  "Samples": {
+    "message": "Amostras"
+  },
+  "Studies Performed": {
+    "message": "Estudos Realizados"
+  },
+  "Published Papers": {
+    "message": "Artigos Publicados"
+  },
+  "Further details can be found in the published papers:": {
+    "message": "Maiores detalhes podem ser encontrados nos artigos publicados:"
+  },
+  "Code Documentation": {
+    "message": "Documentação do Código-Fonte"
+  },
+  "All detailed descriptions of the source-code can be found at": {
+    "message": "Todas as descrições detalhadas do código-fonte podem ser encontradas na"
+  },
+  "Online Documentation": {
+    "message": "Documentação Online"
+  },
+  "generated by": {
+    "message": "gerada pelo"
+  },
+  "Screenshots": {
+    "message": "Screenshots"
+  },
+  "Your browser doesn’t support the this feature.": {
+    "message": "Seu navegador não oferece suporte para esse recurso."
+  },
+  "Power System Platform of Federal University of Uberlândia": {
+    "message": "Plataforma de Sistemas de Potência da Universidade Federal de Uberlândia"
+  }
+}
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current.json b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current.json
new file mode 100644
index 0000000..4671368
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current.json
@@ -0,0 +1,34 @@
+{
+  "version.label": {
+    "message": "Próximo",
+    "description": "The label for version current"
+  },
+  "sidebar.docs.category.Introduction": {
+    "message": "Introdução",
+    "description": "The label for category Introduction in sidebar docs"
+  },
+  "sidebar.docs.category.Power Element Editor": {
+    "message": "Editor de Elementos de Potência",
+    "description": "The label for category Power Element Editor in sidebar docs"
+  },
+  "sidebar.docs.category.Control Element Editor": {
+    "message": "Editor de Elementos de Controle",
+    "description": "The label for category Control Element Editor in sidebar docs"
+  },
+  "sidebar.docs.category.Simulations": {
+    "message": "Simulações",
+    "description": "The label for category Simulations in sidebar docs"
+  },
+  "sidebar.docs.category.Data Visualization": {
+    "message": "Visualização dos Dados",
+    "description": "The label for category Data Visualization in sidebar docs"
+  },
+  "sidebar.docs.category.Power Elements": {
+    "message": "Eelementos de Potência",
+    "description": "The label for category Power Elements in sidebar docs"
+  },
+  "sidebar.docs.category.Control Elements": {
+    "message": "Elementos de Controle",
+    "description": "The label for category Control Elements in sidebar docs"
+  }
+}
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/aboutPSP.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/aboutPSP.md
new file mode 100644
index 0000000..411d55c
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/aboutPSP.md
@@ -0,0 +1,36 @@
+---
+id: aboutPSP
+slug: /
+title: PSP-UFU
+sidebar_label: PSP-UFU
+---
+## Sobre o PSP-UFU
+
+O PSP-UFU (Plataforma de Sistemas de Potência da Universidade Federal de Uberlândia) é um software **multi-plataforma**, **multilíngue**, **livre e de código aberto (FOSS)** com **recursos avançados de GUI (*Graphical User Interface*)** e **ferramentas CAD (*Computer Aided Design*)** para estudos de sistemas elétricos de potência.
+
+O software permite a construção de qualquer rede de transmissão elétrica e sistemas de controle através da inserção de elementos visuais. Para a visualização dos resultados, o programa oferece **elementos de texto vinculados** na tela principal e também **editores de tabelas e gráficos**.
+
+O PSP-UFU tem como objetivo fornecer ferramentas eficientes de simulação para fins de **pesquisa** e **educação**, além de **aplicações industriais** em sistemas elétricos de potência.
+
+Resumidamente, o *software* pode executar os seguintes estudos:
+
+- Fluxo de potência
+- Cálculo de curto-circuito
+- Harmônicos
+- Estabilidade transitória e dinâmica
+
+### Publicações
+Os artigos abaixo possuem maiores detalhes a respeito do PSP-UFU:
+>Oliveira, T. L., Guimarães, G. C., & Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. *International Transactions on Electrical Energy Systems*, e12185. doi: [10.1002/2050-7038.12185](https://doi.org/10.1002/2050-7038.12185)
+
+>Oliveira, T. L., Guimarães, G. C., Silva, L. R., & Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. *The International Journal of Electrical Engineering & Education*, 0020720919879058. doi: [10.1177/0020720919879058](https://doi.org/10.1177/0020720919879058)
+
+A Tese de Doutorado abaixo detalha em profundidade os cálculos envolvidos, arquitetura de software e ferramentas implementadas no PSP-UFU:
+>Oliveira, T. L. (2019). Desenvolvimento de um programa computacional livre, gráfico, e multiplataforma para analisar sistemas elétricos de potência em regime permanente e dinâmico. doi: [10.14393/ufu.te.2019.2444](https://dx.doi.org/10.14393/ufu.te.2019.2444)
+
+### Contribuindo com o projeto
+Se você está interessado em contribuir com o PSP-UFU de alguma forma (desenvolvimento, críticas, sugestões, etc.), entre em contato pelo [GitHub](https://github.com/Thales1330/PSP/issues) ou [Twitter](https://twitter.com/PspUfu).
+
+### Desenvolvedor~~es~~
+[Thales Lima Oliveira](https://github.com/Thales1330)
+
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/bus.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/bus.md
new file mode 100644
index 0000000..d466acc
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/bus.md
@@ -0,0 +1,137 @@
+---
+id: bus
+title: Barramento
+sidebar_label: Barramento
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+>Condutor de baixa impedância ao qual vários circuitos elétricos podem ser conectados em pontos separados.
+Nota - Em muitos casos, o barramento consiste em uma barra. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=151-12-30).
+
+## Barramento no PSP-UFU
+O elemento **barramento**, ou simplesmente **barra**, é um conector ou nó do diagrama unifilar do PSP-UFU. Essa barra pode representar um PAC (Ponto de Acoplamento Comum), um poste de distribuição, uma subestação, um barramento da subestação, entre inúmeros outros tipos pontos de análise e conexão entre elementos.
+
+## Formulário de edição dos barramentos
+:::caution Atenção!
+O barramento deve ser o **primeiro elemento elétrico a ser inserido no diagrama de potência**, uma vez que os demais componentes de potência são conectados nele.
+:::
+
+A imagem abaixo apresenta o formulário de inserção/alteração de dados das barras:
+
+<div><center><img src={useBaseUrl("images/busForm.png")} alt="Formulário dos barramentos no PSP-UFU" title="Formulário dos barramentos no PSP-UFU" /></center></div>
+
+Esse formulário é subdividido em quatro contextos distintos:
+- **Geral**: no qual são inseridas informações gerais da barra e informações do fluxo de carga;
+- **Falta**: local onde o curto-circuito *shunt* deve ser inserido;
+- **Estabilidade**: contendo opções de visualização de dados da barra em gráficos no tempo e inserção de faltas trifásicas no cálculo de estabilidade transitória;
+- **Qualidade de energia**: contém a opção de de visualização da impedância harmônica vista pela barra.
+
+---
+<Tabs
+  groupId="bus-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Falta', value: 'fault'},
+    {label: 'Estabilidade', value: 'stability'},
+    {label: 'Qualidade de energia', value: 'powerQuality'},
+  ]
+}>
+<TabItem value="general">
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+Todos os componentes de potência do PSP-UFU possuem esse campo.
+
+#### Tensão nominal
+Utilizado para cálculo da impedância base de alguns elementos conectados, além do cálculo da relação de transformação dos transformadores conectados entre barras. Sua unidade pode ser selecionada, sendo expressa em V ou em kV.
+
+A modificação desse parâmetro irá alterar toda a tensão do trecho conectado por linhas elétricas, sendo emitido um alerta ao usuário.
+
+#### Tensão controlada
+Caracteriza o barramento como **barra de tensão controlada ([Barra PV](powerFlow))**,
+
+:::caution Atenção!
+Essa opção é somente válida caso alguma máquina síncrona esteja conectado, caso contrário esse valor será ignorado. Caso o limite de potência reativa da máquina síncrona conectada seja ultrapassado esse valor também é ignorado.
+:::
+
+O valor poderá ser inserido em p.u. ou em volts (ou kV caso a tensão nominal esteja nesta unidade).
+
+#### Barra de referência
+Caracteriza o barramento como **barra de referência ([Barra de oscilação](powerFlow))**. Essa opção é somente válida caso esteja conectado um gerador síncrono, caso contrário uma mensagem de erro será exibida ao usuário ao realizar algum dos cálculos do programa.
+
+</TabItem>
+<TabItem value="fault">
+
+#### Inserir falta
+
+Indica se existe um curto-circuito na barra nos estudos de [curto-circuito](fault).
+
+:::warning Cuidado!
+Nota-se que essa opção irá inserir uma falta no cálculo de [curto-circuito](fault) e **não será considerado no estudo de estabilidade**.
+:::
+
+#### Tipo de falta
+Seleciona o tipo de falha shunt da barra:
+- trifásico para a terra
+- fase-fase
+- fase-fase-terra
+- fase-terra
+
+#### Local da falta
+Seleciona a fase em que se situa a falta (ou combinação delas no caso de falha entre duas fases), sendo essa opção desabilitada para o tipo de falta trifásica.
+
+#### Resistência e reatância de falta
+Representam a impedância da falta. Curtos-circuitos francos (sem impedância de falta) são representados inserindo valor zero em ambos os campos.
+
+</TabItem>
+<TabItem value="stability">
+
+#### Imprimir dados
+Mostra os dados da barra editada no gráfico gerado pelo estudo de estabilidade transitória. Os seguintes dados são exibidos:
+- módulo da tensão do barramento
+- ângulo da tensão do barramento.
+
+#### Inserir falta
+Insere uma falta trifásica na barra no cálculo de [estabilidade](stability).
+:::warning Cuidado!
+Nota-se que essa opção irá inserir uma falta no cálculo de [estabilidade](stability) e **não será considerado no estudo de [curto-circuito](fault)**.
+:::
+
+#### Tempo
+Instante no tempo ($t_{falta}$) em que ocorre a falta durante os estudos de [estabilidade](stability). Esse campo e os próximos são somente acessados caso seja inserida uma falta no estudo de estabilidade.
+
+#### Duração da falta
+Duração da falta inserida ($\Delta t_{falta}$) no estudo de estabilidade. A falta irá iniciar no instante $t_{falta}$ e será interrompida em $t_{falta}+\Delta t_{falta}$.
+
+#### Resistência e reatância de falta
+Impedância de falta nos estudos de estabilidade. Uma falta franca é representada pela inserção de um valor de impedância shunt de valor muito baixo ($j10^{-5}  p.u.$).
+
+</TabItem>
+<TabItem value="powerQuality">
+
+#### Imprimir impedância harmônica da barra
+Imprime a impedância harmônica vista por essa barra referente ao estudo de [Resposta na Frequência](harmonics).
+
+</TabItem>
+</Tabs>
+
+---
+
+:::tip Dica
+Resumo de como definir o tipo de barra:
+- Para definir o barramento como sendo uma **[Barra PQ](powerFlow)**, as opções "Barra de referência" e "Tensão controlada" devem estar **desmarcadas**; 
+- Para definir o barramento como sendo uma **[Barra PV](powerFlow)**, deve-se **marcar somente** a opção "Tensão controlada", mantendo a opção "Barra de referência" desmarcada;
+- Para definir o barramento como sendo uma **[Barra de Referência](powerFlow)**, deve-se marcar a opção "Barra de referência". Caso a opção "Tensão controlada" esteja desmarcada será admitido um valor de tensão controlada de $1{,}0~p.u.$
+:::
+
+:::warning Cuidado!
+**O sistema deve possuir somente uma barra de referência**.
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/cadTools.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/cadTools.md
new file mode 100644
index 0000000..e26737b
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/cadTools.md
@@ -0,0 +1,144 @@
+---
+id: cadTools
+title: Ferramentas CAD
+sidebar_label: Ferramentas CAD
+---
+export const Highlight = ({children, color}) => ( <span style={{
+      backgroundColor: color,
+      borderRadius: '2px',
+      color: '#000',
+      padding: '0.2rem',
+    }}>{children}</span> );
+
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+>Desenho assistido por computador (DAC; em inglês: *computer aided design* - CAD) é o nome genérico de sistemas computacionais (*software*) utilizados pela engenharia, geologia, geografia, arquitetura e design para facilitar o projeto e desenho técnicos. No caso do *design*, este pode estar ligado especificamente a todas as suas vertentes (produtos como vestuário, eletroeletrônicos, automobilísticos, etc.), de modo que os jargões de cada especialidade são incorporados na interface de cada programa. [Wikipedia](https://pt.wikipedia.org/wiki/Desenho_assistido_por_computador)
+
+## Ferramentas do menu
+
+Todos os objetos criados contêm vários atributos gráficos, como: posição, orientação, cor, tamanho, etc. Para modificá-los foram inseridas algumas ferramentas intuitivas e de fácil utilização, as quais são: [**selecionar**](cadTools#selecionar), [**mover**](cadTools#mover), [**girar**](cadTools#girar), [**arrastar**](cadTools#arrastar), [**zoom**](cadTools#zoom), [**encaixar**](cadTools#encaixar), [**copiar, colar**](cadTools#copiar-e-colar) e [**excluir**](cadTools#excluir).
+
+Essas ferramentas podem ser acessadas por suas teclas de atalho assim como pelo [submenu Ferramentas](mainScreen#menu-ribbon).
+
+Tais ferramentas permitem a **personalização**, **navegação** e auxiliam a **criação** de redes elétricas de potência, assim como diagramas de controle na plataforma.
+
+### Selecionar
+A seleção de um único objeto é realizada clicando com o botão esquerdo do mouse sobre o elemento, que será identificado pelo contorno em <Highlight color="#80bfff">azul</Highlight> do objeto.
+
+Para a seleção de vários objetos simultaneamente é necessário clicar com o botão esquerdo do mouse em um local vazio da tela e arrastá-lo, criando assim um **retângulo de seleção** que irá acompanhar o ponteiro do mouse enquanto o botão esquerdo estiver pressionado. Ao soltá-lo todos os objetos que intercederem o retângulo serão selecionados.
+
+Mantendo pressionado a tecla Control (Ctrl) do teclado pode-se manter seleções anteriores enquanto novos elementos são selecionados.
+
+Para desmarcar todos os objetos basta clicar em uma área sem objetos.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/select.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/select.webm")} type="video/webm" />
+</video>
+
+### Mover
+Um elemento poderá ser movido para qualquer local por meio da ação *drag-and-drop* (clicando e arrastando) ou pela ferramenta mover.
+
+Ao habilitar o “modo mover”, movendo todos os componentes selecionados à medida que a posição do mouse é alterada. Para desabilitar esse modo basta pressionar a tecla Escape do teclado.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/move.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/move.webm")} type="video/webm" />
+</video>
+
+### Girar
+Alguns objetos poderão ser rotacionados através da ferramenta girar, acessado pela tecla de atalho “R”, girando o objeto no sentido horário, ou “Shift + R”, para o sentido anti-horário (essa opção irá afetar todos os objetos selecionados). O elemento também pode ser rotacionado pelo [submenu Ferramentas](mainScreen#menu-ribbon) ou ao acessar o menu de contexto clicando com o botão direito sobre o elemento.
+
+:::caution Atenção!
+O menu de contexto somente será exibido se o elemento estiver selecionado.
+:::
+
+Ao utilizar essa ferramenta os objetos irão girar 45º a partir do seu ponto de origem. Para atingir a angulação pretendida basta repetir o processo de rotação.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/rotate.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/rotate.webm")} type="video/webm" />
+</video>
+
+### Arrastar
+Três ferramentas muito úteis na navegação do circuito, principalmente em grandes redes, são: arrastar, zoom e encaixar.
+
+A ferramenta arrastar possibilita mover todo o circuito **preservando as posições relativas entre os elementos**. Esta ferramenta pode ser acessada pelo submenu “Ferramentas” ou **mantendo pressionada o *scroll* do mouse**, ativando assim o “modo arrastar”. Ao ativá-lo todo o circuito será movido clicando e arrastando na direção desejada.
+
+Para sair do “modo arrastar” basta pressionar a tecla Escape do teclado.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/drag.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/drag.webm")} type="video/webm" />
+</video>
+
+### Zoom
+O zoom será aplicado ao utilizar o **scroll do mouse** para mais ou menos zoom. A ferramenta zoom irá aproximar ou distanciar o circuito a partir da posição do ponteiro do mouse na tela para encontrar objetos ou destaca-los.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/zoom.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/zoom.webm")} type="video/webm" />
+</video>
+
+### Encaixar
+A ferramenta encaixar irá mover o circuito e aplicar o zoom necessário para que todos os componentes da rede sejam exibidos na tela. Esta ferramenta pode ser acessada através do submenu “Ferramentas” ou pela combinação de teclas “Shift + F”.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/fit.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/fit.webm")} type="video/webm" />
+</video>
+
+### Copiar e Colar
+Duas ferramentas importantes durante o processo de criação da rede elétrica de potência no *software* são copiar e colar. Qualquer circuito criado no PSP-UFU poderá ser duplicado completamente ou parcialmente no mesmo projeto ou em múltiplos através dessas ferramentas.
+
+Para copiar basta selecionar os elementos que se deseja duplicar e acessar a ferramenta no submenu “Ferramentas” ou pressionando a combinação das teclas “Ctrl + C”, então os dados do circuito serão copiados para a área de transferência. Ao copiar um circuito ele poderá ser colado no mesmo projeto ou em um projeto distinto, utilizando o mesmo submenu ou pressionando a combinação das teclas “Ctrl + V”.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/copyPaste.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/copyPaste.webm")} type="video/webm" />
+</video>
+
+### Excluir
+Qualquer objeto poderá ser excluído do projeto por meio da ferramenta deletar, presente no menu Ribbon, pela tecla de atalho Delete ou pelo acesso ao menu de contexto clicando com o botão direito sobre o elemento selecionado.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/delete.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/delete.webm")} type="video/webm" />
+</video>
+
+
+## Personalização gráfica os elementos
+Alguns elementos permitem sua personalização gráfica, como as barras e linhas.
+
+### Barra
+No caso dos barramentos, pode-se alterar o **comprimento da barra** clicando e arrastando no *pickbox* exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.
+
+### Linha
+As linhas podem ser inseridas com **"nós" (pontos de ancoragem)**, personalizando a disposição das linhas no diagrama unifilar. Esses nós são inseridos durante o processo de criação do elemento, anteriormente à seleção da segunda barra.
+
+Após a sua inserção, novos nós podem ser anexados pelo menu de contexto acessado ao clicar com o botão direito sobre a linha selecionada. Sua remoção também é alcançada por uma opção do menu de contexto.
+
+Assim como nos barramentos, a alteração da posição dos nós é obtida clicando e arrastando o *pickbox* exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/graphicPropsChange.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/graphicPropsChange.webm")} type="video/webm" />
+</video>
+
+## Conexão e desconexão de elemetos na barra
+Os elementos podem ser desconectados e reconectados às barras após sua inserção somente pela ferramenta *drag-and-drop*. Para isso, o elemento deve ser selecionado e o seu nó de conexão deve ser arrastado para o local requerido.
+
+:::warning Cuidado!
+O elemento a ser desconectado/reconectado **deve ser previamente selecionado** para efetuar a operação. Caso contrário a barra será movida ou nenhuma operação será realizada.
+:::
+
+No caso de desconexão, o elemento será automaticamente removido da simulação, sendo indicado pela sua cor (o elemento passa a ter cor <Highlight color="#7f7f7f">cinza</Highlight>).
+
+:::caution Atenção!
+Ao reconectar o elemento ao circuito, deve-se **atentar para estado do seu "disjuntor"**, indicado por um quadrado próximo ao seu nó de conexão. Um elemento reconectado volta ao circuito com seu disjuntor aberto (<Highlight color="#ff1919">vermelho</Highlight>), sendo necessário clicar sobre o mesmo a fim de inserir o elemento no circuito.
+:::
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/disconnect.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/disconnect.webm")} type="video/webm" />
+</video>
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/capacitor.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/capacitor.md
new file mode 100644
index 0000000..9c9c0c8
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/capacitor.md
@@ -0,0 +1,60 @@
+---
+id: capacitor
+title: Capacitor
+sidebar_label: Capacitor
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+>Um dispositivo constituído essencialmente por dois eletrodos separados por um dielétrico. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=436-01-03).
+
+## Capacitor no PSP-UFU
+O elemento capacitor representa, geralmente, um banco de capacitores *shunt* no circuito do PSP-UFU.
+
+:::tip Dica
+Um arranjo série de uma [linha](line) sem as susceptâncias *shunt*, um [barramento](bus) e um [capacitor](capacitor) pode ser utilizado para fabricar um filtro passivo nos [estudos harmônicos](harmonics). Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.
+:::
+
+Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.
+
+## Formulário de edição dos capacitores
+A imagem abaixo apresenta o formulário de inserção/alteração de dados dos capacitores:
+
+<div><center><img src={useBaseUrl("images/capacitorForm.png")} alt="Formulário dos capacitores no PSP-UFU" title="Formulário dos capacitores no PSP-UFU" /></center></div>
+
+Além do único contexto geral, pode ser observado o botão "Estabilidade" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do capacitor durante o estudo de [estabilidade](stability).
+
+<div><center><img src={useBaseUrl("images/swCapacitor.png")} alt="Formulário de chaveamento do capacitor" title="Formulário de chaveamento do capacitor" /></center></div>
+
+<Tabs
+  groupId="transformer-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Botão Estabilidade', value: 'stability'},
+  ]
+}>
+<TabItem value="general">
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+Todos os componentes de potência do PSP-UFU possuem esse campo.
+
+#### Potência reativa
+Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou $p.u.$ (na base de potência do sistema). 
+
+</TabItem>
+<TabItem value="stability">
+
+O botão "Estabilidade" irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de [estabilidade](stability).
+
+Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões "Adicionar" e "Remover", respectivamente.
+
+</TabItem>
+</Tabs>
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/const.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/const.md
new file mode 100644
index 0000000..bc2eb39
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/const.md
@@ -0,0 +1,14 @@
+---
+id: const
+title: Constante
+sidebar_label: Constante
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+As constantes são valores numéricos reais que permanecem inalteradas em todo o período de simulação. Esses elementos possuem somente um nó de saída, o qual poderá ser conectado a qualquer outro nó de entrada dos elementos de controle.
+
+:::info Informação
+Esse elemento pode, por exemplo, ser utilizado para definir valores alvos ou deslocamentos de sinal.
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/controlEditor.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/controlEditor.md
new file mode 100644
index 0000000..6be6e71
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/controlEditor.md
@@ -0,0 +1,116 @@
+---
+id: controlEditor
+title: Editor de Controle
+sidebar_label: Editor de Controle
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+Os controles das [máquinas síncronas](syncGenerator) são criados, modificados, importados e exportados em um editor distinto dos elementos de potência, apresentado na figura abaixo.
+
+<div><center><img src={useBaseUrl("images/controlEditor.png")} alt="Editor de elementos de controle do PSP-UFU" title="Editor de elementos de controle do PSP-UFU" /></center></div>
+
+O acesso ao editor é realizado por meio de dois botões presentes no [formulário de edição dos dados de estabilidade dos geradores síncronos](syncGenerator#formulário-de-edição-dos-geradores-síncronos):
+- [Editar AVR](syncGenerator#utilizar-avr-e-regulador-de-velocidade)
+- [Editar regulador de velocidade](syncGenerator#utilizar-avr-e-regulador-de-velocidade)
+
+:::caution Atenção!
+No PSP-UFU a opção de editar o **AVR** engloba mais que somente o controle de tensão da máquina. Nele **deve** ser inserida a malha de controle da máquina assim como a **excitatriz da máquina síncrona**. Outras estratégias de controle (opcionais), como PSS (*Power System Stabilizer*) e/ou controles de sobre e sub excitação, são também implementadas em conjunto (como apresentado na figura anterior).
+:::
+
+:::caution Atenção!
+Assim como no AVR, o **Regulador de Velocidade** engloba mais que a regulação primária da máquina. Nessa opção **deve** ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o **modelo da turbina**. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.
+:::
+
+## Elementos de Controle
+O acesso aos elementos de controle é realizado em uma janela (removível e encaixável) por meio de ícones relacionados.
+
+Uma vez que o usuário clicar no ícone desejado, o elemento de controle acompanhará o ponteiro do mouse até ser efetivamente inserido na posição desejada ao clicar novamente na área de trabalho. Os elementos são então conectados por “linhas de conexão” inseridas ao clicar nos nós dos componentes previamente adicionados, permitindo a construção da rede de controle genérica.
+
+As ferramentas de manipulação e navegação, como arrastar, mover e excluir são [herdadas](cadTools) do [editor de elementos de potência](powerEditor), possuindo comportamento idêntico. A **edição dos dados** dos componentes inseridos também é realizada com **duplo clique** sobre o elemento inserido, exibindo um formulário de edição de dados.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= "/PSP/videos/timelapseControlEditor.mp4" type="video/mp4" />
+  <source src= "/PSP/videos/timelapseControlEditor.webm" type="video/webm" />
+</video>
+
+Os seguintes blocos de controle estão presentes no PSP-UFU:
+- [Entrada e Saída](io)
+- [Função Transferência](transferFunction)
+- Matemática
+	- [Somador](sum)
+	- [Multiplicador](multiplier)
+	- [Divisor](divider)
+	- [Exponencial](exponential)
+- Limitadores
+	- [Limitador absoluto](limiter)
+	- [Limitador de taxa](rateLimiter)
+- Constantes
+	- [Constante](const)
+	- [Ganho](gain)
+- [Expressão Matemática](mathExpression)
+
+## Inicialização do sistema de controle
+Caso as entradas do sistema de controle apresentem valores diferentes de zero ou a primeira operação entre elas não resulte em uma saída nula, o sistema necessita de inicialização, de forma a adequar os valores de entradas e saída dos blocos elementares e dos vetores de estado das [funções transferências](transferFunction) presentes. Tal procedimento é realizado por meio da solução de toda rede de controle até que se obtenha sua convergência, ou seja, a diferença absoluta entre as mesmas saídas de uma solução anterior e uma atual deve ser nula ou muito próxima de zero.
+
+:::info
+O processo de **inicialização é realizada automaticamente** pelo PSP-UFU.
+
+Uma vez que a inicialização é imposta pelas [entradas](io) do controle, erros de convergência podem ocorrer devido à má parametrização dos elementos dos sistemas de **potência e controle**.
+:::
+
+Para otimizar e melhorar a estabilidade do processo de inicialização utilizou-se um passo de integração variável dentro de limites, de forma que o passo aumenta em condições de diferenças menores entre as soluções do sistema de controle e diminui caso essa diferença se torne elevada. A implementação dessa abordagem reduziu significativamente o encerramento do processo com erro causado pela instabilidade numérica, além de acelerar a inicialização.
+
+O fluxograma abaixo evidencia o processo de inicialização implementado no PSP-UFU:
+
+<div><center><img src={useBaseUrl("images/controlInit.png")} alt="Estrutura da inicialização do sistema de controle" title="Estrutura da inicialização do sistema de controle" /></center></div>
+
+## Teste do diagrama de controle
+Na parte inferior esquerda do Editor de Controle está presente o botão "**Testar Sistema...**". O botão acessa o formulário de teste de controles, como é indicado na figura abaixo:
+
+<div><center><img src={useBaseUrl("images/testControl.png")} alt="Formulário de teste de controles" title="Formulário de teste de controles" /></center></div>
+
+Nesse formulário é possível inserir o comportamento de **todas as entradas do diagrama**:
+- **Tipo de entrada**: Define um dos tipos de entradas possíveis no PSP-UFU: "Passo (*Step*)", Rampa (*Ramp*) ou Quadrática (*Quadratic*);
+- **Tempo de início ($t_i$)**: Define o tempo de início na simulação da entrada definida. As entradas iniciais são sempre definidas como zero e assumem valores diferentes após o tempo de início;
+- **Inclinação ($\alpha$)**: Valor relacionado ao tipo de entrada:
+	- Para entrada do tipo "Passo" - O valor da inclinação define o valor final do passo:
+	$$
+	\begin{cases}
+		\text{Se~} t < t_i \rightarrow f(t) = 0,0\\
+		\text{Se~} t \ge t_i \rightarrow f(t) = \alpha
+	\end{cases}
+	$$
+	- Para entrada do tipo "Rampa" - define o coeficiente de inclinação da reta:
+	$$
+	\begin{cases}
+		\text{Se~} t < t_i \rightarrow f(t) = 0,0\\
+		\text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t
+	\end{cases}
+	$$
+	- Para entrada do tipo "Quadrática" - define o coeficiente de crescimento da curva:
+	$$
+	\begin{cases}
+		\text{Se~} t < t_i \rightarrow f(t) = 0,0\\
+		\text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t^2
+	\end{cases}
+	$$
+	
+- **Passo de integração ($h$)**: Passo do método de integração Trapezoidal Implícito;
+- **Tempo de simulação ($t_f$)**: Tempo total de simulação.
+
+Após a simulação os resultados são exibidos no [visualizador de gráficos](graphViewer).
+
+:::caution Atenção!
+A mesma parametrização de entrada é aplicada em todos os blocos de [entrada](io) do sistema.
+:::
+
+:::tip Dica
+É possível aplicar **várias entradas distintas** (inclusive com condicionais!) utilizando o bloco de [expressão matemática](mathExpression).
+:::
+
+Utilizando o bloco de [expressão matemática](mathExpression) é possível inserir entradas complexas, como por exemplo testar o diagrama de blocos (AVR DC1C + PSS1A - [IEEE Std. 421.5-2016](https://doi.org/10.1109/IEEESTD.2016.7553421)) da figura abaixo:
+
+<div><center><img src={useBaseUrl("images/exControl.svg")} alt="Sistema de controle sob teste" title="Sistema de controle sob teste" /></center></div>
+<div><center><img src={useBaseUrl("images/compControl.svg")} alt="Comparação do sistema de controle sob teste" title="Comparação do sistema de controle sob teste" /></center></div>
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/divider.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/divider.md
new file mode 100644
index 0000000..af3bcae
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/divider.md
@@ -0,0 +1,16 @@
+---
+id: divider
+title: Divisor
+sidebar_label: Divisor
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+Esse elemento é bastante simples e cumpre com a função matemáticas de seu nome.
+
+Possui duas entradas de valores reais e sua saída é computada na sequência do nó superior pelo inferior, ou seja, a saída será o valor presente no nó 1 dividido pelo nó 2.
+
+:::caution Atenção!
+Nós não conectados no divisor são considerados como valor nulo. Caso a divisão possua numerador zero, o valor da saída é definida como sendo $10^{15}$.
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/exponential.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/exponential.md
new file mode 100644
index 0000000..a327ef8
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/exponential.md
@@ -0,0 +1,27 @@
+---
+id: exponential
+title: Exponencial
+sidebar_label: Exponencial
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+A função exponencial dos elementos de controle realiza essa operação com uma entrada real, obedecendo a expressão:
+$$
+y_n = \bold{A} e^{\bold{B} u_n}
+$$
+Em que:
+- $u_n$ e $y_n$	é a entrada e a saída do bloco exponencial, respectivamente;
+- $\bold{A}$ e $\bold{B}$	são constantes definidas pelo usuário.
+
+## Formulário de edição de dados do bloco Exponencial
+A figura abaixo apresenta o formulário de edição de dados do bloco exponencial.
+
+<div><center><img src={useBaseUrl("images/expForm.png")} alt="Formulário de edição de dados do bloco exponencial no PSP-UFU" title="Formulário de edição de dados do bloco exponencial no PSP-UFU" /></center></div>
+
+Esse bloco não linear é definido pelas constantes A e B, inseridas pelo usuário.
+
+:::info Informação
+Tais blocos são úteis na representação de não linearidades, como, por exemplo, modelagem da saturação de máquinas de corrente contínuas presentes em alguns reguladores automáticos de tensão.
+:::
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/fault.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/fault.md
new file mode 100644
index 0000000..823771a
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/fault.md
@@ -0,0 +1,169 @@
+---
+id: fault
+title: Curto-Circuito
+sidebar_label: Curto-Circuito
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+O principal objetivo da análise de curto-circuito é o cálculo das correntes e tensões de falta para especificação de disjuntores, transformadores de corrente e a parametrização de relés de proteção. De 70 a 80% das faltas em linhas de transmissão são entre uma fase e terra, as quais ocorrem devido ao centelhamento de apenas uma fase da linha para a torre e então para a terra.O menor número de faltas, cerca de 5%, envolve todas as três fases, chamadas de faltas trifásicas. Os outros tipos de faltas envolvem duas fases e duas fases e a terra.
+
+Todas essas falhas, exceto a trifásica, são assimétricas e causam desequilíbrio entre as fases.
+
+## Cálculo de Curto-Circuito no PSP-UFU
+O primeiro estágio do cálculo de curto-circuito é a determinação das tensões pré-falta, das potências de geração e cargas do sistema. Esses dados são obtidos por meio do estudo de [fluxo de carga](powerFlow).
+
+:::info Informação
+Atualmente, o PSP-UFU fornece resultado para os seguintes tipos de falta:
+- Falta Trifásica (3F-T);
+- Falta Fase-Terra (F-T);
+- Falta Fase-Fase (F-F);
+- Falta Fase-Fase-Terra (F-F-T).
+:::
+
+Os modelos dos elementos elétricos que constituem um sistema de potência para o estudo de curto-circuito são semelhantes aos do fluxo de carga, apresentando algumas divergências para as faltas desbalanceadas (F-T, F-F e F-F-T).
+
+As faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados em conjunto cargas desbalanceadas.
+
+## Execução do cálculo de curto-circuito no PSP-UFU
+Existem duas formas de se calcular o curto-circuito no PSP-UFU:
+- **Falta**: Calcula a falta inserida nas [barras](bus). Nesse tipo de cálculo é possível calcular faltas *shunt* nos [barramentos](bus) balanceadas e desbalanceadas.
+- **Nível de curto-circuito**: Calcula o nível de curto-circuito (falta trifásica) em todos [barramentos](bus) do sistema.
+
+Após a construção do diagrama unifilar no [editor de potência](powerEditor), a execução do cálculo de curto-circuito é realizada no [menu Simulação](mainScreen#menu-ribbon) clicando no botão **Falta**. Para calcular o nível de curto-circuito (falta trifásica) em todos [barramentos](bus) do sistema, basta clicar no botão **Nível de curto-circuito**.
+
+<div><center><img src={useBaseUrl("images/menuSimulationFaulta.svg")} alt="Execução dos cálculos de curto-circuito" title="Execução dos cálculos de curto-circuito" /></center></div>
+
+:::caution Atenção
+É possível calcular as faltas sem a execução do [fluxo de carga](powerFlow), porém **não é recomendável**, visto que os valores das correntes de falta são significativamente alteradas.
+:::
+
+**Outra possibilidade** é a execução por meio do cálculo contínuo, também presente no [menu Simulação](mainScreen#menu-ribbon) e seu acionamento é realizado co clicar no botão **Habilitar solução**. Com essa opção, os cálculos estáticos selecionados nas [configurações de simulação](simulationConfig) são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).
+
+:::warning Cuidado!
+Os cálculos de curtos-circuitos não são habilitados por padrão no cálculo contínuo e devem ser inseridos nas [configurações de simulação](simulationConfig).
+:::
+
+Os resultados do cálculo de curto-circuito são exibidos nos [elementos de texto vinculado](text), ao posicionar o mouse sobre os barramentos e em [relatórios tabulares](tabularReport).
+
+### Erros comuns na execução do cálculo de curto-circuito
+A seguir são apresentados os erros mais comuns relacionados ao calculo de curto-circuito.
+
+#### A seguinte mensagem de erro é exibida: "Falha ao inverter a matriz admitância de sequência zero"
+- **Impossibilidade de circulação da corrente de sequência zero**. Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a matriz admitância de sequência zero é singular. Para contornar esse problema escolha uma das duas soluções abaixo:
+	- Marque a opção "Neutro aterrado" e insira um alto valor de reatância de aterramento ($j9999~p.u.$, por exemplo);
+	- Ou, na barra do gerador, insira um reator de baixo valor de potência reativa ($1{,}0~var$, por exemplo).
+
+## O cálculo de curto-circuito
+Como já foi apresentado anteriormente, as faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados com cargas desbalanceadas.
+
+### Método das componentes simétricas
+Esse método proposto por C. L. Fortescue, permite definir um sistema de n fasores desbalanceados em n – 1 sistemas de n fases balanceados e um sistema de fase zero. O sistema de fase zero é definido por todas as fases de mesmo módulo e ângulo.
+Para um sistema trifásico pode-se definir três componentes de sequência:
+1.	Componentes de sequência positiva, constituindo em três fasores iguais em módulo, 120º defasados entre si, e tendo a mesma sequência de fase que os fasores originais;
+2.	Componentes de sequência negativa, constituindo em três fasores iguais em módulo, 120º defasados entre si, e tendo a sequência de fase oposta à dos fasores originais.
+3.	Componentes de sequência zero, constituindo em três fasores iguais em módulo e com defasagem nula entre si.
+
+Com isso pode-se decompor as tensões de fase em componentes simétricas pelas seguintes equações:
+$$
+\begin{cases}
+\dot{V}_a = \dot{V}_{a1} + \dot{V}_{a2} + \dot{V}_{a0}\\
+\dot{V}_b = \dot{V}_{b1} + \dot{V}_{b2} + \dot{V}_{b0}\\
+\dot{V}_c = \dot{V}_{c1} + \dot{V}_{c2} + \dot{V}_{c0}
+\end{cases}
+$$
+
+A figura abaixo apresenta um exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados.
+
+<div><center><img src={useBaseUrl("images/fortescue.svg")} alt="Exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados" title="Exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados" /></center></div>
+
+Para simplificar os cálculos adota-se um operador “$\overline{a}$”, com o intuito de indicar a rotação de um fasor. Tal operador é um número complexo de módulo unitário e ângulo de 120º:
+$$
+\overline{a} = 1 \angle 120^{\circ} = 1 e^{j2\pi/3} = -0{,}5 + j0{,}866
+$$
+
+Com isso pode-se utilizar as equações (de tensão apresentadas  em conjunto com o operador “$\overline{a}$” para construir a seguinte equação matricial:
+$$
+\begin{bmatrix}
+\dot{V}_a\\
+\dot{V}_b\\
+\dot{V}_c
+\end{bmatrix}
+=
+\overbrace{
+\begin{bmatrix}
+1 & 1 & 1\\
+1 & \overline{a}^2 & \overline{a}\\
+1 & \overline{a} & \overline{a}^2
+\end{bmatrix}
+}^{\left[ \bold{A} \right]}
+\begin{bmatrix}
+\dot{V}_{a0}\\
+\dot{V}_{a1}\\
+\dot{V}_{a2}
+\end{bmatrix}
+$$
+
+Considerando a matriz quadrada da equação anterior sendo $\left[ \bold{A} \right]$, pode-se encontrar as componentes simétricas pré-multiplicando ambos os lados dessa mesma equação por $\left[ \bold{A} \right]^{-1}$.
+
+Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas faltas assimétricas (F-T, F-F e F-F-T) é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.
+
+:::info Informação
+As informações a respeito das particularidades dos modelos para o estudo de curto-circuito são apresentados individualmente nos [elementos de potência](powerEditor#editando-dados-elétricos).
+:::
+
+### Equações do curto-circuito
+Primeiramente será tratado o equacionamento para faltas balanceadas e então os estudos serão estendidos para as faltas desbalanceadas por meio da utilização do método das componentes simétricas.
+
+#### Faltas balanceadas
+Utiliza-se da matriz impedância de barras para o cálculo de curto-circuito, definida pela seguinte equação matricial:
+$$
+[\dot{V}] = [Z_{bus}][\dot{I}]
+$$
+Em que:
+- $[Z_{bus}]$ é a inversa da matriz admitância de barras, chamada de matriz impedância de barras.
+
+Por meio da expansão da equação anterior é possível calcular a corrente de falta trifásica na barra genérica $i$:
+$$
+\dot{I}_f = \frac{\dot{E}_i}{\overline{z}_{ii}+\overline{z}_{f}}
+$$
+Em que:
+- $\dot{I}_f$ é a corrente de falta trifásica na barra $i$
+- $\dot{E}_i$ é a tensão pré-falta na barra $i$
+- $\overline{z}_{ii}$ é a impedância equivalente de Thevenin vista pela barra $i$, retirada da matriz impedância
+- $\overline{z}_{f}$ é a impedância de falta
+
+#### Faltas desbalanceadas
+O desenvolvimento das equações do cálculo de curto-circuito para faltas desbalanceadas é realizado seguindo o seguinte procedimento:
+1.	Definir os diagramas no ponto da falta, mostrando as conexões de todas fases para a falta. Assume-se que apenas impedâncias balanceadas estão presentes em ambos os lados do ponto da falta e o equivalente Thevenin até esse ponto é conhecido;
+2.	Escrever as condições de contorno relacionando as tensões e corrente conhecidas para o tipo de falta estudada;
+3.	Transformar as correntes e tensões do item 2 de a-b-c para o sistema de coordenadas 0-1-2;
+4.	Encontrar a corrente do curto-circuito em estudo baseado no seguinte sistema de equações (para a fase A):
+	$$
+	\begin{cases}
+	\dot{V}_{a1} = \dot{E}_a - \dot{I}_{a1} \overline{z}_1\\
+	\dot{V}_{a2} = - \dot{I}_{a2} \overline{z}_2\\
+	\dot{V}_{a0} = - \dot{I}_{a0} \overline{z}_0
+	\end{cases}
+	$$
+
+A tabela abaixo apresenta as equações para as faltas desbalanceadas após a execução do procedimento apresenteado:
+
+|**Falta**|**Seq. Positiva ($\dot{I}_{f}^{1}$)**|**Seq. Negativa ($\dot{I}_{f}^{2}$)**|**Seq. Zero ($\dot{I}_{f}^{0}$)**|
+|-----|:-----------:|:-----------:|:-------:|
+|**F-T**| $\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f}}$ | $\dot{I}_{f}^{1}$ | $\dot{I}_{f}^{1}$|
+|**F-F**| $\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{f}}$ | $- \dot{I}_{f}^{1}$ | $0{,}0$|
+|**F-F-T**| $\dfrac{\dot{E}_i \left( \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f} \right)}{\overline{z}_{ii}^{1} \overline{z}_{ii}^{2} + 3 \overline{z}_{ii}^{2} \overline{z}_{f} + \overline{z}_{ii}^{2} \overline{z}_{ii}^{0} + 3 \overline{z}_{ii}^{1} \overline{z}_{f} + \overline{z}_{ii}^{1} \overline{z}_{ii}^{0} }$ | $- \dfrac{\dot{E}_i - \overline{z}_{ii}^{1} \dot{I}_{f}^{1}}{\overline{z}_{ii}^{2}}$ | $- \dfrac{\dot{E}_i - \overline{z}_{ii}^{1} \dot{I}_{f}^{1}}{\overline{z}_{ii}^{0} + 3 \overline{z}_{f}}$|
+
+Para obter os valores em a-b-c é usada a equação matricial apresentada anteriormente, encerrando o cálculo de curto-circuito.
+
+## Referências
+1. ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: https://doi.org/10.1002/9781118878286
+1. STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.
+1. ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: https://doi.org/10.1109/9780470545577
+1. FORTESCUE, C. L. Method of Symmetrical Coordinates Applied to the Solution of Polyphase Networks. Trans. AIEE, v. 37, p.1027-1140, 1918. doi: https://doi.org/10.1109/T-AIEE.1918.4765570
+1. ANDERSON, P. M. Analysis of faulted power systems. New York: IEEE Press, 1995.
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/gain.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/gain.md
new file mode 100644
index 0000000..0e4edf1
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/gain.md
@@ -0,0 +1,12 @@
+---
+id: gain
+title: Ganho
+sidebar_label: Ganho
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+O bloco de ganho multiplica o valor de entrada por uma constante definida pelo usuário, possuindo comportamento idêntico à combinação de uma [constante](const) e um bloco [multiplicador](multiplier).
+
+Esse valor de ganho permanece fixo durante todo o tempo de simulação e é identificado no elemento gráfico exibido no editor de elementos de controle.
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/graphViewer.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/graphViewer.md
new file mode 100644
index 0000000..952e252
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/graphViewer.md
@@ -0,0 +1,43 @@
+---
+id: graphViewer
+title: Visualizador de Gráficos
+sidebar_label: Visualizador de Gráficos
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+Para exibição dos resultados originados dos cálculos de [estabilidade](stability) e [varredura de frequências](harmonics), desenvolveu-se um visualizador de gráficos, como mostra a figura abaixo:
+
+<div><center><img src={useBaseUrl("images/graphViewer.png")} alt="Visualizador de gráficos do PSP-UFU" title="Visualizador de gráficos do PSP-UFU" /></center></div>
+
+## Criação, manipulação e navegação de curvas
+Tal visualizador utiliza a biblioteca [wxMathPlot](https://wxmathplot.sourceforge.io/), e permite ao usuário imprimir todas as curvas no tempo originadas dos resultados de [estabilidade](stability) e [varredura de frequências](harmonics), acessados em dados hierárquicos em forma de árvore, identificados pelo tipo e nome do elemento de potência, além do tipo de curva.
+
+:::info Informação
+As curvas podem ser plotadas com **clique duplo** sobre ela ou ao selecionar a caixa "Plotar" nas propriedades do traço.
+:::
+
+Os gráficos impressos podem ser caracterizados pela edição de suas propriedades do traço, ou seja, pode-se alterar a cor, espessura e tipo de traço (seis tipos disponíveis). Por padrão o eixo vertical é o tempo (em estabilidade) ou frequência (varredura de frequências), porém o usuário pode alterar para qualquer curva disponível.
+
+Várias propriedades gerais do gráfico podem ser editadas: título do gráfico, rótulos dos eixos x e y, margens e limites inferiores e superiores de ambos os eixos.
+
+O menu **Visualização** permite a inserção de exibição de legenda, linhas de grade e coordenadas do ponteiro do mouse (utilizada para destacar pontos na curva), além da possibilidade de alternar o tema de impressão entre escuro e claro, facilitando a visualização ou exportação do gráfico.
+
+No menu **Arquivo**, pode-se salvar o gráfico na forma de imagem (*.png*, *.jpg* ou *.bmp*) ou enviá-lo diretamente para a área de transferência. No mesmo menu também é permitido a exportação no formato *[CSV (Comma-separated Values)](https://en.wikipedia.org/wiki/Comma-separated_values)*, o qual poderá ser utilizado em outros visualizadores de gráficos, como o Excel ou o [gnuplot](http://www.gnuplot.info/).
+
+A área de plotagem permite a navegação utilizando ferramentas de zoom, por área de seleção e comandos de mouse, e arrasto das curvas impressas.
+
+:::tip Dica
+Ao clicar com o botão direito sobre a área de plotagem as opções de manipulação são exibidas por meio de um menu de contexto.
+:::
+
+Os seguintes **comandos de mouse** são utilizados para navegação das curvas:
+
+|Comando|Ação|
+|-------|----|
+|Botão esquerdo pressionado + Selecionar área|Seleciona a área de zoom|
+|Botão direito pressionado + Mover mouse|Arrasta a curva|
+|Roda do mouse|Arrasto vertical|
+|Roda do mouse + Shift|Arrasto horizontal|
+|Roda do mouse + Ctrl|Mais/menos zoom|
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/harmSource.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/harmSource.md
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--- /dev/null
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@@ -0,0 +1,41 @@
+---
+id: harmSource
+title: Fonte de Corrente Harmônica
+sidebar_label: Fonte de Corrente Harmônica
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+> (**Fonte de perturbação harmônica**) Parte do sistema elétrico de potência ou das instalações a ele conectadas, que causa distorção harmônica da forma de onda da corrente e / ou tensão. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=614-01-16).
+
+## Fonte de corrente harmônica no PSP-UFU
+As fontes de corrente harmônica são responsáveis pela injeção de correntes harmônicas e são utilizadas pela ferramenta de cálculo de [distorções harmônicas](harmonics). Uma lista de correntes harmônicas pode ser inserida em um mesmo elemento, conforme é exibido na o formulário de edição de dados.
+
+:::caution Atenção!
+A presença da fonte de corrente harmônica **não é considerada** nos estudos de [fluxo de carga](powerFlow), [curto-circuito](fault) e [estabilidade](stability).
+:::
+
+## Formulário de edição das fontes de corrente harmônica
+A imagem abaixo apresenta o formulário de inserção/alteração de dados da fonte de corrente harmônica:
+
+<div><center><img src={useBaseUrl("images/harmSourceForm.png")} alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU" /></center></div>
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+#### Ordem
+Especifica a ordem harmônica a ser inserida na lista de correntes harmônicas do elemento. Caso a ordem harmônica já exista na lista, ela será substituída pelos novos parâmetros.
+
+#### Corrente
+Módulo da corrente harmônica.
+
+#### Unidade
+Unidade do módulo de corrente harmônica, em A ou $p.u.$
+
+#### Ângulo
+Ângulo da corrente harmônica, em graus.
+
+#### Adicionar / Remover
+Insere ou modifica a nova ordem harmônica ao clicar em “Adicionar”. Retira a ordem harmônica selecionada da lista ao clicar em “Remover”.
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/harmonics.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/harmonics.md
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index 0000000..64366d9
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/harmonics.md
@@ -0,0 +1,116 @@
+---
+id: harmonics
+title: Harmônicos
+sidebar_label: Harmônicos
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+Duas ferramentas relacionadas a harmônicos foram desenvolvidas no PSP-UFU:
+- Distorção Harmônica Total de Tensão (THD, do inglês *Total Harmonic Distortion*);
+- Resposta na Frequência.
+
+A ferramenta de **[Distorção Harmônica Total](https://en.wikipedia.org/wiki/Total_harmonic_distortion)** calcula as tensões harmônicas causadas por [fontes de corrente harmônicas](harmSource), assim como o THD de todos os barramentos do sistema.
+
+A ferramenta de **Resposta na Frequência** (ou análise de varredura de frequência) envolve a variação da impedância da rede em um espectro de frequências observado a partir de um certo barramento.
+
+:::info Informação
+A análise de varredura de frequência é amplamente usada no projeto de filtros harmônicos.
+:::
+
+:::tip Dica
+Um arranjo série de uma [linha](line) sem as susceptâncias *shunt*, um [barramento](bus) e um [capacitor](capacitor) pode ser utilizado para fabricar um filtro passivo nos [estudos harmônicos](harmonics). Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.
+:::
+
+:::caution Atenção!
+Na versão atual do programa não são consideradas as alterações das resistências do sistema causado pelo efeito pelicular. Versões futuras irão contemplar tal característica.
+:::
+
+## Execução do cálculo de harmônicos no PSP-UFU
+Após a construção do diagrama unifilar no [editor de potência](powerEditor), a execução do cálculo das distorções harmônicas é realizada no [menu Simulação](mainScreen#menu-ribbon) clicando no botão **Distorções Harmônicas**. Para acessar a ferramenta de de varredura de frequência, basta clicar no botão **Resposta na Frequência**.
+
+<div><center><img src={useBaseUrl("images/menuSimulationHamonics.svg")} alt="Acesso às ferramentas de estudo harmônico" title="Acesso às ferramentas de estudo harmônico" /></center></div>
+
+### Distorções Harmônicas
+Ao clicar sobre o botão "Distorções Harmônicas" as distorções causadas pelas [fontes de corrente harmônica](harmSource) são calculadas em todos os barramentos do sistema.
+
+:::caution Atenção!
+Caso não forem inseridas [fontes de corrente harmônica](harmSource) no sistema de potência, a distorção de tensão de todas as barras será $0{,}0~\%$.
+:::
+
+**Outra possibilidade** é a execução por meio do cálculo contínuo, também presente no [menu Simulação](mainScreen#menu-ribbon) e seu acionamento é realizado co clicar no botão **Habilitar solução**. Com essa opção, os cálculos estáticos selecionados nas [configurações de simulação](simulationConfig) são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).
+
+:::warning Cuidado!
+Os cálculos dos níveis de THD **não são habilitados por padrão** no cálculo contínuo e devem ser inseridos nas [configurações de simulação](simulationConfig).
+:::
+
+Os resultados das distorções harmônicas são exibidos nos [elementos de texto vinculados](text) e ao posicionar o mouse sobre um barramento.
+
+### Resposta na Frequência
+Ao clicar sobre o botão "Resposta na Frequência" será exibido um formulário para inserção dos parâmetros da ferramenta:
+
+<div><center><img src={useBaseUrl("images/injHarmCurrent.png")} alt="Acesso à ferramentas resposta na frequência" title="Acesso à ferramentas resposta na frequência" /></center></div>
+
+#### Frequência inicial
+Define a frequência inicial da varredura.
+
+#### Frequência final
+Define a frequência final da varredura.
+
+#### Passo de frequência
+Define o passo de incremento da frequência. Passos menores irão gerar gráficos mais precisos, porém podem aumentar muito o tempo de execução.
+
+#### Corrente injetada
+Define o barramento o qual será injetada a corrente harmônica para análise.
+:::tip Dica
+Esse campo pode ser entendido como: "*A resposta das impedâncias harmônicas vistas pelos barramentos do sistema caso tenha uma fonte de corrente harmônica na barra selecionada*".
+:::
+
+#### Botão Executar
+Ao clicar no botão executar a varredura será executada e os resultados exibidos no [vizualizador de gráficos](graphViewer), como mostra a imagem abaixo.
+
+<div><center><img src={useBaseUrl("images/freqScanPlot.png")} alt="Exemplo de resposta na frequência" title="Exemplo de resposta na frequência" /></center></div>
+
+:::warning Cuidado!
+Para visualizar a varredura da impedância harmônica de um barramento, essa opção deve ser habilitada em seu [formulário de edição de dados](bus#imprimir-impedância-harmônica-da-barra).
+:::
+
+## Cálculos das ferramentas harmônicas
+
+Ambos cálculos são realizados a partir da seguinte equação matricial:
+$$
+[\dot{I}]^h= [Y_{bus}]^h [\dot{V}]^h
+$$
+Em que:
+- $[\dot{I}]^h$	Vetor das correntes harmônicas injetadas nas barras
+- $[Y_{bus}]^h$	Matriz admitância harmônica de barras
+- $[\dot{V}]^h$	Vetor das tensões harmônicas nas barras
+- $h$	Ordem harmônica
+
+O vetor das correntes harmônicas injetadas é definido no programa utilizando um elemento de potência, chamado “fonte de corrente harmônica”, em que o usuário pode criar uma lista de correntes injetadas (em A e/ou $p.u.$) em um barramento.
+
+O programa define automaticamente as ordens harmônicas na simulação analisando todas as listas de fontes de corrente harmônicas previamente aos cálculos. Com isso são calculadas as admitâncias harmônicas necessárias de cada elemento pela multiplicação das reatâncias indutivas ($x_{l}^1$) e susceptâncias capacitivas ($b_{c}^1$) fundamentais de cada elemento pelas ordens harmônicas:
+$$
+x_{l}^h=h×x_l^1\\
+b_{c}^h=h×b_c^1
+$$
+
+Uma vez calculadas as admitâncias harmônicas, são utilizados os mesmos modelos e algoritmos convencionais para construção da $Y_{bus}^h$.
+
+A equação matricial é resolvida usando o método de eliminação gaussiana para evitar a inversão de cada matriz de admitância harmônica. Este procedimento torna o cálculo das tensões harmônicas computacionalmente eficiente.
+
+Uma vez calculadas as tensões harmônicas, a THD em uma barra genérica $i$ pode ser definida por:
+$$
+\text{THD}_i=\dfrac{\sum_{h=2}^{n} V_{i}^h}{V_i^1}
+$$
+
+Na ferramenta de Resposta na Frequência uma corrente senoidal é injetada na barra em uma faixa de frequências e o conjunto de equações matriciais é usado para calcular a resposta da tensão. Este cálculo é repetido em etapas discretas cobrindo o espectro de frequência especificado.
+
+O programa usa a matriz admitância de sequência positiva e uma corrente injetada de $1{,}0 p.u.$ também de sequência positiva na equação matricial para calcular diretamente as impedâncias em p.u.
+
+## Referências
+1. ARRILLAGA, J.; WATSON, N. R. Power System Harmonics. John Wiley & Sons; Chichester, 2003. doi: https://doi.org/10.1002/0470871229
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/heatmap.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/heatmap.md
new file mode 100644
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+---
+id: heatmap
+title: Mapa de Tensão
+sidebar_label: Mapa de Tensão
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+A ferramenta Mapa de Tensão utiliza o concento de [mapa de calor](https://en.wikipedia.org/wiki/Heat_map) para exibir os valores de tensão de todas as barras do sistema por meio de cores.
+
+<video autoPlay loop muted playsInline controls>
+  <source src= {useBaseUrl("videos/heatmap.mp4")} type="video/mp4" />
+  <source src= {useBaseUrl("videos/heatmap.webm")} type="video/webm" />
+</video>
+
+:::tip Dica
+Essa ferramenta é útil para identificação de barras com sub e sobretensões no sistema de potência, assim como o efeito da inserção / remoção de elementos do sistema.
+:::
+
+A legenda que correlaciona os valores do módulo da tensão com as cores são exibidas à esquerda da área de trabalho.
+
+## Habilitando a ferramenta Mapa de Tensão
+
+A ferramenta pode ser habilitada ou desabilitada no [submenu Ribbon **Ferramentas**](mainScreen#menu-ribbon) clicando no botão **Mapa de tensão** ou pela tecla de atalho "Ctrl + Shift + H".
+
+As configurações da ferramenta podem ser também acessadas no [submenu Ribbon **Ferramentas**](mainScreen#menu-ribbon) clicando no botão **Configurações do projeto**
+
+<div><center><img src={useBaseUrl("images/heatmapForm.png")} alt="Configurações da ferramenta Mapa de Tensão" title="Configurações da ferramenta Mapa de Tensão" /></center></div>
+
+:::info Informação
+Na versão atual do programa, somente as opções do mapa de tensão são exibidas nas configurações de projeto.
+:::
+
+Os limites superiores e inferiores, em $p.u.$ são editados nesse formulário.
+
+Alternativamente, o programa pode calcular automaticamente os limites. Nesse caso, o limite superior será a maior tensão do circuito e o limite inferior será a menor tensão do circuito.
+
+:::warning Cuidado!
+O cálculo automático dos limites pode prejudicar a identificação de barras com sub e sobretensões no sistema.
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/indMotor.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/indMotor.md
new file mode 100644
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--- /dev/null
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+---
+id: indMotor
+title: Motor de Indução
+sidebar_label: Motor de Indução
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+> Uma máquina assíncrona da qual apenas um enrolamento é energizado. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=411-31-10).
+
+## Motor de indução trifásico no PSP-UFU
+No PSP-UFU, os motores de indução são contemplados nos estudos de [fluxo de carga](powerFLow) e no estudo de [estabilidade](stability).
+
+:::info Informação
+Os dados de estabilidade da máquina de indução são utilizados em conjunto com o fluxo de carga, **calculando de forma correta a potência reativa das máquinas** e consequentemente a tensão no barramento conectado.
+:::
+
+:::warning Cuidado!
+Ao marcar a opção "Calcular a potência reativa no fluxo de carga", devem ser inseridos os dados corretos na aba "Estabilidade", caso contrário, será atribuída uma potência reativa incorreta no motor.
+:::
+
+### Motor de indução trifásico no cálculo do fluxo de carga
+A figura abaixo apresenta o modelo do motor de indução trifásico (MIT) de gaiola simples.
+
+<div><center><img src={useBaseUrl("images/indMotorModel.svg")} alt="Circuito equivalente do motor de indução" title="Circuito equivalente do motor de indução" /></center></div>
+
+As potências ativa e reativa podem ser calculadas em relação às variáveis e parâmetros do motor em $p.u.$ como:
+
+$$
+P = \frac{V^2 \left\{ \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]+K_1 \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\
+Q = \frac{-V^2 \left\{ K_1 \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]- \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\
+~\\
+K_1=x_2+x_m
+$$
+
+Como pode ser observado nas equações acima, existem quatro variáveis e somente duas equações. Na prática, as variáveis podem ser reduzidas a três, uma vez que o módulo da tensão (V) é obtido nos resultados do [fluxo de potência](powerFlow). Para resolver as equações é necessário definir uma variável adicional. A variável escolhida como fixa no PSP-UFU é a potência ativa (P), por fornecer resultados numericamente corretos e adequados para motores em situações de estabilidade.
+
+Portanto, nesse modelo estático, a potência ativa é mantida constante durante o cálculo do fluxo de carga e o escorregamento (s) é atualizado em cada iteração. A equação da potência ativa pode ser reescrita em relação ao escorregamento:
+
+$$
+\left(\frac{r_2}{s} \right)^2 A + \left(\frac{r_2}{s} \right) B + C = 0
+$$
+Em que:
+$$
+A = P \left( r_1^2 + K_3^2 \right) - V^2 r_1\\
+B = 2P(r_1 K_2 + K_3 K_4) - V^2 \left(K_2 + K_1 K_3 \right)\\
+C = P \left( K_2^2 + K_4^2 \right) - V^2 K_1 K_4\\
+K_2 = -x_1 K_1 -x_2 x_m\\
+K_3 = x_m + x_1\\
+K_4 = r_1 K_1
+$$
+
+Esse modelo pode ser inserido na solução do [fluxo de carga](powerFlow) seguindo os seguintes passos:
+
+1. As constantes $K_1$ a $K_4$ são inicialmente calculadas . Esses valores são mantidos constantes durante toda a solução;
+2. Em cada iteração são calculados os coeficientes $A$, $B$ e $C$ utilizando o valor atualizado de $V$;
+3. A equação quadrática é resolvida e dois valores de $\left(\frac{r_2}{s} \right)$ são obtidos, em que o maior deles é escolhido por estar na região estável da característica toque-escorregamento do motor;
+4. Utilizando o novo valor de $\left(\frac{r_2}{s} \right)$, a potência reativa ($Q$) é obtida. O vetor de potências é então atualizado e os procedimentos convencionais de solução do fluxo de potência são realizados.
+
+Os passos de 2 a 4 são repetidos até que se obtenha a convergência.
+
+:::caution Atenção!
+No PSP-UFU, os motores de indução não são considerados no cálculo de curto-circuito.
+:::
+
+### Motor de indução trifásico no estudo de estabilidade
+Uma importante carga dinâmica são os motores de indução, uma vez que correspondem a uma parcela significativa das cargas presentes no sistema elétrico. O [modelo da máquina de indução apresentado anteriormente](indMotor#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga), a qual pode ser utilizada tanto como motor quanto como gerador, é bem estabelecida na literatura.
+
+Como descrito na [seção anterior](indMotor#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga), a inicialização dessa máquina é realizada em conjunto com o fluxo de potência, visto que a potência reativa exigida pela máquina de indução é dependente dos parâmetros do motor, assim como a tensão do seu barramento. Essa abordagem é necessária, pois métodos convencionais conduzem a resultados errôneos em sistemas altamente carregados.
+
+É necessário expressar a equação de movimento da máquina de indução em termos de torque e não potência, como é realizado com as máquinas síncronas. A simetria do rotor também faz com que a posição angular não seja importante e o escorregamento ($s$) é utilizado no lugar da velocidade ($\omega$), em que:
+
+$$
+s = frac{\Omega_0 - \omega}{\Omega_0}
+$$
+
+Desprezando as perdas por atrito e ventilação e a potência no eixo suave, as equações mecânicas do motor são expressas da seguinte forma:
+
+$$
+T_m = A -Bs + Cs^2\\
+T_e = \frac{Re\left\{ \dot{E}\dot{I}^* \right\}}{\Omega_b}\\
+\frac{ds}{dt} = \frac{\left( T_m - T_e \right)}{2H}
+$$
+Em que:
+- $T_m$ é o torque mecânico;
+- $T_e$ é o torque elétrico;
+- $H$ é a inércia do conjunto motor - carga mecânica
+
+Os termos $A$, $B$ e $C$ são termos que definem o comportamento do torque mecânico da carga de acordo com o escorregamento. O torque mecânico normalmente varia com a **velocidade**, podendo ser expressa proporcionalmente com a seguinte equação quadrática:
+
+$$
+T_m \propto a + b\omega + c\omega^2
+$$
+Em que:
+$$
+\begin{cases}
+	A \propto a + b + c \\
+	B \propto b + 2c \\
+	C \propto c
+\end{cases}
+$$
+
+As equações elétricas do motor de indução de gaiola simples são baseadas no circuito equivalente da [figura anterior](indMotor#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga). De forma semelhante ao modelo transitório da máquina síncrona, o motor de indução pode ser modelado pelo circuito equivalente de Thevenin de tensão transitória $E'$ atrás de uma resistência do estator $r_1$ e uma reatância transitória $x'$. A reatância transitória aparente de rotor bloqueado é dada por:
+$$
+x' = x_1 + \frac{x_2 x_m}{x_2 + x_m}
+$$
+
+A constante de tempo de circuito aberto do rotor ($T_0'$) é:
+$$
+T_0' = \frac{x_2 + x_m}{\Omega_b r_2}
+$$
+
+E a reatância de circuito aberto é:
+$$
+x_0 = x_1 + x_m
+$$
+
+Uma vez que as reatâncias não são afetadas pela posição do rotor, as EADs do motor de indução podem ser expressar diretamente por componentes reais ($r$) e imaginárias ($m$). Portanto, a descrição completa desse modelo é representada pelo seguinte sistema de equações algébrico-diferenciais:
+$$
+V_r - E_r' = r_1 I_r - x' I_m\\
+V_m - E_m' = r_1 I_m - x' I_m\\
+\frac{dE_r'}{dt} = \Omega_b s E_m' - \frac{E_r' + \left( x_0 - x' \right) I_m}{T_0'}\\
+\frac{dE_m'}{dt} = \Omega_b s E_r' - \frac{E_m' i \left( x_0 - x' \right) I_r}{T_0'}
+$$
+
+## Formulário de edição dos geradores síncronos
+A imagem abaixo apresenta o formulário de inserção/alteração de dados dos motores de indução:
+
+<div><center><img src={useBaseUrl("images/indMotorForm.png")} alt="Formulário dos motores de indução no PSP-UFU" title="Formulário dos motores de indução no PSP-UFU" /></center></div>
+
+No formulário pode ser observado o botão "Chaveamento" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do motor durante o estudo de [estabilidade](stability).
+
+<div><center><img src={useBaseUrl("images/indMotorSw.png")} alt="Formulário de chaveamento do motores de indução" title="Formulário de chaveamento do motores de indução" /></center></div>
+
+<Tabs
+  groupId="indMotor-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Estabilidade', value: 'stability'},
+    {label: 'Botão Chaveamento', value: 'switching'},
+  ]
+}>
+<TabItem value="general">
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+Todos os componentes de potência do PSP-UFU possuem esse campo.
+
+#### Potência nominal
+Potência nominal do gerador, inserida em MVA, kVA ou VA.
+
+Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.
+
+#### Potências ativa e reativa
+Potências ativa (inserida em W, kW, MW ou $p.u.$) e reativa (inserida em var, kvar, Mvar ou $p.u.$) do motor.
+
+:::caution Atenção!
+Caso a opção “Calcular a potência reativa no fluxo de carga” esteja ativada, o campo de potência reativa é desativado para edição.
+:::
+
+#### Calcular a potência reativa no fluxo de carga
+Caso essa opção seja marcada, o programa irá utilizar os dados fornecidos no formulário de estabilidade para calcular a potência reativa do motor durante o processo iterativo do fluxo de carga.
+
+:::warning Cuidado!
+Caso essa opção não seja utilizada o motor será considerado uma [carga de potência constante](load) no estudo de [fluxo de carga](powerFlow).
+
+A não utilização dessa opção poderá gerar erros de regime permamente no estudo de [estabilidade](stability).
+:::
+
+#### Utilizar potência nominal como base
+Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a potência base do sistema.
+
+</TabItem>
+<TabItem value="stability">
+
+#### Imprimir dados da máquina de indução
+Exibe os dados do gerador síncrono nos gráficos no tempo. Os seguintes dados são exibidos:
+- tensão terminal
+- potências ativa e reativas
+- corrente, torque elétrico
+- torque mecânico
+- velocidade
+- escorregamento
+
+#### Inércia
+Inércia do conjunto motor-carga, em s.
+
+#### Resistência e reatância do estator
+Resistência e reatância do estator do motor de indução, em $p.u.$
+
+#### Resistência e reatância do rotor
+Resistência e reatância do rotor do motor de indução, em $p.u.$
+
+#### Reatância de magnetização
+Reatância do ramo de magnetização do motor de indução, em $p.u.$
+
+#### Utilizar fator de gaiola
+Permite a inserção de um fator de gaiola ($K_g$) no motor de indução. Tal fator eleva o valor da resistência do rotor em condições de grande escorregamento, como ocorre em motores de gaiola dupla ou gaiola profunda. A resistência do rotor segue a seguinte equação:
+
+$$
+r_2 = r_{2_0} \left( 1+K_g s \right)
+$$
+Em que:
+- $r_{2_0}$ é a resistência do rotor inserida no formulário.
+
+#### Característica da carga
+Descreve a característica do torque da carga mecânica no eixo do motor seguindo a equação do torque mecânico.
+$$
+T_m \propto \bold{a} + \bold{b}\omega + \bold{c}\omega^2
+$$
+
+:::warning Cuidado!
+A soma dos coeficientes ($a + b + c$) deve ser unitária.
+:::
+
+</TabItem>
+<TabItem value="switching">
+
+O botão "Chaveamento" irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do gerador durante o estudo de [estabilidade](stability).
+
+Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões "Adicionar" e "Remover", respectivamente.
+
+</TabItem>
+</Tabs>
+
+## Referências
+1. SÁNCHEZ, J. C.; OLIVARES, T. I. A.; ORTIZ, G. R.; VEGA, D. R. Induction Motor Static Models for Power Flow and Voltage Stability Studies. In: IEEE Power and Energy Society General Meeting, 2012, San Diego. doi: https://doi.org/10.1109/PESGM.2012.6345618
+2. IEEE Std 399-1997. IEEE Recommended Practice for Industrial and Commercial Power Systems Analysis (Brown Book). IEEE, New York, ago. 1998. doi: https://doi.org/10.1109/IEEESTD.1998.88568
+3. MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: https://doi.org/10.1007/978-3-642-13669-6
+4. ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: https://doi.org/10.1002/9781118878286
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/inductor.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/inductor.md
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@@ -0,0 +1,56 @@
+---
+id: inductor
+title: Indutor
+sidebar_label: Indutor
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+>Um reator destinado à conexão de derivação em uma rede para compensar a corrente capacitiva. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=421-01-03).
+
+## Indutor no PSP-UFU
+O elemento indutor (ou reator) representa, geralmente, reator *shunt* no circuito do PSP-UFU.
+
+Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.
+
+## Formulário de edição dos indutores
+A imagem abaixo apresenta o formulário de inserção/alteração de dados dos indutores:
+
+<div><center><img src={useBaseUrl("images/indForm.png")} alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU" /></center></div>
+
+Além do único contexto geral, pode ser observado o botão "Estabilidade" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do indutor durante o estudo de [estabilidade](stability).
+
+<div><center><img src={useBaseUrl("images/indSw.png")} alt="Formulário de chaveamento do indutor" title="Formulário de chaveamento do indutor" /></center></div>
+
+<Tabs
+  groupId="transformer-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Botão Estabilidade', value: 'stability'},
+  ]
+}>
+<TabItem value="general">
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+Todos os componentes de potência do PSP-UFU possuem esse campo.
+
+#### Potência reativa
+Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou $p.u.$ (na base de potência do sistema). 
+
+</TabItem>
+<TabItem value="stability">
+
+O botão "Estabilidade" irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de [estabilidade](stability).
+
+Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões "Adicionar" e "Remover", respectivamente.
+
+</TabItem>
+</Tabs>
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/installation.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/installation.md
new file mode 100644
index 0000000..4b937e9
--- /dev/null
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@@ -0,0 +1,77 @@
+---
+id: installation
+title: Instalação
+sidebar_label: Instalação
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+## Download
+
+O download do PSP-UFU pode ser realizado pelo ["*last release*" do repositório do PSP-UFU no GitHub](https://github.com/Thales1330/PSP/releases/latest).
+
+Ao acessar ao link, role a página até encontrar os *Assets*:
+
+<div><center><img src={useBaseUrl("images/download.png")} alt="Assets download" title="Informações da Figura" /></center></div>
+
+Como o PSP-UFU é um *software* multiplataforma, você deve baixar o arquivo correto:
+- Para o sistema operacional [Windows](installation#windows) baixe os arquivos *.exe* (recomendado) ou *.zip*.
+- Para os sistemas [Linux](installation#linux) baseados no Debian (por exemplo, Ubuntu, Mint) *64 bits* baixe o pacote de instalação *.deb*.
+
+:::tip Dica
+Se você está interessado em contribuir com o PSP-UFU construindo-o para outros sistemas operacionais (principalmente macOS!) entre em contato pelo [GitHub](https://github.com/Thales1330/PSP/issues) ou [Twitter](https://twitter.com/PspUfu).
+:::
+
+Na sequência são apresentados o passo-a-passo para instalação dos SOs [Windows](installation#windows) e [Linux](installation#linux). Também são apresentadas as soluções para os erros mais recorrentes.
+
+## Windows
+Após baixar o arquivo *.exe* avance em cada passo do instalador. O executável irá solicitar automaticamente a instalação do **Microsoft Visual C++ Redistribuível** e, caso não esteja instalado no seu computador, realize os procedimentos do programa.
+
+Caso o executável falhe em abrir, baixe o arquivo *.zip* e descompacte em alguma pasta do seu sistema. Nesse caso é necessário baixar e instalar manualmente o [**Microsoft Visual C++ Redistribuível 32 bits**](https://aka.ms/vs/16/release/vc_redist.x86.exe).
+
+Após a conclusão da instalação abra o PSP-UFU para verificar a correta instalação. O executável está presente na pasta "*<local de instalação>/PSP-UFU/bin/PSP-UFU.exe*".
+
+### Problemas recorrentes no Windows
+- Ao baixar o navegador pode **falsamente** sinalizar que é um arquivo perigoso, basta clicar em "Manter arquivo";
+- Ao abrir o instalador, novamente o Windows sinaliza que o arquivo é perigoso, basta clicar em "Executar assim mesmo";
+- Ao abrir o programa e alterar o seu idioma, uma mensagem de erro é exibida. Para resolver esse problema basta executar o PSP-UFU como administrador (clicar com botão direito no atalho do programa e posteriormente em executar como administrador). Altere novamente o idioma e reinicie o programa;
+- Ao criar um novo projeto a tela de trabalho não é exibida ou é exibida uma tela preta. Vá nas opções gerais e altere o renderizador para "Device Context". Caso apareça uma mensagem de erro, execute o programa como administrador e repita o processo. Reinicie o programa.
+
+
+## Linux
+A instalação no sistema Linux é facilmente realizado pelo pacote de instalação automático *.deb*.
+:::caution Atenção!
+Note que essa versão para Linux está defasada e algumas funcionalidades do programa podem não estar presentes.
+
+Caso queira contribuir com o projeto e construir a versão atual no sistema Linux baseado em Debian ou em outras distribuições entre em contato pelo [GitHub](https://github.com/Thales1330/PSP/issues) ou [Twitter](https://twitter.com/PspUfu).
+:::
+
+### Problemas recorrentes no Linux
+Pode ser necessário adicionar o [diretório das bibliotecas do wxWidgets](https://wiki.wxwidgets.org/Compiling_and_getting_started#Running_wxWidgets_projects) nas variáveis de ambiente.
+Para isso, siga **UM** dos métodos apresentados abaixo:
+
+#### *Método I (recomendado)*
+Insira o seguinte comando no arquivo **~/.bashrc**:
+```shell
+echo "export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:/usr/local/lib" >> ~/.bashrc
+```
+
+#### *Método II (não recomendado)*
+:::warning Cuidado!
+Utilizando esse método os passos abaixo devem ser executados **todas as vezes** que você quiser abrir o PSP-UFU.
+:::
+1. Abra o terminal e acesse o diretório onde está instalado o PSP-UFU:
+```shell
+cd /usr/local/bin
+```
+2. Insira as bibliotecas compartilhadas do wxWidgets nas variáveis de ambiente:
+```shell
+export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:/usr/local/lib
+```
+3. Então, execute o PSP-UFU na **mesma seção** do terminal:
+```shell
+./PSP-UFU
+```
+
+:::caution Atenção!
+Verifique se o computador possui as unidades de vídeo instaladas corretamente, caso contrário, o programa não exibirá a área de trabalho para criar e editar diagramas de linha única ou diagramas de blocos de controle.
+:::
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/io.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/io.md
new file mode 100644
index 0000000..5af53bd
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/io.md
@@ -0,0 +1,43 @@
+---
+id: io
+title: Entrada / Saída
+sidebar_label: Entrada / Saída
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+As entradas e saída são variáveis relacionadas às máquinas síncronas. Cada tipo de entrada e saída está relacionada a um regulador da máquina: **Regulador de Tensão (AVR, do inglês *Automatic Voltage Regulator*)** e **Regulador de Velocidade (RV)**.
+
+:::warning Cuidado!
+O sistema de controle deve ter **ao menos uma entrada e uma saída**.
+:::
+
+As entradas e saídas do sistema de controle são definidas por esses blocos, os quais são distintos para cada tipo de escopo. A figura abaixo apresenta um exemplo de um formulário de edição das entradas e saídas de um AVR:
+
+<div><center><img src={useBaseUrl("images/ioForm.png")} alt="Formulário de edição de entradas e saídas de sistemas de controle de um AVR" title="Formulário de edição de entradas e saídas de sistemas de controle de um AVR" /></center></div>
+
+## Variáveis de entrada e saída
+
+O usuário deve sinalizar o tipo de bloco entre entrada e saída e, então, selecionar em uma lista a variável desejada. As seguintes variáveis de controle estão atualmente disponíveis no programa:
+
+- **Tensão terminal** (*entrada: AVR*): Módulo da tensão no barramento da máquina síncrona, em $p.u.$, variável no tempo. Essa variável é normalmente utilizada no cálculo do erro da tensão de referência do AVR;
+- **Velocidade** (*entrada: AVR e RV*): Velocidade da máquina síncrona, em rad/s, variável no tempo. Normalmente utilizada no cálculo do erro de velocidade nos reguladores de velocidade, além de entrada do PSS em AVRs;
+- **Potência ativa e reativa** (*entrada: AVR*): Potência ativa fornecida pela máquina síncrona, em $p.u.$, variável no tempo. Normalmente utilizada como entrada do PSS (potência ativa) e controle de sub e sobrecorrente de excitação nos AVRs;
+- **Tensão terminal inicial** (*entrada: AVR*): Módulo da tensão no barramento da máquina síncrona prévia ao estudo dinâmico originado do fluxo de carga, em $p.u.$, fixo no tempo. Essa variável está normalmente associada à referência de tensão do AVR;
+- **Velocidade inicial** (*entrada: AVR e RV*): Velocidade do sistema ($2 \pi f_{ref}$), definida nas [opções de simulação](simulationConfig), em rad/s, fixa no tempo. Normalmente utiliza-se essa variável como referência de velocidade em RVs e normalização da velocidade;
+- **Potência mecânica inicial** (*entrada: RV*): Potência mecânica inicial, calculada após a inicialização das máquinas síncronas com os dados originados do fluxo de carga, em $p.u.$, fixa no tempo. Normalmente é utilizada como referência de potência mecânica nos reguladores de velocidade;
+- **Variação de velocidade e potência ativa** (*entrada: AVR*): Cálculo da variação dessas entradas entre os passos de integração normalizada pelo passo de integração, conforme a equação:
+	$$
+	\Delta x = \frac{x_n-x_{n-1}}{h}
+	$$
+	Em que:
+	- $\Delta x$	é a variação da entrada (velocidade ou potência ativa);
+	- $x_n$ e $x_{n-1}$	é a variável no passo atual e anterior, respectivamente;
+	- $h$	é o passo de integração.
+	
+	:::info Informação
+	A normalização é necessária para a correta utilização da razão de passo de controle, definida nas [configurações de simulação](simulationConfig).
+	:::
+- **Tensão de campo** (*saída: AVR*): Define a tensão aplicada ao campo na máquina síncrona, em $p.u.$ Utilizada como saída dos AVRs, controlando principalmente tensão no barramento conectado e/ou fator de potência da máquina;
+- **Potência mecânica** (*saída: RV*): Define qual a potência mecânica aplicada no eixo da máquina síncrona. Utilizada como saída dos reguladores de velocidade, controlando principalmente a potência ativa injetada pela máquina e sua frequência.
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/limiter.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/limiter.md
new file mode 100644
index 0000000..adb4e04
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/limiter.md
@@ -0,0 +1,35 @@
+---
+id: limiter
+title: Limitador absoluto
+sidebar_label: Limitador absoluto
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+Os blocos de limite absoluto cumprem a função de restringir os valores de entrada de acordo com os limites máximos e mínimos inseridos pelo usuário. Caso a entrada ultrapasse algum dos limites impostos, a saída será igual a esse valor limite até que a entrada retorne para a faixa permitida.
+
+As restrições máximas e mínimas de valores dos sistemas de controle são modeladas utilizando um Limitador, cuja implementação é bastante simples e segue a seguinte expressão:
+$$
+y_n = 
+\begin{cases}
+L_{sup}{,}	& \text{se } u_n > L_{sup} \\
+u_n{,}		& \text{se } L_{inf} \le u_n \le L_{sup}\\
+L_{inf}{,}	& \text{se } u_n < L_{inf}
+\end{cases}
+$$
+Em que:
+- $y_n$	Éé o valor de saída atual do bloco
+- $L_{sup}$ e $L_{inf}$	são os limites superior e inferior, respectivamente
+- $u_n$	é o valor de entrada atual do bloco
+
+## Formulário de edição do bloco Limitador
+A figura abaixo apresenta o formulário de edição de dados do bloco limitador.
+
+<div><center><img src={useBaseUrl("images/limiterForm.png")} alt="Formulário de edição de dados do bloco limitador no PSP-UFU" title="Formulário de edição de dados do bloco limitador no PSP-UFU" /></center></div>
+
+Esse bloco não linear é definido pelo **limite superior e inferior**, inseridos pelo usuário.
+
+:::info Informação
+Tais blocos são bastante encontrados nos sistemas de controle para limitar a atuação do controlador, além de representar limites físicos de modelos implementados, como, por exemplo, valores máximos de corrente de excitação ou potência mecânica fornecida pelas turbinas.
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/line.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/line.md
new file mode 100644
index 0000000..eaece87
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/line.md
@@ -0,0 +1,120 @@
+---
+id: line
+title: Linha
+sidebar_label: Linha
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+>Um meio de transmissão fabricado usado para transmitir energia eletromagnética entre dois pontos com um mínimo de radiação. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=704-02-02).
+
+## Linha no PSP-UFU
+As linhas no PSP-UFU são modelos $\pi$ equilibradas. Podem ser utilizadas como linhas de transmissão e distribuição de sistemas elétricos de potência.
+
+A figura abaixo mostra o modelo $\pi$ implementado no PSP-UFU:
+
+<div><center><img src={useBaseUrl("images/lineModel.svg")} alt="Modelo PI da linha" title="Modelo PI da linha" /></center></div>
+
+Em que:
+- $\bold{r_L}$	é a resistência da linha;
+- $\bold{x_L}$	é a reatância indutiva da linha;
+- $\bold{b_{L}^{d}}$	é a susceptância capacitiva *shunt* da linha.
+
+:::caution Atenção
+O modelo utilizado no PSP-UFU é **equilibrado** e **não possui impedâncias mútuas** entre as fases.
+:::
+
+A linha pode ser inseridas com pontos de ancoragem, ou "nós", para maior personalização gráfica do elemento, como apresentado nas [Ferramentas CAD](cadTools#linha).
+
+:::warning Cuidado!
+A linha deve ser inserida entre duas barras de **mesma tensão nominal**. Caso você tente inseri-la entre barras de tensão diferentes, uma mensagem de erro será exibida.
+:::
+
+:::tip Dica
+Um arranjo série de uma [linha](line) sem as susceptâncias *shunt*, um [barramento](bus) e um [capacitor](capacitor) pode ser utilizado para fabricar um filtro passivo nos [estudos harmônicos](harmonics). Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.
+:::
+
+## Formulário de edição das linhas
+
+A imagem abaixo apresenta o formulário de inserção/alteração de dados das linhas de transmissão:
+
+<div><center><img src={useBaseUrl("images/lineForm.png")} alt="Formulário das linhas no PSP-UFU" title="Formulário das linhas no PSP-UFU" /></center></div>
+
+Esse formulário é subdividido em dois contextos distintos:
+- **Geral**: no qual são inseridas informações gerais da linha e informações do fluxo de carga;
+- **Falta**: local onde as impedâncias de sequência zero são inseridas.
+
+:::info Informação
+Os parâmetros necessários para construção da linha segundo seu modelo $\pi$ são inseridos na aba Geral, utilizados para construção da matriz admitância de sequência positiva e negativa.
+
+Dados adicionais de impedâncias de sequência zero necessário para o cálculo de curtos-circuitos desbalanceados são editados na aba Falta, utilizados na construção da matriz admitância de sequência zero.
+:::
+
+Além desses dois contextos, pode ser observado o botão "Estabilidade" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da linha durante o estudo de [estabilidade](stability).
+
+<div><center><img src={useBaseUrl("images/lineSw.png")} alt="Formulário de chaveamento da linha" title="Formulário de chaveamento da linha" /></center></div>
+
+<Tabs
+  groupId="line-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Falta', value: 'fault'},
+    {label: 'Botão Estabilidade', value: 'stability'},
+  ]
+}>
+<TabItem value="general">
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+Todos os componentes de potência do PSP-UFU possuem esse campo.
+
+#### Tensão nominal
+Campo de informação *não editável* que apresenta a tensão nominal da linha. Para alterar esse campo é necessário editar o campo correspondente do [barramento](bus#geral) conectado.
+
+
+#### Potência nominal
+Potência nominal da linha, inserida em MVA, kVA ou VA.
+
+Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.
+
+#### Parâmetros do modelo $\pi$
+Resistência ($r_L$), Reatância indutiva ($x_L$) série e Susceptância capacitiva shunt **total** ($2b_{L}^{d}$) presentes no modelo $\pi$ da linha.
+
+Esses parâmetros são inseridos em $p.u.$, $\Omega$ ou $\Omega/km$ ($S$ ou $S/km$ para $b_{L}^{d}$).
+
+#### Comprimento da linha
+Utilizado para calcular o valor dos parâmetros da linha inseridos em $\Omega/km$ (ou $S/km$).
+
+:::info Informação
+O comprimento da linha é ignorado caso não sejam utilizadas as unidades por quilometro.
+:::
+
+#### Utilizar potência nominal como base
+Caso essa opção seja marcada, o programa irá utilizar a potência nominal da linha como base para a conversão das unidades, caso contrário será usada a [potência base do sistema](simulationConfig).
+
+</TabItem>
+<TabItem value="fault">
+
+#### Parâmetros do modelo $\pi$ de sequência zero
+Esses parâmetros são necessários para o correto cálculo das correntes de [falta desbalanceadas](fault) (fase-fase, fase-fase-terra e fase-terra) e devem ser inseridos em $p.u.$
+
+:::warning Cuidado!
+A não inserção desses dados acarretarão em resultados incorretos para [faltas desbalanceadas](fault).
+:::
+
+</TabItem>
+<TabItem value="stability">
+
+O botão "Estabilidade" irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção da linha durante o estudo de [estabilidade](stability).
+
+Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões "Adicionar" e "Remover", respectivamente.
+
+</TabItem>
+</Tabs>
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/load.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/load.md
new file mode 100644
index 0000000..0f2b6f3
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/load.md
@@ -0,0 +1,146 @@
+---
+id: load
+title: Carga
+sidebar_label: Carga
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+>1. a potência ativa, reativa ou aparente gerada, transmitida ou distribuída dentro de um sistema;
+1. a potência demandada por um grupo de consumidores classificados de acordo com suas particularidades e características, por exemplo, carga de aquecimento, carga reativa diurna, etc. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=601-01-15).
+
+## Carga no PSP-UFU
+
+As cargas são os elementos de potência consumidores de potência genéricos. Nos estudos de fluxo de carga é possível modelar a carga como **potência ou impedância constante** e em estabilidade pode-se compor a carga na proporção desejada em três tipos (**carga ZIP**): potência constante, impedância constante e corrente constante.
+
+As cargas de potência constante nos estudos de fluxo de carga são inseridos no vetor de potências e permanecem fixos durante toda a simulação.
+
+Os elementos shunt, como reatores, banco de capacitores e cargas de impedância constante, são representados simplesmente por suas admitâncias em derivação. Como os parâmetros de entrada geralmente se constituem de suas potências ativa (P) e reativa (Q) nominais (no caso de reatores e capacitores, P=0), deve-se, portanto, encontrar sua admitância com a equação abaixo para inseri-la na matriz admitância:
+
+$$
+\overline{y} = \frac{P-jQ}{\dot{V}^2}
+$$
+
+:::caution Atenção!
+Para os estudos de **curto-circuito** e **harmônicos** as cargas são modeladas como impedância constante, independentemente da sinalização indicada em seu formulário de edição de dados.
+:::
+
+### Carga no estudo de estabilidade
+A modelagem das cargas do sistema de forma exata se torna impraticável em um problemas de estabilidade, visto à grande quantidade e variedade dos componentes envolvidos, além de apresentar modificações em suas composições causado por vários fatores, como tempo, condições climáticas e economia. Portanto várias aproximações devem ser utilizadas na formulação de um modelo de carga nos estudos aqui postos.
+
+Uma forma interessante de se representar cargas dependentes da tensão são as cargas ZIP, as quais são funções quadráticas, possuindo três parcelas: impedância constante (Z), corrente constante (I) e potência constante (P). As potências ativa e reativa são obtidas pelas seguintes equações:
+$$
+P_L=P_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+P_{I0} \left(\frac{V_i}{V_i0} \right)+P_{P0}
+$$
+$$
+Q_L=Q_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+Q_{I0} \left(\frac{V_i}{V_i0} \right)+Q_{P0}
+$$
+
+Em que:
+
+$P_L$ e $Q_L$	são as potências ativa e reativa da carga ZIP, respectivamente, conectadas em uma barra genérica i;
+
+$V_i$ é a tensão atual de uma barra genérica i;
+
+$V_{i0}$ é a tensão inicial de uma barra genérica i, obtida do estudo de fluxo de carga;
+
+$P_{Z0}$ e $Q_{Z0}$ são as parcelas de impedância constante para potências ativa e reativa, respectivamente;
+
+$P_{I0}$ e $Q_{I0}$ são as parcelas de corrente constante para potências ativa e reativa, respectivamente;
+
+$P_{P0}$ e $Q_{P0}$ são as parcelas de potência constante para potências ativa e reativa, respectivamente.
+
+Os valores das parcelas de potência da carga ZIP são obtidos utilizando as potências da carga após a convergência do cálculo do fluxo de carga ($P_{L0}$, para a potência ativa e $Q_{L0}$, para a potência reativa), utilizando as seguintes expressões:
+
+$$
+\begin{cases}
+P_{Z0}=\displaystyle \frac{k_{PZ}}{100} \frac{P_{L0}}{V_{i0}^2}\\
+P_{I0}=\displaystyle \frac{k_{PI}}{100} \frac{P_{L0}}{V_{i0}}\\
+P_{P0}=\displaystyle \frac{k_{PP}}{100} P_{L0}
+\end{cases}
+$$
+
+Os valores de $k_{PZ}$, $k_{PI}$ e $k_{PP}$ representam a composição em impedância, corrente e potência constantes, respectivamente, da parcela ativa da carga, os quais são inseridos pelo usuário na forma de porcentagem, sendo a soma desses três valores necessariamente igual a 100%. Os parâmetros $k_{QZ}$, $k_{QI}$ e $k_{QP}$ podem ser interpretados de forma análoga, porém compõe a parcela reativa da carga.
+
+O comportamento das potências, retiradas de simulações no PSP-UFU, de cada parcela que compõe a carga ZIP são evidenciadas na figura abaixo.
+
+<div><center><img src={useBaseUrl("images/loadModel.svg")} alt="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente" title="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente" /></center></div>
+
+Cargas de corrente e potência constantes possuem problemas em tensões muito baixas. À medida que a tensão diminui as correntes dessas cargas não reduz, como pode ser observado na figura acima, resultando em perda de precisão e problemas na convergência de processos iterativos. Para contornar esse problema utiliza-se uma tensão pré-definida pelo usuário ($V_{low}$), a qual as cargas (ou parcelas) de corrente e potência constantes são modeladas como impedância constante, resultando no comportamento de potência e corrente apresentados na figura acima.
+
+## Formulário de edição das cargas
+A imagem abaixo apresenta o formulário de inserção/alteração de dados das cargas:
+
+<div><center><img src={useBaseUrl("images/loadForm.png")} alt="Formulário das cargas no PSP-UFU" title="Formulário das cargas no PSP-UFU" /></center></div>
+
+Esse formulário é subdividido em dois contextos distintos:
+- **Geral**: no qual são inseridas informações gerais das cargas, informações e o tipo de carga no fluxo de carga;
+- **Estabilidade**: contendo opções de visualização de dados da carga em gráficos no tempo e opções de parametrização da carga ZIP.
+
+Além desses dois contextos, pode ser observado o botão "Chaveamento" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da carga durante o estudo de [estabilidade](stability).
+
+<div><center><img src={useBaseUrl("images/loadSw.png")} alt="Formulário de chaveamento da carga" title="Formulário de chaveamento da carga" /></center></div>
+
+<Tabs
+  groupId="load-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Estabilidade', value: 'stability'},
+    {label: 'Botão Chaveamento', value: 'switching'},
+  ]
+}>
+<TabItem value="general">
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+Todos os componentes de potência do PSP-UFU possuem esse campo.
+
+#### Potência ativa
+Parcela de potência ativa da carga. Pode ser inserido em MW, kW W ou $p.u.$ (na base do sistema).
+
+#### Potência reativa
+Parcela de potência reativa da carga. Pode ser inserido em Mvar, kvar var ou $p.u.$ (na base do sistema).
+
+#### Tipo de carga (fluxo de carga)
+Tipo da carga para o estudo de fluxo de carga, podendo ser selecionado dois tipos: Potência constante e Impedância constante. As cargas de potência constantes são inseridas nos vetores de potência e permanecem invariantes no cálculo, enquanto que para as cargas de impedância constante determina-se o valor da impedância utilizando a potência e tensão nominal, a qual é inserida na matriz admitância.
+
+</TabItem>
+<TabItem value="stability">
+
+#### Imprimir dados
+Exibe os dados da carga nos gráficos no tempo.
+Os seguintes dados são exibidos:
+- módulo da tensão
+- corrente
+- potências ativa e reativa.
+
+#### Utilizar composição de carga ZIP
+Permite compor a carga ZIP para estudos de estabilidade. Caso essa opção não esteja selecionada é usada a composição global definida nas [configurações de simulação](simulationConfig).
+
+:::caution Atenção
+Por padrão, todas as cargas no PSP-UFU são definidas como impedância constante no estudo de estabilidade.
+:::
+
+:::tip Dica
+Caso todas as barras possuírem o mesmo comportamento de carga ZIP, a parametrização poderá ser realizada nas [configurações de simulação](simulationConfig) para todas as cargas, sem a necessidade de parametrização individual.
+:::
+
+#### Composição de potência ativa e reativa
+Valores da composição da carga ZIP, inseridas para potência ativa e reativa separadamente, em porcentagem. A soma da composição deve ser igual a 100%, caso contrário será exibida uma mensagem de erro ao usuário.
+
+</TabItem>
+<TabItem value="switching">
+
+O botão "Chaveamento" irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de [estabilidade](stability).
+
+Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões "Adicionar" e "Remover", respectivamente.
+
+</TabItem>
+</Tabs>
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/mainScreen.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/mainScreen.md
new file mode 100644
index 0000000..87ba944
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/mainScreen.md
@@ -0,0 +1,127 @@
+---
+id: mainScreen
+title: Tela Principal
+sidebar_label: Tela Principal
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+## Organização da tela principal
+A tela principal do PSP-UFU é dividida em três seções: *[Menu "Ribbon"](mainScreen#menu-ribbon)*, *[Área de Trabalho](mainScreen#área-de-trabalho)* e *[Barra de Status](mainScreen#barra-de-status)*.
+
+<div><center><img src={useBaseUrl("images/mainScreen.svg")} alt="Tela principal" title="Tela principal do PSP-UFU no Windows" /></center></div>
+
+Resumidamente, no **Menu Ribbon** são acessadas as ferramentas do PSP-UFU; na **Área de Trabalho** são criados os diagramas unifilares de potência; e na **Barra de Status** são exibidas importantes informações acerca do estado da Área de Trabalho.
+
+
+### Menu Ribbon
+>Em informática o ribbon (faixa, em inglês) é um formato de apresentação de interface baseada na GUI onde a barra de ferramentas é mostrada através de uma barra mais larga com icones maiores possibilitando o uso dos aplicativos por dispositivos *touch screen*.
+[Wikipedia](https://pt.wikipedia.org/wiki/Ribbon_(inform%C3%A1tica))
+
+O menu Ribbon é composto de três submenus: “Arquivo”, “Ferramentas” e “Simulação”.
+
+<div><center><img src={useBaseUrl("images/menuFile.png")} alt="Menu Arquivo" title="Menu Arquivo" /></center></div>
+
+Em **Arquivo** estão as ferramentas gerais do programa, as quais permitem ao usuário criar e salvar novos projetos, além daquelas que possibilitam carregar os projetos já existentes e importar arquivos de outros programas. Esse submenu também possui opções gerais do programa e acesso a informações sobre o programa.
+
+<div><center><img src={useBaseUrl("images/menuTools.png")} alt="Menu Ferramentas" title="Menu Ferramentas" /></center></div>
+
+Em **Ferramentas** estão presentes os itens específicos, os quais estão relacionados à criação e manipulação da rede elétrica de potência, relatórios tabulares de saída de dados e acesso ao formulário de criação de gráficos no tempo. Algumas configurações específicas do projeto ativo são acessadas por esse menu.
+
+<div><center><img src={useBaseUrl("images/menuSimulation.png")} alt="Menu Simulação" title="Menu Simulação" /></center></div>
+
+Finalmente, o submenu **Simulação** proporciona o acesso do usuário aos cálculos realizados pelo *software* e suas configurações, além da ferramenta de “solução contínua”, a qual resolve os cálculos estáticos (fluxo de potência, curto-circuito e harmônicos) após quaisquer mudanças na rede de potência, como remoção de componente ou alteração de seus dados. Todas as configurações de simulação do projeto ativo são acessadas por esse menu.
+
+### Área de Trabalho
+Na área de trabalho estão presentes os projetos em execução, os quais são divididos por abas identificadas pelo nome do projeto (caso já esteja gravado no disco). É nessa área que são inseridos e excluídos os elementos elétricos, cujas ferramentas são acessadas por meio do [Menu Ribbon](mainScreen#menu-ribbon).
+
+As edições gráficas dos elementos, como posição, conexão e tamanho dos barramentos, são realizadas por meio da ação *drag-and-drop* (arrastar e soltar) com a utilização do mouse. A edição dos dados elétricos é feita por meio de formulários de dados acessados clicando duas vezes sobre os elementos.
+
+<div><center><img src={useBaseUrl("images/workspace.png")} alt="Área de Trabalho" title="Área de Trabalho" /></center></div>
+
+### Barra de Status
+A barra de status é responsável por informações interessantes sobre as circunstâncias atuais do programa: modo de operação do mouse (editar, mover, arrastar), zoom aplicado e posição do mouse na área de trabalho, além de dados sobre as ações dos usuários, como: inserção de elementos, informação sobre copiar e colar, etc.
+
+:::tip Dica
+**Sempre fique atendo à barra de status**, pois são fornecidas informações importantes acerca da operação do programa. Caso tenha alguma dúvida siga as instruções apresentadas nesse componente.
+:::
+
+## Configurações gerais
+As configurações gerais do programa são acessadas no submenu Ribbon **Arquivo**. Essas configurações são aplicadas para todos os projetos e permanecem gravadas no disco.
+
+<div><center><img src={useBaseUrl("images/generalSettings.png")} alt="Configurações gerais" title="Configurações gerais" /></center></div>
+
+:::warning Cuidado
+Algumas configurações de segurança do seu computador (principalmente em sistemas Windows) podem gerar uma mensagem de erro ao confirmar as alterações das configurações gerais.
+
+Para resolver esse problema basta **executar o PSP-UFU como administrador** (clicar com botão direito no atalho do programa e posteriormente em executar como administrador). Altere novamente as configurações e reinicie o programa.
+:::
+
+### Idioma
+Atualmente os seguintes idiomas estão disponíveis no PSP-UFU:
+- Inglês
+- Português
+
+:::info Informação
+O programa deve ser reiniciado para surtir efeito da alteração do idioma.
+:::
+
+### Renderização
+Define como os elementos gráficos do [editor de potência](powerEditor) e do [editor de controle](controlEditor) são desenhados na tela. Atualmente duas opções estão disponíveis:
+- [OpenGL](https://www.opengl.org/about/)
+- [Device Context](https://docs.wxwidgets.org/3.0/classwx_graphics_context.html)
+
+:::warning Cuidado!
+o OpenGL pode não ser suportado pelo seu computador. Nesse caso, ao criar um novo projeto a tela de trabalho não é exibida ou é exibida uma tela preta. A solução é alterar o renderizador para "Device Context". 
+
+Caso apareça uma mensagem de erro, execute o programa como administrador e repita o processo.
+:::
+
+:::info Informação
+O programa deve ser reiniciado para surtir efeito de alteração do renderizador.
+:::
+
+## Arquivos de projeto do PSP-UFU
+As opções de criação, gravação e abertura de projetos no disco, assim como importação de arquivos de outros programas estão presentes no submenu Ribbon **Arquivo**.
+
+### Novo projeto
+A criação de um novo projeto é realizada clicando no botão **Novo projeto** no [submenu **Arquivo**](mainScreen#menu-ribbon). Essa ação cria um sistema em branco na área de trabalho, local onde é possível inserir os elementos elétricos por meio do [submenu **Ferramentas**](mainScreen#menu-ribbon) ou pelas [teclas de atalho](powerEditor#teclas-de-atalho).
+
+Esse sistema em branco pode ser ciado utilizando o [Editor de Potência](powerEditor).
+
+### Salvar e Salvar como...
+A opção "Salvar" sobrepõe as alterações realizadas no projeto aberto e grava no disco. A opção "Salvar como..." cria um novo arquivo e grava o projeto no disco com o auxílio de uma janela de seleção de pasta (e definição do nome do arquivo).
+
+:::info Informação
+Para projetos que estão sendo gravados pela primeira vez a opção "Salvar" se comporta de forma idêntica à opção "Salvar como...".
+:::
+
+:::tip Dica
+O PSP-UFU grava os arquivos com a extensão *.psp*. Esses arquivos nada mais são que arquivos de texto utilizando a [linguagem de marcação](https://en.wikipedia.org/wiki/Markup_language) XML ([eXtensible Markup Language](https://en.wikipedia.org/wiki/XML)), que define uma série de regras de formatação dos dados de forma que eles são tanto legíveis por humanos quanto por máquinas.
+
+Portanto, os dados elétricos contidos neles podem ser facilmente identificados e alterados, caso necessário.
+:::
+
+### Abrir projeto
+Essa opção abre os projetos gravados no disco por meio com o auxílio de uma janela de seleção de arquivos. 
+
+### Importar projeto
+O PSP-UFU permite a importação de arquivos dos seguintes programas:
+- [ANAREDE](http://www.cepel.br/pt_br/produtos/programas-computacionais-por-categoria/menu/anarede-analise-de-redes-eletricas.htm)
+- [Matpower](https://matpower.org/)
+
+Para importação dos arquivos do **ANAREDE** são utilizados tanto o arquivo de dados elétricos (*.pwf*) quanto de dados gráficos dos elementos (*.lst*).
+
+Uma vez que o arquivo do **Matpower** (*.m*) não possui dados gráficos dos elementos elétricos, o diagrama unifilar é automaticamente gerado pelo PSP-UFU ao importá-lo.
+
+#### Geração automática do *layout* de diagramas unifilares
+Para criar o *layout* automático, utilizou-se a teoria dos grafos baseada no posicionamento direcionado à força e aplicação de grafos com arestas ponderadas. De acordo com a teoria dos grafos, o sistema é modelado como vértices e arestas. Nesse contexto, os barramentos são os vértices e os ramos do sistema (linhas e transformadores) são as arestas. As localizações de elementos de derivação, assim como os nós dos elementos, são automaticamente controladas pelo PSP-UFU.
+
+A ferramenta de *layout* automático é composta por um processo iterativo e os resultados são mais refinados com um número maior de iterações. Os melhores resultados são obtidos utilizando o número de iterações igual ou superior a cinco vezes o número de barramentos do sistema.
+
+:::caution Atenção!
+O *layout* automático ainda está em desenvolvimento e algumas melhorias podem ser feitas para evitar cruzamentos e sobreposições, porém, em seu estado atual, é totalmente funcional e satisfatório.
+:::
+
+:::tip Dica
+Uma vez que o Matpower pode converter os formatos CDF (*Common Data Format*) e PSS/E RAW para arquivos *.m*, estes podem ser importados de forma indireta para o PSP-UFU.
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/mathExpression.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/mathExpression.md
new file mode 100644
index 0000000..a9f62f4
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/mathExpression.md
@@ -0,0 +1,157 @@
+---
+id: mathExpression
+title: Expressão Matemática
+sidebar_label: Expressão Matemática
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+O bloco de expressão matemática utiliza a biblioteca externa [fparser](http://warp.povusers.org/FunctionParser/) e permite ao usuário inserir expressões matemáticas genéricas de qualquer complexidade combinadas a estruturas condicionais em um diagrama de controle, aumentando significativamente a capacidade de generalização do programa.
+
+## Sintaxe das expressões matemáticas
+As sintaxes das operações (para expressões `A` e `B`) são apresentadas na tabela abaixo:
+
+|Operador|Expressão|
+|--------|---------|
+|`( )`	|Expressão em parênteses primeiro
+|`A unit`	|Um multiplicador de unidade (se um estiver sido adicionado)
+|`A^B`	|Exponenciação (`A` elevado à potência `B`)
+|`-A`	|Operação unária de oposto
+|`!A`	|Operação lógica unária de negação (resulta em $1$ se `int(A)` é $0$, senão $0$)
+|`A*B` `A/B` `A%B`	|Multiplicação, divisão e módulo
+|`A+B` `A-B`	|Adição e subtração
+|`A=B` `A<B` `A<=B` `A!=B` `A>B` `A>=B`	|Comparação entre `A` e `B` (resulta em $1$ ou $0$)
+|`A&B`	|Resulta em $1$ se `int(A)` e `int(B)` forem diferentes de $0$, senão $0$
+|`A`\|`B`	|Resulta em $1$ se `int(A)` ou `int(B)` forem diferentes de $0$, senão $0$
+
+As **funções matemáticas** suportadas pela biblioteca, as quais podem ser utilizadas na construção de diagrama de blocos, são descritas na tabela abaixo:
+
+|Expressão|Descrição
+|---------|---------|
+|`abs(A)`|Valor absoluto de `A`. Para números reais, se `A` é negativo, retorna `–A`, senão retorna `A`. Para números complexos a expressão é equivalente a `hypot(real(x),imag(x))`.
+|`acos(A)`|Arco cosseno de `A`. Retorna o ângulo em radianos.
+|`acosh(A)`|O mesmo que `acos()` mas para cosseno hiperbólico
+|`arg(A)`|Ângulo de fase de um número complexo `A`.
+|`asin(A)`|Arco seno de `A`. Retorna o ângulo em radianos.
+|`asinh(A)`|Mesmo que `asin()`, mas para seno hiperbólico
+|`atan(A)`|Arco tangente de `A`. Retorna o ângulo em radianos.
+|`atan2(A,B)`|Arco tangente de `A/B`, a qual os sinais dos dois argumentos para determinar o quadrante do resultado. Retorna a solução de do seguinte sistema: `hypot(A,B)*sin(x)=A, hypot(A,B)*cos(x)=B`. O valor retornado varia de $-\pi$ a $\pi$.
+|`atanh(A)`|Mesmo que `atan()`, mas para tangente hiperbólica.
+|`cbrt(A)`|Raiz cúbica de `A`.
+|`conj(A)`|Conjugado complexo de `A`.
+|`ceil(A)`|Teto de `A`. Arredonda para o próximo maior inteiro.
+|`cos(A)`|Cosseno de `A`. Retorna o ângulo em radianos.
+|`cosh(A)`|Mesmo que `cos()`, mas para cosseno hiperbólico
+|`cot(A)`|Cotangente de `A`. Retorna o ângulo em radianos.
+|`csc(A)`|Cossecante de `A`. Retorna o ângulo em radianos.
+|`exp(A)`|Exponencial de `A`. Retorna o valor de e elevado a potência A.
+|`exp2(A)`|Exponencial de `A` na base $2$.
+|`floor(A)`|Piso de `A`. Arredonda para o próximo menor inteiro.
+|`hypot(A,B)`|Função de distância Euclidiana.
+|`if(A,B,C)`|Se `int(A)` é diferente de $0$, retorna o valor de `B`, senão retorna `C`.
+|`imag(A)`|Retorna a parte imaginária do número complexo `A`.
+|`int(A)`|Arredonda `A` para o inteiro mais próximo.
+|`log(A)`|Logaritmo natural (base $e$) de `A`.
+|`log2(A)`|Logaritmo na base $2$ de `A`.
+|`log10(A)`|Logaritmo na base $10$ de A.
+|`max(A,B)`|Se `A>B`, o resultado é `A`, senão é `B`.
+|`min(A,B)`|Se `A<B`, o resultado é `A`, senão é `B`.
+|`polar(A,B)`|Retorna o número complexo de magnitude `A` e ângulo de fase `B` (em radianos).
+|`pow(A,B)`|Exponenciação (`A` elevado à potência `B`)
+|`real(A,B)`|Retorna a parte real do número complexo `A`.
+|`sec(A)`|Secante de `A`.
+|`sin(A)`|Seno de `A`. Retorna o ângulo em radianos.
+|`sinh(A)`|Mesmo que `sin()`, mas para seno hiperbólico.
+|`sqrt(A)`|Raiz quadrada de `A`.
+|`tan(A)`|Tangente de `A`. Retorna o ângulo em radianos.
+|`tanh(A)`|Mesmo que `tan()`, mas para tangente hiperbólica.
+|`trunc(A)`|Valor truncado de `A`. Retorna o número inteiro de `A` sem a parcela fracionada.
+
+É possível atribuir novas variáveis utilizando a seguinte sintaxe: 
+
+``` cpp
+<nome da variável> := <expressão>; <função>
+```
+
+Por exemplo:
+``` cpp
+comprimento := sqrt(x*x+y*y); 2*comprimento*sin(comprimento)
+```
+
+:::tip Dica
+O caractere de espaço e de nova linha são ignorados na interpretação da expressão, portanto para maior organização, o código anterior pode ser escrito da seguinte forma:
+``` cpp
+comprimento := sqrt(x*x+y*y);
+2*comprimento*sin(comprimento)
+```
+:::
+
+:::warning Cuidado!
+Note que a expressão que determina o valor de saída do bloco **não possui** o caractere `;` em seu final.
+:::
+
+## Formulário de edição de dados do bloco de Expressão Matemática
+O formulário de inserção e edição das entradas do bloco e da expressão matemática genérica, assim como ferramentas de auxílio de sua construção é apresentado na figura abaixo.
+
+<div><center><img src={useBaseUrl("images/mathExpressionForm.png")} alt="Formulário de edição de dados do bloco de expressão matemática no PSP-UFU" title="Formulário de edição de dados do bloco de expressão matemática no PSP-UFU" /></center></div>
+
+No campo de “Variáveis de entrada” é inserida uma lista com os nomes das entradas separados por espaços. Um número qualquer de entrada pode ser definido nessa lista esses nomes são apresentados no ícone gráfico presente no editor de controle, além de serem destacados na expressão inserida pelo usuário. O número de entradas e saídas se comporta de forma semelhante aos [blocos somadores](sum).
+
+Abaixo do campo das variáveis de entrada está presente o local para inserção da expressão matemática. A sintaxe da expressão inserida pelo usuário possui realce (por meio de diferentes formas e cores da fonte) para números, operadores, variáveis de entrada, funções e constantes, facilitando a criação, manipulação e identificação de erros de digitação e lógica.
+
+:::info Informação
+Como ferramenta de auxílio ao usuário foi desenvolvida uma verificação da expressão inserida.
+:::
+
+Tal ferramenta irá encontrar erros e indicará ao usuário qual o tipo do erro, além da sua localização, destacando-o. A figura anterior exemplifica a identificação automática do erro pelo PSP-UFU, assim como sua posição na expressão inserida.
+
+## Exemplo de Expressão Matemática
+A corrente de campo em $p.u.$ pode ser estimada de forma aproximada utilizando as potências ativa ($P$) e reativa ($Q$), assim como as reatâncias transitórias de eixo direto ($x_d$) e em quadratura ($x_q$) e o módulo da tensão terminal ($V$) da máquina:
+$$
+i_f \approx \sqrt{ \left( V + Q' \right)^2 + P^2} + \left(\frac{x_d}{x_q} +1{,}0 \right) \times \frac{Q' \left( V + Q' \right) + {P'}^2}{\sqrt{\left( V + Q'\right)^2 + P^2}}
+$$
+Em que:
+- $P' = x_q \times \frac{P}{V}$
+- $Q' = x_q \times \frac{Q}{V}$
+
+Primeiramente, deve-se inserir no campo "variáveis de entrada" as grandezas fornecidas pelo PSP-UFU. Nesse caso, como pode ser observado no bloco de [entrada / saída](io), todas as variáveis necessárias são fornecidas: $P$, $Q$ e $V$. Essas variáveis devem ser inseridas separadas por espaço: `p q v`.
+
+Utilizando três blocos de [entrada / saída](io) pode-se fornecer tais dados ao conectá-los ao bloco de expressão matemática.
+
+As reatâncias transitórias de eixo direto ($x_d$) e em quadratura ($x_q$) podem ser definidas diretamente na expressão matemática:
+``` cpp
+xd := 0.146;
+xq := 0.0969;
+```
+
+Os valores de $P'$ e $Q'$ podem ser também calculados:
+``` cpp
+yp := xq * p / v;
+yq := xq * q / v;
+```
+
+Com todos os dados necessários pode-se calcular a corrente de campo:
+``` cpp
+i_f := sqrt((v + yq)^2 + p^2) + (xd / xq + 1.0) *
+       ((yq * (v + yq) + yp^2) / sqrt((v + yq)^2 + p^2));
+```
+
+Finalmente, pode-se definir a saída do bloco (não inserindo `;`):
+``` cpp
+i_f
+```
+
+Portanto, para as entradas definidas como `p q v`, a expressão matemática total para cálculo da corrente de campo será:
+``` cpp
+xd := 0.146;
+xq := 0.0969;
+yp := xq * p / v;
+yq := xq * q / v;
+i_f := sqrt((v + yq)^2 + p^2) + (xd / xq + 1.0) *
+       ((yq * (v + yq) + yp^2) / sqrt((v + yq)^2 + p^2));
+i_f
+```
+
+## Referências
+- NIEMINEN, J.; YLILUOMA, J. Function Parser for C++ v4.5.2. Disponível em: <http://warp.povusers.org/FunctionParser/>. 
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/multiplier.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/multiplier.md
new file mode 100644
index 0000000..f21610b
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/multiplier.md
@@ -0,0 +1,16 @@
+---
+id: multiplier
+title: Multiplicador
+sidebar_label: Multiplicador
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+Esse elemento é bastante simples e cumpre com a função matemáticas de seu nome.
+
+Possui duas entradas de valores reais e sua saída é computada na sequência do nó superior pelo inferior, ou seja, a saída será o valor presente no nó 1 multiplicado pelo nó 2.
+
+:::caution Atenção!
+Nós não conectados no multiplicador são assumidos com o valor unitário e para o divisor é considerado valor nulo.
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/powerEditor.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/powerEditor.md
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+---
+id: powerEditor
+title: Editor de Potência
+sidebar_label: Editor de Potência
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+O Editor de Elementos de Potência no PSP-UFU permite a inserção, manipulação e edição de elementos elétricos de potência por meio de um diagrama unifilar.
+
+>(**Diagrama unifilar**) Um diagrama do sistema no qual as linhas polifásicas são representados por sua única linha equivalente. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=601-02-04).
+
+:::info Informação
+Todos os elementos inseridos pode ser manipulados e editados utilizando as [Ferramentas CAD](cadTools).
+:::
+
+## Novo projeto
+A criação de um novo projeto é realizada clicando no item **Novo projeto** no [submenu **Arquivo**](mainScreen#menu-ribbon). Essa ação cria um sistema em branco na área de trabalho, local onde é possível inserir os elementos elétricos por meio do [submenu **Ferramentas**](mainScreen#menu-ribbon) ou pelas [teclas de atalho](powerEditor#teclas-de-atalho).
+
+Abaixo são apresentados os elementos elétricos de potência que podem ser inseridos no PSP-UFU:
+- [Barramento](bus)
+- Elementos de ramo
+	- [Linha](line)
+	- [Transformador](transformer)
+- Elementos Shunt
+	- [Carga](load)
+	- [Capacitor](capacitor)
+	- [Indutor](inductor)
+	- [Fonte de Corrente Harmônica](harmSource)
+- Máquinas
+	- [Gerador Síncrono](syncGenerator)
+	- [Compensador Síncrono](syncMotor)
+	- [Motor de Indução](indMotor)
+
+<div><center><img src={useBaseUrl("images/powerElements.svg")} alt="Elementos de potência" title="Elementos de potência" /></center></div>
+
+## Inserindo elementos
+Por meio do submenu **Ferramentas** pode-se inserir os elementos de potência, como é apresentado na imagem abaixo.
+
+<div><center><img src={useBaseUrl("images/addPowerElements.png")} alt="Adicionar elementos de potência" title="Adicionar elementos de potência" /></center></div>
+
+No caso de inserção de um barramento, o usuário deve apenas clicar na posição desejada, já para os outros componentes será solicitado que o usuário selecione em uma ou duas barras, o qual o novo componente será conectado.
+
+Sempre que são inseridos novos elementos, um formulário de dados é exibido e pode-se editar os dados elétricos e acessar formulários adicionais, como dados de estabilidade.
+
+:::caution Atenção
+O primeiro elemento que a ser inserido no projeto **deve** ser um barramento. Os demais elementos elétricos devem ser conectados em uma ou mais barras.
+:::
+
+Além dos elementos de potência pode-se inserir os elementos de *[texto vinculado](text)*, os quais estão associados a uma grandeza que se deseja vizualizar diretamente na [Área de Trabalho](mainScreen#área-de-trabalho).
+
+:::warning Cuidado!
+Ao inserir um barramento, a mesma irá acompanhar o ponteiro do mouse até que você clique na posição desejada. Para os demais elementos **deve-se clicar em uma mais barras** para exibir o elemento inserido.
+
+**Sempre fique atento às instruções da [Barra de Status](mainScreen#barra-de-status).**
+:::
+
+<video autoPlay loop muted playsInline controls>
+  <source src= "/PSP/videos/timelapseBuild.mp4" type="video/mp4" />
+</video>
+
+### Teclas de atalho
+É possível inserir todos os elementos de potência por meio de teclas de atalho, facilitando e agilizando a criação dos diagramas unifilares no PSP-UFU. A tabela abaixo apresenta tais atalhos:
+
+| Elemento             | Atalho    | Elemento             | Atalho    |
+| -------------------- | :-------: | -------------------- | :-------: |
+| Barramento           | B         | Carga                | Shift + L |
+| Linha                | L         | Capacitor            | Shift + C |
+| Transformador        | T         | Indutor              | Shift + I |
+| Gerador              | G         | Corrente harmônica   | Shift + H |
+| Motor de indução     | I         | Texto                | A         |
+| Compensador síncrono | K         |
+
+## Editando dados elétricos
+Ao inserir um elemento, seu respectivo formulário de edição é exibido para inserção dos dados elétricos d elemento. Todos os dados podem ser **alterados** ao **clicar duas vezes sobre o elemento** ou por opção do **menu de contexto** exibido ao clicar com o botão direito sobre o elemento.
+
+Cada formulário de edição de dados elétricos dos elementos possuem suas particularidades e são descritos em documentação específica: [Barramento](bus), [Linha](line), [Transformador](transformer), [Carga](load), [Capacitor](capacitor), [Indutor](inductor), [Fonte de Corrente Harmônica](harmSource), [Gerador Síncrono](syncGenerator), [Compensador Síncrono](syncMotor), [Motor de Indução](indMotor).
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/powerFlow.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/powerFlow.md
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+---
+id: powerFlow
+title: Fluxo de Potência
+sidebar_label: Fluxo de Potência
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+Um estudo fundamental no planejamento da expansão e operação de um sistema elétrico é o **fluxo de potência (ou fluxo de carga)** uma vez que a operação satisfatória desse sistema depende do conhecimento dos efeitos da interligação, de novas cargas, de novas centrais geradoras e de novas linhas antes que elas sejam instaladas. Esse estudo tem como objetivo encontrar o fluxo de potência nos ramos e as tensões nodais do Sistema Elétrico de Potência (SEP) impostas pela geração e a carga.
+
+As equações formuladas a partir dos modelos dos elementos presentes no estudo de fluxo de carga são não lineares e não possuem solução analítica explícita. Portanto é necessária a utilização de métodos numéricos iterativos para solução do problema de fluxo de carga.
+
+## Formulação do problema de fluxo de carga
+O problema de fluxo de carga pode ser representado por um sistema de equações e inequações algébricas não-lineares que correspondem às leis de Kirchhoff e a um conjunto de restrições de operação impostos pelos componentes de uma rede elétrica.
+Na formulação do problema a cada barra da rede são associadas quatro variáveis, sendo que duas delas surgem como dados e duas como incógnitas (em uma barra de índice $i$):
+- $V_i$	é a magnitude da tensão na barra $i$;
+- $θ_i$	é o ângulo da tensão na barra $i$;
+- $P_i$	é a injeção líquida de potência ativa na barra $i$;
+- $Q_i$	é a injeção líquida de potência reativa na barra $i$.
+
+Relativo às variáveis que são incógnitas e os dados do sistema, podem ser definidos **três tipos de barras**:
+- **Barra PQ**:	$P_i$ e $Q_i$ são dados, $V_i$ e $θ_i$ são calculados;
+- **Barra PV**:	$P_i$ e $V_i$ são dados, $Q_i$ e $θ_i$ são calculados;
+- **Barra de Referência**	$V_i$ e $θ_i$ são dados, $P_i$ e $Q_i$ são calculados.
+
+:::info Informação
+As **barras PQ** geralmente são representadas pelos barramentos de carga, os quais não há geração e controle de tensão.
+
+As **barras PV** se caracterizam pelo controle de tensão mediante a injeção ou absorção de potência reativa por meio do controle da excitação de uma máquina síncrona.
+
+A **barra de Referência** (ou de folga, de oscilação) tem como função, assim como o próprio nome diz, servir de referência de tensão e ângulo do sistema. Essa barra é necessariamente geradora, uma vez que ela é responsável pelo equilíbrio do balanço de potência do sistema.
+:::
+
+:::caution Atenção!
+O tipo de barra deve ser definido no [elemento barramento](bus).
+
+Note que **o sistema deve possuir somente uma barra de referência**.
+:::
+
+Como mencionado anteriormente, as equações são não-lineares e a solução analítica não é prática. As soluções dessas equações seguem **[processos iterativos](https://en.wikipedia.org/wiki/Iterative_method)**, em que são atribuídos valores estimados (ou iniciais) para as barras com tensões desconhecidas e, baseado na potência ativa e reativa e módulo da tensão especificados, calcula-se por meio das equações previamente apresentadas as novas tensões complexas em cada nó do sistema.
+
+Na sequência, esse conjunto de valores para as tensões em cada barra é utilizado para novamente calcular outro grupo de tensões. Cada cálculo de um novo conjunto de tensões é chamado iteração. O processo iterativo é repetido até que as mudanças em todas as barras sejam menores do que um valor pré-estipulado, obtendo assim a convergência.
+
+## Execução do fluxo de carga no PSP-UFU
+Após a construção do diagrama unifilar no [editor de potência](powerEditor), a execução do fluxo de carga é realizada no [menu Simulação](mainScreen#menu-ribbon) clicando no botão **Fluxo de carga**.
+
+<div><center><img src={useBaseUrl("images/menuSimulationPF.svg")} alt="Menu Simulação" title="Menu Simulação" /></center></div>
+
+:::tip Dica
+Caso o fluxo de carga tenha sido executado com sucesso, as setas de potência serão exibidas, a barra de status indicará sucesso na operação e os [elementos de texto](text) serão atualizados.
+:::
+
+**Outra possibilidade** é a execução por meio do cálculo contínuo, também presente no [menu Simulação](mainScreen#menu-ribbon) e seu acionamento é realizado co clicar no botão **Habilitar solução**. Com essa opção, os cálculos estáticos selecionados nas [configurações de simulação](simulationConfig) são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).
+
+:::caution Atenção!
+Evite construir o circuito com o cálculo contínuo habilitado, uma vez que configurações temporárias podem levar a erros de execução da simulação.
+
+Para desabilitar o cálculo contínuo clique no botão **Desabilitar solução**.
+:::
+
+Os resultados do fluxo de carga são exibidos nos [elementos de texto vinculado](text), ao posicionar o mouse sobre os elementos e em [relatórios tabulares](tabularReport).
+
+### Erros comuns na execução do fluxo de carga
+A seguir são apresentados os erros mais comuns relacionados ao fluxo de carga.
+
+#### A seguinte mensagem de erro é exibida: "O número máximo de iterações foi alcançado"
+Essa mensagem de erro é exibida quando o método de solução numérica selecionado nas [configurações de simulação](simulationConfig) atinge o número máximo de iterações inserido. As seguintes situações podem ocasionar esse erro:
+- **Os parâmetros do circuito estão incorretos**. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.
+- **Algum elemento possui parâmetros discrepantes dos demais**. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.
+- **O número máximo de iterações está muito baixo**. Alguns circuitos exigem um número maior de iterações, portanto altere o valor do máximo de iterações nas [configurações de simulação](simulationConfig). Também tente alternar entre os métodos numéricos de solução disponíveis.
+- **Os parâmetros de simulação estão inadequados**. Caso um parâmetro do método de solução esteja inadequado, como fator de aceleração ou tolerância, o cálculo pode não alcançar a convergência. Altere esses parâmetros nas [configurações de simulação](simulationConfig).
+
+#### Os dados de saída são exibidos como "NaN" ou "nan"
+Isso ocorre devido a erros de operações matemáticas nos cálculos de fluxo de carga. "NaN" significa *Not a Number*.
+- **Algum barramento está isolado**. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.
+- **Os parâmetros do circuito estão incorretos**. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.
+- **Algum elemento possui parâmetros discrepantes dos demais**. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.
+
+## Métodos de solução numérica do fluxo de carga no PSP-UFU
+Os métodos implementados no programa para solução do problema de fluxo de carga no PSP-UFU são **Gauss-Seidel (GS)** e **Newton-Raphson (NR)**. Além desses métodos clássicos, um método híbrido pode ser utilizado (definido nas [configurações de simulação](simulationConfig)), em que é utilizado inicialmente o GS e na sequência o NR, aumentando a chance de convergência do NR.
+
+### Gauss-Seidel
+O método de Gauss-Seidel tem sido bastante utilizado nas últimas décadas para solução do problema de fluxo de carga, uma vez que não há a necessidade de fatorar a matrizes, reduzindo o esforço computacional. Atualmente, restrições computacionais são menos problemáticas e outros métodos são normalmente escolhidos, porém o Gauss-Seidel ainda possui valor didático e, visto que o PSP-UFU também possui fins educacionais, optou-se pela implementação desse método.
+
+Para iniciar as iterações do método são necessários valores iniciais para as tensões ($\dot{V}_i^0$) que devem ser calculadas. Normalmente para barras do tipo PQ tem-se $\dot{V}_i^0=1{,}0+j0{,}0~p.u.$ e para barras do tipo PV $\dot{V}_i^0=V_i^{esp}+j0{,}0$ [p.u.], em que $V_i^{esp}$ é o módulo da tensão especificada para a barra PV. A barra de referência tem o módulo e ângulo de tensão fixos e não participam no processo iterativo.
+
+O método de Gauss-Seidel mostra um **número excessivo de iterações** e, com o intuito reduzi-los, multiplicam-se as correções de tensões por uma constante. Essa operação amplia o valor da correção, trazendo a tensão para mais perto do valor do valor final. Os multiplicadores que realizam essa convergência melhorada são chamados de fatores de aceleração. Para qualquer sistema existem valores ótimos para os fatores de aceleração e uma escolha inadequada pode resultar em uma convergência mais lenta ou torná-la impossível. Normalmente é utilizado um fator de aceleração igual a 1,6, valor definido como padrão no programa. A utilização do fator de aceleração é realizada por meio da equação:
+$$
+\dot{V}_{i_{AC}}^{(v + 1)} = \alpha \left( \dot{V}_{i}^{(v + 1)}- \dot{V}_{i_{AC}}^{(v)} \right) + \dot{V}_{i_{AC}}^{(v)}
+$$
+Em que:
+- $\dot{V}_{i_{AC}}^{(v + 1)}$ é a tensão complexa da iteração atual com fator de aceleração aplicado
+- $\dot{V}_{i}^{(v + 1)}$ é a tensão complexa da iteração atual
+- $\dot{V}_{i_{AC}}^{(v)}$ é a tensão complexa da iteração anterior com fator de aceleração aplicado
+- $\alpha$ é o fator de aceleração
+
+O fluxograma abaixo demonstra como o método de Gauss-Seidel foi implementado no PSP-UFU:
+
+<div><center><img src={useBaseUrl("images/gaussSeidel.svg")} alt="Método numérico de Gauss-Seidel para fluxo de carga" title="Método numérico de Gauss-Seidel para fluxo de carga" /></center></div>
+
+### Newton-Raphson
+O método de Newton-Raphson (também conhecido como método de Newton ou Newton-Fourier) para solução do fluxo de carga é descrito em vários livros e artigos. Atualmente é o algoritmo mais utilizado para solução do fluxo de carga. Para casos bem condicionados, esse método geralmente converge em 4 a 5 iterações, porém existe a possibilidade da técnica contornar o ponto da solução sem nunca atingi-la, o que também justifica implementação do método de Gauss-Seidel no *software*.
+
+A expansão da [série de Taylor](https://en.wikipedia.org/wiki/Taylor_series) para uma função de duas ou mais variáveis é a base do método de Newton-Raphson para resolver o problema do fluxo de carga. Fazendo a expansão em série de Taylor, para duas equações e duas incógnitas ($f_1\left( x_{1}, x_{2} \right)$ e $f_2\left( x_{1}, x_{2} \right)$), sem listar as derivadas parciais maiores que 1 na forma matricial, obtêm-se:
+$$
+\begin{bmatrix}
+K_1 - f_1\left( x_{1}^{(0)}, x_{2}^{(0)} \right)\\
+K_2 - f_2\left( x_{1}^{(0)}, x_{2}^{(0)} \right)
+\end{bmatrix}
+=
+\begin{bmatrix}
+\displaystyle\frac{\partial f_1}{\partial x_1} & \displaystyle\frac{\partial f_1}{\partial x_2}\\
+\displaystyle\frac{\partial f_2}{\partial x_1} & \displaystyle\frac{\partial f_2}{\partial x_2}
+\end{bmatrix}
+\begin{bmatrix}
+\Delta x_{1}^{(0)}\\
+\Delta x_{2}^{(0)}
+\end{bmatrix}
+$$
+Em que:
+- $K$ é o resultado da equação $f\left( x_{1}, x_{2} \right)$
+- $x^{(0)}$ é a estimativa iniciais de $x$
+- $\Delta x$ é o valor acrescido de $x^{(0)}$ que resulta em $x$, ou seja: $f\left( x_{1}, x_{2} \right) = f_1\left( x_{1}^{(0)} + \Delta x_1, x_{2}^{(0)} + \Delta x_2 \right)$
+
+Essa expressão pode ser resumida em:
+$$
+\begin{bmatrix}
+\Delta K_1^{(0)}\\
+\Delta K_2^{(0)}
+\end{bmatrix}
+=
+\begin{bmatrix}
+J
+\end{bmatrix}^{(0)}
+\begin{bmatrix}
+\Delta x_{1}^{(0)}\\
+\Delta x_{2}^{(0)}
+\end{bmatrix}
+$$
+Em que $[J]$ é a **Matriz Jacobiana**
+
+Com a equação acima é possível calcular os valores de $\Delta x_{1}^{(0)}$ e $\Delta x_{2}^{(0)}$. Entretanto, esses valores somados às estimativas iniciais não determinam a solução correta, sendo necessário repetir o processo de determinação das constantes, formação da matriz jacobiana e solução da equação acima, o qual será refeito determinando novas estimativas $\Delta x_{1}^{(1)}$ e $\Delta x_{2}^{(1)}$.
+
+Esse processo é repetido até que as correções se tornem tão pequenas que satisfaçam uma precisão escolhida.
+
+Para aplicar o método de Newton-Raphson à solução das equações do fluxo de carga, utiliza-se as equações que representam as potências ativa e reativa injetadas em uma barra.
+
+Assim como no método de Gauss-Seidel, a barra de referência é omitida da solução iterativa para determinar as tensões, pois a tensão complexa dessa barra é especificada. Como é conhecido o valor da potência ativa injetada ($P_i^{esp}$) nas barras do tipo PQ e PV, além da potência reativa injetada ($Q_i^{esp}$) nas barras PQ pode-se definir $P_i^{esp}$ e $Q_i^{esp}$ como os valores de $K$. Os valores estimados do módulo e ângulo da tensão correspondem aos valores estimados para $x_1$ e $x_2$.
+
+O jacobiano consiste nas derivadas parciais de $P_i$ e $Q_i$ em relação a cada uma das variáveis das equações injeção de potência líquida na barra $i$. A matriz coluna formada por $\Delta x_{1}$ e $\Delta x_{2}$ corresponde às correções de ângulo ($\Delta \theta_i$) e módulo ($\Delta V_i$) das tensões de barra.
+
+Com isso pode-se escrever a equação matricial de um sistema de $n$ barras, em que a barra número 1 corresponde à barra de referência e as barras de número 2 a n são barras do tipo PQ:
+$$
+\begin{array}{r}
+\text{NPQ} + \text{NPV}
+\begin{cases}
+~\\
+~
+\end{cases}\\
+~\\
+\text{NPQ}
+\begin{cases}
+~\\
+~
+\end{cases}
+\end{array}
+\begin{bmatrix}
+\Delta P_2\\
+\vdots\\
+\Delta P_n\\
+\Delta Q_2\\
+\vdots\\
+\Delta Q_n\\
+\end{bmatrix}
+=
+\begin{bmatrix}
+\displaystyle\frac{\partial P_2}{\partial \theta_2} & \dots & \displaystyle\frac{\partial P_2}{\partial \theta_n} & \displaystyle\frac{\partial P_2}{\partial V_2} & \dots & \displaystyle\frac{\partial P_2}{\partial V_n}\\
+\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
+\displaystyle\frac{\partial P_n}{\partial \theta_2} & \dots & \displaystyle\frac{\partial P_n}{\partial \theta_n} & \displaystyle\frac{\partial P_n}{\partial V_2} & \dots & \displaystyle\frac{\partial P_n}{\partial V_n}\\
+\displaystyle\frac{\partial Q_2}{\partial \theta_2} & \dots & \displaystyle\frac{\partial Q_2}{\partial \theta_n} & \displaystyle\frac{\partial Q_2}{\partial V_2} & \dots & \displaystyle\frac{\partial Q_2}{\partial V_n}\\
+\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
+\displaystyle\frac{\partial Q_n}{\partial \theta_2} & \dots & \displaystyle\frac{\partial Q_n}{\partial \theta_n} & \displaystyle\frac{\partial Q_n}{\partial V_2} & \dots & \displaystyle\frac{\partial Q_n}{\partial V_n}
+\end{bmatrix}
+\begin{bmatrix}
+\Delta \theta_2\\
+\vdots\\
+\Delta \theta_2\\
+\Delta V_2\\
+\vdots\\
+\Delta V_n\\
+\end{bmatrix}
+$$
+Em que:
+- $\text{NPQ}$ é o número de barras PQ
+- $\text{NPV}$ é o número de barras PV
+
+O processo iterativo se inicia calculando as potências ativas ($P_i^{calc}$) para as barras PQ e PV e as potências reativas ($Q_i^{calc}$) para as barras PQ, ambas as equações utilizando as estimativas iniciais das tensões complexas. Calcula-se, então, as correções de potência ($\Delta P$ e $\Delta Q$):
+$$
+\Delta P = P_i^{esp} - P_i^{calc}\\
+\Delta Q = Q_i^{esp} - Q_i^{calc}
+$$
+
+O passo seguinte é a formação da matriz jacobiana. Com isso é possível calcular as correções de módulo e ângulo das tensões de todas as barras (com exceção da barra de referência). Para tanto, no PSP-UFU utiliza-se o método de [Eliminação Gaussiana](https://en.wikipedia.org/wiki/Gaussian_elimination) e em sequência a substituição regressiva. Esse procedimento diminui o esforço computacional, uma vez que a inversão da matriz jacobiana em todas as iterações é evitada.
+
+Com as correções de módulo e ângulo das tensões das barras calculados, aplicam-se as seguintes equações:
+$$
+	\theta_i^{(v+1)}=\theta_i^{(v)}+\Delta \theta_i^{(v)}\\
+	V_i^{(v+1)}=V_i^{(v)}+\Delta V_i^{(v)}
+$$
+
+O processo é então reiniciado e será repetido até que se obtenha a convergência, quando as correções se tornam tão pequenas que satisfaçam uma tolerância pré-estipulada.
+
+O fluxograma abaixo mostra o método de Newton-Raphson para solução do fluxo de carga implementado.
+
+<div><center><img src={useBaseUrl("images/newtonRaphson.svg")} alt="Método numérico de Newton-Raphson para fluxo de carga" title="Método numérico de Newton-Raphson para fluxo de carga" /></center></div>
+
+## Controles e limites em um problema de fluxo de carga
+Nas barras de [geração](syncGenerator) e também naquelas em que se encontra um [compensador síncrono](syncMotor) conectado, o controle da magnitude da tensão no barramento é realizado por meio do ajuste da corrente de campo das máquinas síncronas, as quais podem operar sobrexcitadas (injetando reativos) ou subexcitadas (absorvendo reativos). Os valores limites de potência reativa que podem ser injetadas ou absorvidas dependem da máquina síncrona em estudo. Esses limites são incluídos no fluxo de carga com a criação de dois novos parâmetros, a potência reativa máxima ($Q_i^{máx}$) e potência reativa mínima ($Q_i^{min}$), sendo esses valores a soma dos limites individuais das máquinas em uma mesma barra genérica $i$.
+
+A manutenção da potência reativa dentro dos limites é realizada pela troca do tipo de barra, ou seja, as barras violadoras que controlam a tensão (PV), passam a ser barras de carga (PQ), cuja potência é fixada como o limite que seria ultrapassado ($Q_i^{máx}$ ou $Q_i^{min}$) e a tensão deixa de ser controlada partir de então.
+
+A verificação de violação e troca de tipo de barra pode ser realizada a cada iteração ou ao final da convergência do cálculo. No PSP-UFU foi implementada a última estratégia, uma vez que separa os conceitos de cálculo e de verificação de limites, tornando mais fácil o desenvolvimento de novos métodos numéricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situação não violadora o cálculo iterativo deve ser retomado até que obtenha novamente a convergência.
+
+## Referências
+1. MONTICELLI, A. J. Fluxo de Carga em Redes de Energia Elétrica. São Paulo: Edgar Blücher, 1983.
+1. STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.
+1. MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: https://doi.org/10.1007/978-3-642-13669-6
+1. ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: https://doi.org/10.1002/9781118878286
+1. TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton’s Method. IEEE Transaction on Power Apparatus and Systems, v. PAS-86, n. 11, nov. 1967. doi: https://doi.org/10.1109/TPAS.1967.291823
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/rateLimiter.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/rateLimiter.md
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+---
+id: rateLimiter
+title: Limitador de taxa
+sidebar_label: Limitador de taxa
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+Os blocos de limite de taxa têm como objetivo conter a taxa de crescimento ou decrescimento da sua entrada a valores inseridos pelo usuário. A taxa de variação é calculada a partir dos valores de entrada atual e passado e, caso seja violada, será definida uma saída que respeite essas taxas (de crescimento ou decrescimento).
+
+A taxa ($r$) é calculada de acordo com a equação:
+$$
+r = \frac{u_n - y_{n-1}}{h}
+$$
+Em que:
+- $u_n$	é a entrada atual do bloco
+- $y_{n-1}$	é o valor de saída anterior do bloco
+- $h$	é o passo de integração
+
+Obtida a taxa, o bloco irá resultar em um valor calculado pelas seguintes condições:
+$$
+\begin{cases}
+h \times R_{cre}{,}	& \text{se } r > R_{cre} \\
+u_n{,}		& \text{se } R_{dec} \le r \le R_{cre}\\
+h \times R_{dec}{,}	& \text{se } r < R_{dec}
+\end{cases}
+$$
+Em que:
+- $R_{cre}$ e $R_{dec}$	são as taxas de crescimento e decaimento, respectivamente
+
+## Formulário de edição de dados do Limitador de taxa
+A figura abaixo apresenta o formulário de edição de dados do bloco limitador de taxa.
+
+<div><center><img src={useBaseUrl("images/rateLimitForm.png")} alt="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU" title="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU" /></center></div>
+
+Esse bloco é definido pelo **limite superior (taxa de crescimento máxima) e inferior (taxa de decrescimento máxima)**, inseridos pelo usuário.
+
+:::info Informação
+Esses blocos são úteis para limitar variações bruscas na entrada que possam comprometer o controle ou limites físicos de modelos de turbina, por exemplo.
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/simulationConfig.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/simulationConfig.md
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+---
+id: simulationConfig
+title: Configurações da Simulação
+sidebar_label: Configurações da Simulação
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+As parametrizações e configurações das simulações são acessadas no [menu Simulação](mainScreen#menu-ribbon) clicando no ícone do botão **Configurações de simulação**.
+
+<div><center><img src={useBaseUrl("images/menuSimulationSettings.svg")} alt="Menu Simulação" title="Menu Simulação" /></center></div>
+
+## Formulário de edição das configurações de simulação
+
+A imagem abaixo apresenta o formulário de edição das configurações de simulação:
+
+<div><center><img src={useBaseUrl("images/simulationConfigForm.png")} alt="Configurações de simulação" title="Configurações de simulação" /></center></div>
+
+Esse formulário é subdividido em quatro contextos distintos:
+- **Geral**: no qual são inseridas informações gerais e comuns a várias simulações;
+- **Fluxo de carga**: local onde é selecionado o [método de solução numérica](powerFlow#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu), assim como seus parâmetros;
+- **Estabilidade**: contendo opções de simulação no tempo e parâmetros do [método de integração](stability#integração-numérica);
+- **Carga ZIP**: contém as opções gerais das [cargas do tipo ZIP](load#carga-no-estudo-de-estabilidade) no estudo de [estabilidade](stability).
+
+---
+<Tabs
+  groupId="simulationConfigs-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Fluxo de carga', value: 'powerFlow'},
+    {label: 'Estabilidade', value: 'stability'},
+    {label: 'Carga ZIP', value: 'zipLoad'},
+  ]
+}>
+<TabItem value="general">
+
+#### Potência base
+Potência base do sistema utilizada para conversão dos dados reais em $p.u.$ e vice-versa. Pode ser inserido em VA, kVA ou MVA.
+
+:::tip Dica
+A potência base de cada elemento pode ser distinta da potência base do sistema. Para isso, basta marcar a opção "Utilizar potência nominal \[*do elemento*\] como base", presente em seus respectivos [formulários de edição da dados](powerEditor).
+:::
+
+#### Frequência do sistema
+Define a frequência nominal do sistema.
+
+:::warning Cuidado!
+Ao alterar a frequência nominal atente-se ao campo "[Frequência de circuito aberto](syncGenerator#frequência-de-circuito-aberto)" das máquinas síncronas.
+:::
+
+#### Cálculo contínuo
+Habilita ou desabilita o cálculo contínuo para os cálculos de [curto-circuito](fault), [nível de curto-circuito](fault) e [distorções harmônicas](harmonics). O cálculo contínuo para o [fluxo de carga](powerFlow) é sempre habilitado.
+
+:::tip Dica
+Para habilitar o cálculo contínuo aperte o botão **Habilitar solução** presente no [menu Simulação](mainScreen#menu-ribbon).
+
+Com essa opção, os cálculos estáticos selecionados são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).
+:::
+
+</TabItem>
+<TabItem value="powerFlow">
+
+#### Método de solução
+Define o método de solução numérica para o estudo de [fluxo de carga](powerFlow). Na versão atual, está presente no PSP-UFU os seguintes métodos:
+- [Gauss-Seidel](powerFlow#gauss-seidel)
+- [Newton-Raphson](powerFlow#newton-raphson)
+- [Gauss-Newton híbrido](powerFlow#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu)
+
+#### Fator de aceleração
+Fator utilizado para acelerar a convergência nos métodos de Gauss. A utilização do fator de aceleração é realizada por meio da equação:
+$$
+\dot{V}_{i_{AC}}^{(v + 1)} = \alpha \left( \dot{V}_{i}^{(v + 1)}- \dot{V}_{i_{AC}}^{(v)} \right) + \dot{V}_{i_{AC}}^{(v)}
+$$
+Em que:
+- $\dot{V}_{i_{AC}}^{(v + 1)}$ é a tensão complexa da iteração atual com fator de aceleração aplicado
+- $\dot{V}_{i}^{(v + 1)}$ é a tensão complexa da iteração atual
+- $\dot{V}_{i_{AC}}^{(v)}$ é a tensão complexa da iteração anterior com fator de aceleração aplicado
+- $\alpha$ é o fator de aceleração
+
+Essa opção é habilitada somente para os métodos de [Gauss-Seidel](powerFlow#gauss-seidel) e [Gauss-Newton híbrido](powerFlow#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu).
+
+#### Tolerância
+É o valor de erro entre as iterações (em $p.u.$) no qual é obtida a convergência do método numérico.
+
+Para o método de [Gauss-Seidel](powerFlow#gauss-seidel) o erro é calculado pelo maior valor do módulo da diferença entre a tensão na iteração anterior e atual ($e = \left| V_{i}^{(v + 1)} - V_{i}^{(v)} \right|$); enquanto que no método de [Newton-Raphson](powerFlow#newton-raphson) esse erro é o módulo do maior valor de ajuste de potência ($|\Delta P|$ ou $|\Delta Q|$).
+
+#### Iterações máx
+É o número máximo de iterações que um método poderá alcançar. Caso esse valor seja alcançado, o cálculo de [fluxo de carga](powerFlow) é interrompido com erro.
+
+#### Ângulo da barra de referência
+É o valor do ângulo da [barra de referência](powerFlow), em graus.
+
+#### Inércia do Newton
+É o valor que multiplica as correções de potência. Valores maiores que $1{,}0$ incrementam a correção e menores que $1{,}0$ diminuem a correção.
+
+:::info Informação
+Valores menores que $1{,}0$ *podem* auxiliar na convergência, porém aumentam o número de iterações e consequentemente o tempo de processamento do método.
+:::
+
+Essa opção é habilitada somente para o método de [Newton-Raphson](powerFlow#newton-raphson).
+
+#### Tolerância do Gauss
+Define a tolerância do Gauss-Seidel para o método híbrido. Essa opção é habilitada somente para o método de [Gauss-Newton híbrido](powerFlow#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu).
+
+Uma vez que o Newton-Raphson é consideravelmente sensível às condições iniciais, as primeiras iterações são calculadas utilizado o Gauss-Seidel até que o erro fique menor que a tolerância especificada no campo “Tolerância do Gauss”. A partir desse ponto o cálculo é realizado utilizando o método de Newton-Raphson até que se obtenha a convergência com um erro menor que a tolerância estipulada no campo “Tolerância”.
+
+:::info Informação
+Tal opção auxilia na convergência de sistemas impossíveis de resolver utilizando Newton-Raphson convencional.
+:::
+
+</TabItem>
+<TabItem value="stability">
+
+#### Passo de integração
+Define o passo de integração para o método de integração [Trapezoidal Implícito](stability#integração-numérica).
+
+:::caution Atenção!
+Valores muito elevados poderão gerar erros de simulação, enquanto valores muito pequenos aumentarão significativamente o tempo de processamento do cálculo de estabilidade.
+:::
+
+#### Tempo de simulação
+Define o tempo total de simulação, em segundos.
+
+#### Tolerância
+Tolerância do processo iterativo (em $p.u.$) do método de integração [Trapezoidal Implícito](stability#integração-numérica).
+
+#### Iterações máx
+Número máximo de iterações do processo iterativo do método de integração [Trapezoidal Implícito](stability#integração-numérica). Caso o número de iterações ultrapasse esse valor, o cálculo de estabilidade será encerrado com erro.
+
+#### Razão de passo dos controles
+Define a quantidade passos de integração que o sistema de controle será submetido para $1$ passo de integração do sistema de potência.
+
+Por exemplo, se o passo de integração for definido em $0{,}01~s$ e a razão de passo como $10$, o valor efetivo do passo de integração para os sistemas de controle será de $0{,}001~s$ (10 vezes menor).
+
+:::info Informação
+A utilização desse campo se justifica em situações de grandes diferenças nas constantes de tempo das máquinas síncronas (geralmente maiores) e dos sistemas de controle (geralmente menores).
+:::
+
+:::tip Dica
+Um valor de razão de passo de controles igual a $1$ não irá alterar o passo de integração do sistema de controle.
+:::
+
+#### Tempo de impressão
+É o intervalo de tempo o qual o programa irá armazenar os dados para impressão em [gráficos no tempo](graphViewer).
+
+:::warning Cuidado!
+Valores muito pequenos (menores que $0{,}001~s$) podem aumentar significativamente o tempo de processamento do cálculo de estabilidade.
+
+O tempo de impressão **deve ser maior ou igual** ao passo de integração.
+:::
+
+</TabItem>
+<TabItem value="zipLoad">
+
+#### Utilizar composição geral para todas as cargas do sistema
+Habilita a composição da [carga ZIP](load#carga-no-estudo-de-estabilidade) para todas as cargas que **não** possuem parametrização individual.
+
+#### Composição da carga (potências ativa e reativa)
+Define a composição, em porcentagem, das parcelas de Impedância, Corrente e Potência constantes da carga. Essa composição pode ser implementada individualmente para a potência ativa e reativa.
+
+:::caution Atenção!
+A soma das parcelas da carga ZIP para uma potência deve ser 100%, caso contrário será exibida uma mensagem de erro.
+:::
+
+#### Subtensão a qual as cargas serão modeladas como impedância constante
+Define o valor percentual da tensão ($V_{low}$) que, para valores inferiores a esse, a carga passa a ser considerada como impedância constante pura.
+
+:::info Informação
+Cargas de corrente e potência constantes possuem problemas em tensões muito baixas. À medida que a tensão diminui as correntes dessas cargas não reduz, resultando em perda de precisão e problemas na convergência de processos iterativos. Para contornar esse problema utiliza-se uma tensão pré-definida ($V_{low}$), a qual as cargas (ou parcelas) de corrente e potência constantes são modeladas como impedância constante.
+:::
+
+</TabItem>
+</Tabs>
+
+---
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/stability.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/stability.md
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+---
+id: stability
+title: Estabilidade
+sidebar_label: Estabilidade
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+Estabilidade de sistemas de potência é a habilidade de um Sistema Elétrico de Potência (SEP), para uma certa condição de operação, de reestabelecer um estado de operação de equilíbrio após ser submetido a um distúrbio físico. A estabilidade é reconhecida como um importante problema para a segurança da operação desde a década de 1920. Muitos *blackouts* de grande porte causados pela instabilidade do SEP têm ilustrado a importância desse fenômeno.
+
+<div><center><img src={useBaseUrl("images/stability.svg")} alt="Estabilidade" title="Estabilidade" /></center></div>
+
+Normalmente são realizados dois tipos de estudos de estabilidade (uma classificação mais aprofundada pode ser observada [nesse artigo](https://doi.org/10.1109/TPWRS.2004.825981)). A recuperação de uma rápida e brusca perturbação é chamada de **estabilidade transitória**, e sua solução é geralmente obtida no domínio do tempo. Outro estudo é denominado **estabilidade dinâmica**, utilizado para descrever a resposta do sistema frente a pequenas perturbações ao longo de muito tempo, o qual pode ser resolvido tanto no domínio da frequência quanto no domínio do tempo.
+
+:::info Informação
+No PSP-UFU, a estabilidade dinâmica é tratada como uma extensão da estabilidade transitória, em que os efeitos de pequenas constantes de tempo devem ser ignorados.
+:::
+
+O modelo para representação do SEP em um estudo de estabilidade transitória é feito por meio do sistema de equações algébrico-diferenciais (EADs), em que o estado inicial é considerado estável e originado de um estudo de [fluxo de carga](powerFlow). A solução das equações se dá por meio de um [método de integração numérica](https://en.wikipedia.org/wiki/Numerical_integration) e, então, é observada a resposta do sistema de potência.
+
+Caso a solução no domínio do tempo, após alguma perturbação, leve um ou mais parâmetros do sistema a valores fisicamente impossíveis ou, até mesmo, a solução das equações diferenciais não obtenha convergência em processos iterativos, o sistema é dito **instável**. Em contrapartida, caso as variáveis de estado dos elementos do sistema oscilem após uma perturbação e se estabeleçam em um novo ponto de operação, mesmo que distinto dos valores iniciais, o sistema é considerado **estável**.
+
+## Execução do estudo de estabilidade no PSP-UFU
+Após a construção do diagrama unifilar no [editor de potência](powerEditor), assim como os [controles](controlEditor) das [máquinas síncronas](syncGenerator), a execução da estabilidade é realizada no [menu Simulação](mainScreen#menu-ribbon) clicando no ícone do botão **Executar Estabilidade**. Ao clicar na parte inferior do botão "Executar Estabilidade" será exibido um menu suspenso com a opção **Lista de eventos de estabilidade**.
+
+<div><center><img src={useBaseUrl("images/menuSimulationStability.svg")} alt="Menu Simulação" title="Menu Simulação" /></center></div>
+
+Ao executar o cálculo de estabilidade, uma caixa com o status do processo de simulação será exibida, indicando primeiramente a inicialização do estudo e posteriormente o tempo de simulação calculado.
+
+Ao clicar na lista de eventos de estabilidade, será exibida uma janela com a descrição dos eventos de estabilidade inseridos.
+
+<div><center><img src={useBaseUrl("images/stabList.png")} alt="Lista de eventos de estabilidade" title="Lista de eventos de estabilidade" /></center></div>
+
+### Inserindo um evento de estabilidade
+Os distúrbios mais comuns aplicados em estudos de estabilidade transitória são faltas e chaveamentos. Tais operações são facilmente realizadas no PSP-UFU:
+- **Faltas**: Curtos-circuitos trifásicos podem ser inseridos nos [barramentos](bus) do sistema por meio da inclusão da impedância de falta na matriz admitância de barras da mesma maneira realizada em um elemento *shunt*. Tal valor é definido pelo usuário e caso seja um curto-circuito franco, um valor não nulo, mas suficientemente próximo de zero, é aplicado, de forma que a tensão no barramento é levada a zero durante o distúrbio.
+- **Chaveamento de ramo**: Da mesma forma que as faltas, o chaveamento de ramos é realizado por meio da alteração na matriz admitância, removendo ou inserindo os parâmetros do elemento a ser chaveado.
+	Cada [elemento de potência](powerEditor) possui um botão de "Chaveamento" ou "Estabilidade", em que pode ser inserido os tempos de remoção e/ou inserção do componente.
+- **Chaveamento de máquinas**: A remoção de uma máquina síncrona é efetivada com a retirada de sua participação no vetor de correntes, além da remoção de sua admitância fictícia.
+	Assim como os [elementos de potência](powerEditor), as [máquinas síncronas](syncGenerator) possui um botão "Chaveamento", em que pode ser inserido os tempos de remoção e/ou inserção do componente.
+	
+	:::caution Atenção!
+	Mesmo que removida da barra, os parâmetros das [máquinas síncronas](syncGenerator) continuam a ser calculados com a corrente do estator nula, podendo fornecer resultados em uma eventual reconexão.
+	:::
+
+:::tip Dica
+Eventos nos **sistemas de controle** podem ser facilmente introduzidos com o bloco de [expressão matemática](mathExpression). Nesse caso, tais eventos **não** serão exibidos na lista de eventos de estabilidade.
+:::
+
+### Erros comuns na execução do estudo de estabilidade
+A seguir são apresentados os erros mais comuns relacionados ao estudo de estabilidade.
+
+#### A seguinte mensagem de erro é exibida: "Não foi possível construir a matriz admitância"
+Essa mensagem de erro é exibida quando não é possível construir a matriz admitância de barras. As seguintes situações podem ocasionar esse erro:
+- **Os parâmetros do circuito estão incorretos**. Caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.
+- **Algum barramento está isolado**. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.
+- **Algum elemento possui parâmetros discrepantes dos demais**. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.
+
+#### A seguinte mensagem de erro é exibida: "Erro ao inicializar os valores de saturação do..."
+Essa mensagem de erro é exibida quando não é possível calcular os fatores de saturação da [máquina síncrona](syncGenerator). As seguintes situações podem ocasionar esse erro:
+- **O fator de saturação é menor que 1,2**. Esse valor deve ser maior que 1,2, ou irá gerar erros na simulação. Caso não seja informado, a saturação da máquina não é considerada nos cálculos.
+- **Os parâmetros da máquina síncrona estão incorretos**. Caso parâmetros muito fora dos valores padrão das máquinas síncronas sejam inseridos, o cálculo dos fatores de saturação pode divergir. Verifique se os dados foram inseridos corretamente.
+
+#### A seguinte mensagem de erro é exibida: "Erro ao inicializar o AVR / regulador de velocidade..."
+Essa mensagem de erro é exibida quando não é possível [inicializar o sistema de controle](controlEditor#inicialização-do-sistema-de-controle) de uma [máquina síncrona](syncGenerator). As seguintes situações podem ocasionar esse erro:
+- **O sistema de controle está vazio**. Caso esteja habilitado o AVR e/ou o regulador de velocidade e o controle não foi inserido, esse erro pode ser acionado. Insira o controle da máquina ou desmarque a opção de utilização do AVR e/ou regulador de velocidade.
+- **O sistema de controle não possui ao menos uma entrada e uma saída**. O sistema de controle deve ter ao menos uma [entrada e uma saída](io), caso contrário apresentará erro de execução.
+- **O passo de integração está muito pequeno**. Caso o passo de integração esteja muito pequeno, os cálculos gerarão erros e irão divergir. Reduza o passo de integração nas [configurações de simulação](simulationConfig).
+
+#### A seguinte mensagem de erro é exibida: "Impossível resolver as máquinas do sistema"
+Essa mensagem de erro é exibida quando não é possível resolver as EADs das máquinas síncronas inseridas no sistemas. As seguintes situações podem ocasionar esse erro:
+- **Os parâmetros da máquina síncrona estão incorretos**. Caso parâmetros muito fora dos valores padrão das máquinas síncronas sejam inseridos, o cálculo das EADs pode se tornar impossível. Verifique se os dados foram inseridos corretamente.
+- **O passo de integração está muito pequeno**. Caso o passo de integração esteja muito pequeno, os cálculos gerarão erros e irão divergir. Reduza o passo de integração nas [configurações de simulação](simulationConfig).
+- **Os parâmetros do circuito estão incorretos**. Caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.
+- **Algum barramento está isolado**. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.
+- **Algum elemento possui parâmetros discrepantes dos demais**. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.
+
+#### A seguinte mensagem de erro é exibida: "Erro ao inicializar o escorregamento do motor..."
+Essa mensagem de erro é exibida quando não é possível calcular o valor de escorregamento inicial do motor de indução. As seguintes situações podem ocasionar esse erro:
+- **Os parâmetros de estabilidade do motor estão incorretos**. Caso parâmetros muito fora dos valores padrão dos motores de indução sejam inseridos, o cálculo do escorregamento pode se tornar impossível. Verifique se os dados foram inseridos corretamente.
+
+## Estrutura da ferramenta de estabilidade
+A estabilidade de um SEP é um problema dinâmico e necessita de modelos mais elaborados de elementos de potência comparados àqueles apresentados nos outros estudos. Esses modelos são descritos individualmente, com destaque às [máquinas síncronas](syncGenerator#gerador-síncrono-no-estudo-de-estabilidade), [cargas ZIP](load#carga-no-estudo-de-estabilidade) e [motores de indução](indMotor#motor-de-indução-trifásico-no-estudo-de-estabilidade).
+
+A representação dos demais componentes do sistema elétrico: [linhas de transmissão](line), [transformadores](transformer) e elementos *shunt* (com exceção de cargas ZIP), que formam a rede de transmissão ou distribuição balanceada, é realizada utilizando os mesmos modelos do [fluxo de carga](powerFlow).
+
+Os modelos dinâmicos para a análise no domínio do tempo são na forma de um sistema de equações algébrico-diferenciais (EADs), descritas a seguir:
+$$
+\dfrac{dx}{dt}= f(x,y,u)\\
+g = f(x,y)
+$$
+Em que:
+- $x$	são as variáveis de estado
+- $y$	são as variáveis de algébricas
+- $u$	são as variáveis de entrada
+
+As variáveis de entrada são inseridas pelo usuário e permanecem constantes durante todo o processo de cálculo. Para o problema de estabilidade, as variáveis algébricas iniciais correspondem às tensões e ângulos das barras calculadas no [fluxo de carga](powerFlow). As variáveis de estado iniciais são estimadas após a convergência do fluxo de potência (inicialização dos elementos dinâmicos).
+
+Na sequência é apresentado, na forma de fluxogramas, a estrutura do módulo de estabilidade implementado no PSP-UFU. O fluxograma abaixo mostra a estrutura geral da ferramenta de estabilidade.
+
+<div><center><img src={useBaseUrl("images/stabGeneral.svg")} alt="Estrutura geral da ferramenta de estabilidade" title="Estrutura geral da ferramenta de estabilidade" /></center></div>
+
+A inicialização da rede elétrica e dos elementos dinâmicos, assim como o precesso iterativo para solução das EADs são apresentados nos dois fluxogramas conseguintes.
+
+<div><center><img src={useBaseUrl("images/stabInit.svg")} alt="Inicialização da ferramenta de estabilidade" title="Inicialização da ferramenta de estabilidade" /></center></div>
+
+<div><center><img src={useBaseUrl("images/stabIterative.svg")} alt="Processo iterativo para cálculo dos elementos dinâmicos" title="Estrutura geral da ferramenta de estabilidade" /></center></div>
+
+### Integração numérica
+Para resolver as equações diferenciais da máquina é necessário um método de integração numérica. Tais métodos são classificados em dois grupos: métodos explícitos e métodos implícitos. Os métodos explícitos, devido à sua formulação, calculam diretamente o estado do sistema em um instante de tempo posterior, enquanto métodos implícitos envolvem estados atuais e posteriores em suas equações, exigindo, portanto, um processo iterativo.
+
+As constantes de tempo presentes no estudo de estabilidade têm uma grande variação em seu valor (podem variar de $10^{-3}~s$ a $10~s$). Isso torna o sistema de equações diferenciais da máquina síncrona como um sistema rígido (*[stiff equation](https://en.wikipedia.org/wiki/Stiff_equation)*). Caso a análise da estabilidade numérica tanto das equações diferenciais rígidas quanto do método de integração obtenham o mesmo comportamento, o método é chamado de absolutamente estável, ou A-estável.
+
+Métodos de integração numérica explícitos, como por exemplo o [Runge-Kutta](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods) de quarta ordem, não podem ser A-estáveis e, portanto, normalmente possuem comportamento ruim em problemas com equações diferenciais rígidas. Por outro lado, métodos implícitos podem ser A-estáveis. Um método implícito adequado para solução do comportamento dinâmico de sistemas elétricos é o **Trapezoidal Implícito**, por possuir as seguintes vantagens:
+- É numericamente estável (A-estável);
+- É bastante rápida;
+- Possui boa precisão (dependendo somente do passo de integração utilizado).
+
+Tal método foi implementado no PSP-UFU tanto para solução das equações diferenciais da máquina síncrona quanto nas [funções transferência](transferFunction) do [sistema de controle](controlEditor). Sua formulação é dada pela seguinte expressão:
+$$
+y_{n+1}=y_n + 0{,}5 h \left(y_{n+1}' + y_{n}' \right)
+$$
+Em que:
+- $h$	é o passo de integração
+- $y_n$	é o valor do estado no passo anterior (instante $t_n$)
+- $y_{n+1}$	é o valor do estado no passo atual (instante $t_{n+1}$)
+- $y_{n}'$	é a equação diferencial no passo anterior
+- $y_{n+1}'$	é a equação diferencial no passo atual
+
+Nota-se nessa equação que o método é implícito devido ao termo $n+1$ aparecer em ambos os lados da equação, por isso é necessário um processo iterativo para sua solução.
+
+## Referências
+1. MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: https://doi.org/10.1007/978-3-642-13669-6
+1. KUNDUR, P.; et al. Definition and classification of power system stability. IEEE Transactions Power Systems, v. 19, n. 2, mai 2004, p. 1387-1401. doi: https://doi.org/10.1109/TPWRS.2004.825981
+1. ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: https://doi.org/10.1002/9781118878286
+1. PARK, R. H. Two-reactions Theory of Synchronous Machine. Generalized Method of Analysis – Part I. AIEE Transactions, v. 48, n. 3, jul 1929. doi: https://doi.org/10.1109/T-AIEE.1929.5055275
+1. ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: https://doi.org/10.1109/9780470545577
+1. SAUER, P. W.; PAI, M. A. Power System Dynamics and Stability. Pretience Hall, Upper Saddle River, 1998.
+1. KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994.
+1. DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: https://doi.org/10.1109/TPAS.1972.293341
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/sum.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/sum.md
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--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/sum.md
@@ -0,0 +1,26 @@
+---
+id: sum
+title: Somador
+sidebar_label: Somador
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+Como o próprio nome sugere, os somadores são elementos de controle capazes de somar qualquer quantidade de valores reais.
+
+## Formulário de edição de dados do somador
+
+A quantidade de entradas, assim como seus sinais são definidas pelo usuário, como mostra a figura abaixo.
+
+<div><center><img src={useBaseUrl("images/sumForm.png")} alt="Formulário de edição de dados do somador no PSP-UFU" title="Formulário de edição de dados do somador no PSP-UFU" /></center></div>
+
+No seu único campo “**Sinais**” é inserida uma lista de sinais **separados por espaços** contendo os símbolos “+” para uma entrada positiva e “-” para negativa.
+
+:::info Informação
+O usuário pode inserir duas ou mais entradas de sinais no bloco.
+:::
+
+Uma vez inseridas novas entradas, nós correspondentes serão incluídos no elemento gráfico do somador, o qual poderá ser conectado por meio das linhas de conexão.
+
+De forma semelhante, o usuário poderá excluir entradas de sinais pré-existentes, as quais serão removidas no ícone gráfico pela remoção do nó e da linha (caso exista alguma conectada).
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/syncGenerator.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/syncGenerator.md
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+---
+id: syncGenerator
+title: Gerador Síncrono
+sidebar_label: Gerador Síncrono
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+> Uma máquina de corrente alternada na qual a frequência das tensões geradas e a velocidade da máquina estão em uma proporção constante. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=411-31-08).
+
+## Gerador Síncrono no PSP-UFU
+Os geradores síncronos são a fonte de potência para o cálculo do fluxo de carga e estabilidade, além de serem um dos principais contribuintes para as correntes de falta.
+
+:::caution Atenção!
+Somente barramentos que possuem esse elemento conectado podem ser considerados [barras de referência](powerFlow). [Barras do tipo PV](powerFlow) devem conter um gerador síncrono ou um [motor síncrono (compensador síncrono)](syncMotor).
+:::
+
+O formulário de dados é dividido em dois, sendo o primeiro responsável pelos dados gerais, de [fluxo de carga](powerFlow) e de [falta](fault) e um segundo pelos dados de [estabilidade](stability) . Esse último também permite o acesso aos controles das máquinas síncronas manipulados pelo editor de controle.
+
+:::info Informação
+Os dados referentes às **impedâncias de sequência positiva do gerador síncrono** inseridas no [contexto (aba) "Falta"](syncGenerator#falta) são utilizados **tanto para os estudos de [curto-circuito](fault) quanto para os estudos de [harmônicos](harmonics)**.
+
+Esses dados são ignorados nos estudos de [fluxo de carga](powerFlow) (*não são utilizado nesse estudo*) e [estabilidade](stability) (*são utilizados dados inseridos em formulário específico*).
+:::
+
+### Gerador Síncrono no fluxo de carga
+O gerador síncrono é a fonte de potência do PSP-UFU no estudo de fluxo de carga. Seu comportamento difere de acordo com o tipo de barra conectada:
+- [Barra de referência](powerFlow): Os dados de potência ativa e reativa inseridos são desprezados, uma vez que esse elemento será utilizado para completar o balanço de potência do estudo de fluxo d carga;
+- [Barra PV](powerFlow): O dado de potência ativa é considerado, porém o dados de potência reativa são desprezados. O valor da potência reativa é utilizado para manter o módulo da tensão constante no barramento conectado;
+- [Barra PQ](powerFlow): Para os geradores conectados a essa barra tanto a potência ativa quanto reativa inseridas são consideradas.
+
+:::caution Atenção
+Caso o limite de potência reativa seja excedido, o programa automaticamente transforma a [Barra PV](powerFlow) conectada em uma [Barra PQ](powerFlow), utilizando o valor limite de potência reativa que seria ultrapassado.
+::: 
+
+### Gerador Síncrono no estudo de curto-circuito
+Enquanto os geradores no estudo de fluxo de carga são modelados somente por correntes injetadas nas barras, para o curto-circuito utiliza-se uma tensão atrás de uma impedância. A figura abaixo mostra o caminho da corrente e o circuito equivalente de cada sequência nos geradores.
+
+<div><center><img src={useBaseUrl("images/sycGeneratorSeq.svg")} alt="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero" title="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero" /></center></div>
+
+As tensões geradas são somente de sequência positiva, uma vez que o gerador fornece sempre tensões trifásicas equilibradas. Portanto a rede de sequência positiva é composta de uma tensão pré-falta atrás de uma impedância de sequência positiva. As redes de sequência negativa e zero não contêm forças eletromotrizes, porém incluem as impedâncias do gerador de sequência negativa e zero.
+
+A corrente que circula na impedância $\overline{z}_n$ entre o neutro e a terra é $3\dot{I}_{a0}$. Pela figura acima (c), observa-se que a queda de tensão de sequência zero do ponto a para terra ($\dot{V}_{a0}$) é:
+
+$$
+\dot{V}_{a0} = -3\dot{I}_{a0}\overline{z}_n - \dot{I}_{a0}\overline{z}_{g0}
+$$
+
+A rede de sequência zero, que é um circuito monofásico pelo qual se supõe que circule apenas corrente de sequência zero e deve, portanto, ter uma impedância definida pela seguinte equação:
+
+$$
+\overline{z}_{0} = 3\overline{z}_n + \overline{z}_{g0}
+$$
+
+:::tip Dica
+Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a seguinte mensagem de erro pode ser exibida:
+
+<p><center>"<i>Falha ao inverter a matriz admitância de sequência zero</i>"</center></p>
+
+Isso ocorre porque a matriz admitância de sequência zero é singular. Para contornar esse problema escolha **uma** das duas soluções abaixo:
+1. Marque a opção "Neutro aterrado" e insira um alto valor de reatância de aterramento ($j9999~p.u.$, por exemplo);
+2. **Ou**, na barra do gerador, insira um [reator](inductor) de baixo valor de potência reativa ($1,0~var$, por exemplo).
+:::
+
+### Gerador Síncrono no estudo de estabilidade
+A relação de valores observados em testes adequados (definidos na [IEEE Std. 115-2019](https://doi.org/10.1109/IEEESTD.2020.9050934)), denominados parâmetros padrões, são utilizados para modelar a máquina síncrona no estudo de estabilidade do PSP-UFU.
+
+Os parâmetros da máquina síncrona que influenciam rapidamente no decaimento de valores são chamados **subtransitórios** (indicados por $''$), aqueles que influenciam mais lentamente são chamados **transitórios** (indicados por $'$) e, finalmente, aqueles que influenciam continuamente são chamados de parâmetros **síncronos** (sem indicação de sobrescrito).
+
+Um conjunto de equações algébrico-diferenciais determinam o comportamento da máquina síncrona no estudo de estabilidade:
+
+$$
+E_{q}' - V_q = r_aI_q - x_{ds}'Id\\
+E_{d}' - V_d = r_aI_d - x_{qs}'Iq\\
+E_{q}'' - V_q = r_aI_q - x_{ds}''Id\\
+E_{d}'' - V_d = r_aI_d - x_{qs}''Iq\\
+~\\
+\frac{dE_{q}'}{dt} = \frac{V_{fd} + \left( x_d - x_{d}' \right)I_d - s_d E_{q}'}{T_{d0}'}\\
+\frac{dE_{d}'}{dt} = \frac{- \left( x_q - x_{q}' \right)I_q - s_q E_{d}'}{T_{q0}'}\\
+
+\frac{dE_{q}''}{dt} = \frac{s_dE_{q}' + \left( x_{d}' - x_{d}'' \right)I_d - s_d E_{q}''}{T_{d0}''}\\
+\frac{dE_{d}''}{dt} = \frac{s_dE_{d}' + \left( x_{q}' - x_{q}'' \right)I_q - s_d E_{d}''}{T_{q0}''}\\
+~\\
+\frac{d\omega}{dt} = \frac{\omega_r}{2H} \left[ P_m - P_e - D_a \left( \omega - \omega_r \right) \right]\\
+\frac{d\delta}{dt} = \Omega_b\left( \omega - \omega_r \right)
+$$
+
+As duas últimas equações diferenciais são as equações mecânicas da máquina; e as demais são equações elétricas (consulte essa [tese](https://dx.doi.org/10.14393/ufu.te.2019.2444) para maiores detalhes acerca dos parâmetros dessas equações).
+
+Utilizando as equações transitórias e subtransitórias podem-se definir cinco modelos de distintas complexidades.
+
+:::caution Atenção!
+O modelo da máquina síncrona é **selecionado automaticamente** de acordo com os dados fornecidos ao programa.
+:::
+
+Na sequência são apresentados tais modelos, incluídos os efeitos da saturação magnética, em conjunto com seus diagramas de blocos:
+
+- **Modelo 1**: Corresponde a uma tensão constante atrás de uma reatância transitória de eixo direto ($x_{d}'$), não exigindo equações diferenciais;
+- **Modelo 2**: São representados os efeitos transitórios de eixo direto, sendo necessária a solução de uma equação diferencial ($\frac{dE_{q}'}{dt}$), cujo diagrama de blocos é apresentado na figura abaixo:
+
+<div><center><img src={useBaseUrl("images/model2SyncGenerator.svg")} alt="Diagrama de blocos do Modelo 2 das máquinas síncronas" title="Diagrama de blocos do Modelo 2 das máquinas síncronas" /></center></div>
+
+- **Modelo 3**: São representados os efeitos transitórios de eixo direto e em quadratura, exigindo duas equações diferenciais ($\frac{dE_{q}'}{dt}$ e $\frac{dE_{d}'}{dt}$), cujo diagrama de blocos é apresentado na figura abaixo:
+
+<div><center><img src={useBaseUrl("images/model3SyncGenerator.svg")} alt="Diagrama de blocos do Modelo 3 das máquinas síncronas" title="Diagrama de blocos do Modelo 3 das máquinas síncronas" /></center></div>
+
+- **Modelo 4**: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de três equações diferenciais ($\frac{dE_{q}'}{dt}$, $\frac{dE_{q}''}{dt}$ e $\frac{dE_{d}''}{dt}$), cujo diagrama de blocos é apresentado na figura abaixo:
+
+<div><center><img src={useBaseUrl("images/model4SyncGenerator.svg")} alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas" /></center></div>
+
+- **Modelo 5**: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de quatro equações diferenciais ($\frac{dE_{q}'}{dt}$, $\frac{dE_{d}'}{dt}$, $\frac{dE_{q}''}{dt}$ e $\frac{dE_{d}''}{dt}$), cujo diagrama de blocos é apresentado na figura abaixo:
+
+<div><center><img src={useBaseUrl("images/model5SyncGenerator.svg")} alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas" /></center></div>
+
+:::info Informação
+Em todos os modelos as equações diferenciais mecânicas são solucionadas.
+:::
+
+#### Saturação
+Para representar matematicamente o efeito da saturação nas equações das máquinas síncronas são introduzidos “fatores de saturação” que modificam as impedâncias do circuito equivalente, os quais dependem de uma reatância de dispersão efetiva, chamada de reatância de Potier ($x_p$).
+
+Tal reatância pode ser obtida por meio de ensaios (utilizando curvas de saturação de circuito aberto e carga de fator de potência zero) ou estimadas de forma aproximada por outros parâmetros da máquina. A reatância de dispersão ($x_l$), aqui substituída de forma aproximada por $x_p$, representa a parcela da reatância da máquina originados do fluxo magnético que percorrem o ar na maioria de seu caminho e, portanto, é independente da saturação.
+
+O método implementado no programa permite reproduzir a saturação em ambos os eixos (direto e em quadratura), diferindo entre si devido à divergência no tamanho do entreferro. É assumido que a soma vetorial das duas componentes do fluxo magnético saturado está em fase com a f.m.m. e proporcional à Tensão de Potier ($E_p$, a qual é a tensão atrás da reatância de Potier).
+
+Para isso, são utilizados internamente dois fatores de saturação, sendo um no eixo direto ($s_d$) e outro no eixo em quadratura ($s_q$). Esses fatores de saturação são **automaticamente calculados** a cada passo de integração e dependem da curva de saturação da máquina definida pelo [fator de saturação inserido no formulário de edição de dados](syncGenerator#fator-de-saturação).
+
+Portanto as reatâncias saturadas, que devem ser inseridas nas equações algébricas da máquina, são definidas pelas seguintes equações:
+$$
+x_{ds}=\frac{x_d-x_p}{s_d +x_p}\\
+x_{qs}=\frac{x_q-x_p}{s_q +x_p}
+$$
+
+Essas equações também são utilizadas para as reatâncias transitórias e subtransitórias, visto que o valor da reatância de Potier (ou de dispersão) não é alterada.
+
+#### Barramento infinito
+Algumas referências incluem um modelo sem equações diferenciais, em que a máquina é somente representada por uma tensão constante atrás de uma reatância transitória de eixo direto. Tal é utilizado na representação de um **barramento infinito**, o qual é normalmente constituído de um subsistema muito maior àquele simulado.
+
+No PSP-UFU a **representação de um barramento infinito** pode ser obtido por meio da utilização de uma máquina representada pelo Modelo 1 cujo valor da constante de inércia (H) é infinito ou muito grande ($9999~s$, por exemplo) em relação às demais máquinas do sistema, e o valor de $x_{d}'$ deve ser um valor muito pequeno ($10^{-3}~p.u.$, por exemplo).
+
+#### Centro de inércia
+Normalmente utiliza-se a velocidade de referência como sendo a síncrona e portanto, nesse caso, $\omega_r = \omega_b = 1,0~p.u.$ Essa abordagem, adotada por vários livros de estabilidade, considera como referência uma máquina fictícia girando sempre na velocidade síncrona independente das perturbações aplicadas no sistema. No PSP-UFU foi implementado o conceito de centro de inércia (COI, do inglês, *Center of Inertia*), que constitui uma soma ponderada das velocidades das máquinas presentes no sistema:
+
+$$
+\omega_r=\frac{\left( \sum_{i=1}^{n} H_i \omega_i \right)}{\left( \sum_{i=1}^{n} H_i \right)}
+$$
+Em que:
+$n$	é o número de máquinas síncronas **conectadas** no sistema.
+
+A aplicação do COI resulta em dados de saída, como o ângulo do rotor, mais fáceis de serem analisados. Na implementação realizada no programa a utilização ou não desse recurso é opcional e pode ser [definida pelo usuário](simulationConfig).
+
+## Formulário de edição dos geradores síncronos
+A imagem abaixo apresenta o formulário de inserção/alteração de dados dos geradores síncronos:
+
+<div><center><img src={useBaseUrl("images/syncGeneratorForm.png")} alt="Formulário dos geradores síncronos no PSP-UFU" title="Formulário dos geradores síncronos no PSP-UFU" /></center></div>
+
+Um segundo pelos dados de estabilidade, como mostra a figura abaixo, acessado ao clicar no botão "Estabilidade" do formulário principal. Nele é possível também acessar aos controles das máquinas síncronas manipulados pelo [editor de controle](controlEditor).
+
+<div><center><img src={useBaseUrl("images/syncGeneratorStabForm.png")} alt="Formulário de estabilidade dos geradores síncronos no PSP-UFU" title="Formulário de estabilidade dos geradores síncronos no PSP-UFU" /></center></div>
+
+No formulário de estabilidade pode ser observado o botão "Chaveamento" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do gerador durante o estudo de [estabilidade](stability).
+
+<div><center><img src={useBaseUrl("images/syncGeneratorSw.png")} alt="Formulário de chaveamento do gerador síncrono" title="Formulário de chaveamento do gerador síncrono" /></center></div>
+
+<Tabs
+  groupId="syncGenerator-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Falta', value: 'fault'},
+    {label: 'Botão Estabilidade', value: 'stability'},
+    {label: 'Botão Chaveamento', value: 'switching'},
+  ]
+}>
+<TabItem value="general">
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+Todos os componentes de potência do PSP-UFU possuem esse campo.
+
+#### Potência nominal
+Potência nominal do gerador, inserida em MVA, kVA ou VA.
+
+Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.
+
+#### Potências ativa e reativa
+Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do gerador.
+
+Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.
+
+:::caution Atenção!
+Caso mais de um gerador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.
+:::
+
+#### Potências reativas máxima e mínima
+Limites de potência reativa máxima e mínima do gerador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.
+
+#### Utilizar potência nominal como base
+Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a [potência base do sistema](simulationConfig).
+
+</TabItem>
+<TabItem value="fault">
+
+#### Impedâncias de sequência
+Valores de resistência e reatância para cálculo das correntes de falta. São inseridos dados de sequência positiva, negativa e zero.
+
+#### Impedância de aterramento
+Valores utilizados para o cálculo das correntes de falta do tipo fase-terra e fase-fase-terra. Caso o neutro do gerador não seja aterrado, o valor inserido nesse campo é ignorado.
+
+#### Neutro aterrado
+Indica se o neutro do gerador é aterrado.
+
+</TabItem>
+<TabItem value="stability">
+
+#### Imprimir dados da máquina síncrona
+Exibe os dados do gerador síncrono nos gráficos no tempo. Os seguintes dados são exibidos:
+- tensão terminal
+- potências ativa e reativas
+- potência mecânica
+- frequência
+- tensão de campo
+- ângulo $\delta$.
+
+#### Utilizar AVR e regulador de velocidade
+Esses campos acionam ou inibem os controles da máquina síncrona, assim como modelos de turbina e excitatrizes.
+
+Caso selecionadas os controles podem ser inseridos e modificados pelos botões “Editar AVR” e “Editar regulador de velocidade”, os quais acessam o [editor de controle do PSP-UFU](controlEditor).
+
+#### Resistência de armadura
+Resistência de armadura da máquina síncrona, em $p.u.$
+
+#### Reatância de Potier
+Reatância de Potier para cálculo da saturação da máquina, em $p.u.$
+
+#### Fator de saturação
+Valor utilizado no cálculo da curva de saturação.
+
+Representa o valor (em $p.u.$) de corrente de campo necessária para atingir 1,2 p.u. de tensão terminal.
+
+:::caution Atenção!
+Esse valor deve ser **maior que 1,2**, ou irá gerar erros na simulação. Caso não seja informado, a saturação da máquina não é considerada nos cálculos.
+:::
+
+#### Frequência de circuito aberto
+Indica a velocidade da máquina no caso de início da simulação desconectada da rede.
+
+:::info Informação
+Essa informação é particularmente útil na análise de conexão de geradores dessincronizados na rede.
+:::
+
+#### Reatâncias síncronas
+Valores de reatância síncrona (regime permanente) da máquina. Os valores de eixo direto e em quadratura devem ser iguais ou muito próximos para representação de uma máquina de polos lisos, enquanto para polos salientes esses valores são distintos.
+
+#### Reatâncias e constantes de tempo transitórias
+Parâmetros transitórios da máquina síncrona em $p.u.$ ou segundos.
+
+:::warning Cuidado!
+O valor da reatância transitória de eixo direto deve ser diferente de zero ou levará o programa a erro.
+:::
+
+De acordo com a quantidade de parâmetros inseridos é definido internamente pelo programa qual o modelo a ser utilizado.
+
+#### Reatâncias e constantes de tempo subtransitórias
+Parâmetros subtransitórios da máquina síncrona em $p.u.$ ou segundos, representando em detalhes a presença de enrolamentos amortecedores. Assim como os dados transitórios, esses parâmetros definem o modelo da máquina.
+
+</TabItem>
+<TabItem value="switching">
+
+O botão "Chaveamento" irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do gerador durante o estudo de [estabilidade](stability).
+
+Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões "Adicionar" e "Remover", respectivamente.
+
+</TabItem>
+</Tabs>
+
+## Acesso aos controles da máquina síncrona
+Como já [mencionado anteriormente](syncGenerator#utilizar-avr-e-regulador-de-velocidade), os reguladores de velocidade e tensão da máquina síncrona podem ser acionados ou inibidos por meio das caixas de seleção "[Utilizar AVR e regulador de velocidade](syncGenerator#utilizar-avr-e-regulador-de-velocidade)". Ambas as opções irão acessar o [editor de controles](controlEditor).
+
+O acesso aos controles do **AVR** poderão então ser criados e manipulados ao clicar no botão "Editar AVR", assim como o **Regulador de Velocidade** é acessado no botão "Editar regulador de velocidade".
+
+:::caution Atenção!
+No PSP-UFU a opção de editar o **AVR** engloba mais que somente o controle de tensão da máquina. Nele **deve** ser inserida a malha de controle da máquina assim como a **excitatriz da máquina síncrona**. Outras estratégias de controle (opcionais), como PSS (*Power System Stabilizer*) e/ou controles de sobre e sub excitação, são também implementadas em conjunto.
+:::
+
+:::caution Atenção!
+Assim como no AVR, o **Regulador de Velocidade** engloba mais que a regulação primária da máquina. Nessa opção **deve** ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o **modelo da turbina**. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.
+:::
+
+## Referências
+1. MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: https://doi.org/10.1007/978-3-642-13669-6
+2. ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: https://doi.org/10.1002/9781118878286
+3. KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994.
+4. DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: https://doi.org/10.1109/TPAS.1972.293341
+5. IEEE Std 1110-2002 IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses. IEEE, New York, nov. 2003. doi: https://doi.org/10.1109/IEEESTD.2003.94408
+6. KIMBARK, E. W. Power System Stability: Volume III – Synchronous Machine. New York: Wiley-IEEE Press, 1995.
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/syncMotor.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/syncMotor.md
new file mode 100644
index 0000000..fb5d005
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/syncMotor.md
@@ -0,0 +1,85 @@
+---
+id: syncMotor
+title: Compensador Síncrono
+sidebar_label: Compensador Síncrono
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+> Uma máquina síncrona operando sem carga mecânica e fornecendo ou absorvendo energia reativa. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=411-34-03).
+
+## Compensador Síncrono no PSP-UFU
+O compensador síncrono nada mais é que um motor síncrono operando sem carga em seu eixo com o intuito de fornecer ou absorver potência reativa na rede. Em relação ao [fluxo de carga](powerFlow), no PSP-UFU esse elemento tem comportamento idêntico ao [gerador síncrono](syncGenerator), porém a sua potência ativa é inserida nos algoritmos de solução com sinal negativo.
+
+Sua parametrização para o estudo de [fluxo de carga](powerFlow), [falta](fault) e [harmônicos](harmonics) é idêntica ao [gerador síncrono](syncGenerator).
+
+:::warning Cuidado!
+O compensador síncrono **não foi implementado no estudo de [estabilidade](stability)** e sua presença no circuito pode causar erros durante os cálculos.
+
+Não inclua esse elemento para os estudos de [estabilidade](stability). Versões futuras do PSP-UFU irão contemplar esse elemento para esse estudo.
+:::
+
+:::tip Dica
+É possível utilizar um [gerador síncrono](syncGenerator) com potência ativa *nula* ou *negativa* para representar o compensador síncrono nos estudos de [estabilidade](stability).
+:::
+
+## Formulário de edição dos compensadores síncronos
+A imagem abaixo apresenta o formulário de inserção/alteração de dados dos compensadores síncronos:
+
+<div><center><img src={useBaseUrl("images/syncMotorForm.png")} alt="Formulário dos compensadores síncronos no PSP-UFU" title="Formulário dos compensadores síncronos no PSP-UFU" /></center></div>
+
+
+<Tabs
+  groupId="syncMotor-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Falta', value: 'fault'},
+  ]
+}>
+<TabItem value="general">
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+Todos os componentes de potência do PSP-UFU possuem esse campo.
+
+#### Potência nominal
+Potência nominal do compensador, inserida em MVA, kVA ou VA.
+
+Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.
+
+#### Potências ativa e reativa
+Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do compensador.
+
+Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.
+
+:::caution Atenção!
+Caso mais de um compensador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.
+:::
+
+#### Potências reativas máxima e mínima
+Limites de potência reativa máxima e mínima do compensador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.
+
+#### Utilizar potência nominal como base
+Caso essa opção seja marcada, o programa irá utilizar a potência nominal do compensador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a [potência base do sistema](simulationConfig).
+
+</TabItem>
+<TabItem value="fault">
+
+#### Impedâncias de sequência
+Valores de resistência e reatância para cálculo das correntes de falta. São inseridos dados de sequência positiva, negativa e zero.
+
+#### Impedância de aterramento
+Valores utilizados para o cálculo das correntes de falta do tipo fase-terra e fase-fase-terra. Caso o neutro do gerador não seja aterrado, o valor inserido nesse campo é ignorado.
+
+#### Neutro aterrado
+Indica se o neutro do gerador é aterrado.
+
+</TabItem>
+</Tabs>
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/tabularReport.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/tabularReport.md
new file mode 100644
index 0000000..abef142
--- /dev/null
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@@ -0,0 +1,20 @@
+---
+id: tabularReport
+title: Relatórios Tabulares
+sidebar_label: Relatórios Tabulares
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+Para visualização de todos os resultados dos cálculos de [fluxo de carga](powerFlow) e [curto-circuito](fault), desenvolveu-se um visualizador de relatórios tabulares:
+
+<div><center><img src={useBaseUrl("images/tabReport.png")} alt="Visualizador de dados tabulados no PSP-UFU" title="Visualizador de dados tabulados no PSP-UFU" /></center></div>
+
+Tal visualizador pode ser acessado no [submenu Ribbon](mainScreen#menu-ribbon) **Ferramentas** e então no botão **Relatório de dados**, sendo exibido o formulário organizado em abas.
+
+As grandezas de alguns dados desse relatório podem ser modificadas, sendo selecionadas no cabeçalho da tabela, como mostra a figura acima.
+
+:::tip Dica
+Os valores podem ser exportados para editores de texto e tabelas externos, selecionando e copiando (Ctrl+C) para a área de transferência.
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/text.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/text.md
new file mode 100644
index 0000000..54079a4
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/text.md
@@ -0,0 +1,57 @@
+---
+id: text
+title: Texto Vinculado
+sidebar_label: Texto Vinculado
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+O elemento de texto vinculado (ao elemento de potência) pode ser inserido no editor de [elementos de potência](powerEditor). Esse elemento de texto pode ser vinculado a inúmeros resultados e parâmetros do sistema de potência.
+
+Esse elemento é útil para identificação e análise dos resultados de elementos de potência.
+
+:::tip Dica
+Outra aplicação importante é a atualização de seu valor ao aplicar a “solução contínua” no projeto, auxiliando na visualização das mudanças de estado da rede elétrica.
+
+Também pode-se utilizar os resultados apresentados por esses elementos juntamente com o diagrama unifilar na criação de relatórios do projeto.
+:::
+
+<div><center><img src={useBaseUrl("images/textExp.png")} alt="Exemplo do elemento de texto no PSP-UFU" title="Exemplo de elemento de texto no PSP-UFU" /></center></div>
+
+## Inserindo elementos de texto no PSP-UFU
+Por meio do submenu **Ferramentas** pode-se inserir os elementos de texto, como é apresentado na imagem abaixo.
+
+<div><center><img src={useBaseUrl("images/addTextElement.svg")} alt="Adicionar elementos de texto" title="Adicionar elementos de texto" /></center></div>
+
+Ao selecionar o botão "Texto" ou utilizar a **tecla de atalho "A"**, o usuário deve clicar na posição desejada da [área de trabalho](powerEditor). Com isso, será adicionado um elemento temporário com o valor "Texto". Para parametrizar esse elemento clique duas vezes sobre ele.
+
+:::caution Atenção
+O elemento de texto, assim como os [elementos de controle](controlEditor) não exibem automaticamente o formulário de edição de dados ao inseri-lo pela primeira vez.
+:::
+
+## Formulário de edição de dados do elemento de texto
+A figura abaixo apresenta o formulário de edição de dados desse componente gráfico.
+
+<div><center><img src={useBaseUrl("images/textForm.png")} alt="Formulário do elemento de texto no PSP-UFU" title="Formulário do elemento de texto no PSP-UFU" /></center></div>
+
+Sua parametrização é realizada por campos que associam o componente aos elementos elétricos de potência, além do formato dos resultados numéricos. Os campos são liberados a medida que os dados anteriores são inseridos e uma mensagem de erro é apresentada ao usuário caso algum parâmetro necessário não seja preenchido.
+
+O processo de associação se inicia na escolha do tipo de elemento de potência será vinculado ao texto. Em seguida é exibida uma lista com o nome de todos os componentes previamente inseridos no editor de elementos de potência do tipo definido no campo anterior, associando, assim, o texto ao elemento.
+
+Em “Tipo de texto” define-se qual será a saída apresentada. Tal campo depende do tipo de elemento selecionado, como mostra a tabela abaixo. Alguns outros dados são necessários para os elementos de ramo, como barra de origem e destino, com o intuito de representar o fluxo correto de potência e corrente. Finalmente o usuário deve inserir a quantidade de casas decimais para dados numéricos. A pré-visualização pode ser observada no formulário.
+
+|Elemento de potência|Tipo de texto|Unidades disponíveis|
+|:------------------:|-------------|--------------------|
+|[Barra](bus)|Nome<br/>Tensão<br/>Ângulo<br/>Corrente de falta<br/>Tensão de falta<br/>Nível de curto-circuito<br/>THD de tensão|-<br/>$p.u.$, V, kV<br/>Graus, Radianos<br/>$p.u.$, A, kA<br/>$p.u.$, V, kV<br/>$p.u.$, VA, kVA, MVA<br/>%|
+|[Gerador](syncGenerator)|Nome<br/>Potência ativa<br/>Potência reativa<br/>Corrente de falta|-<br/>$p.u.$, W, kW, MW<br/>$p.u.$, var, kvar, Mvar<br/>$p.u.$, A, kA|
+|[Linha](line) e [transformador](transformer)|Nome<br/>Fluxo de potência ativo<br/>Fluxo de potência reativo<br/>Perdas<br/>Corrente<br/>Corrente de falta|-<br/>$p.u.$, W, kW, MW<br/>$p.u.$, var, kvar, Mvar<br/>$p.u.$, W, kW, MW<br/>$p.u.$, A, kA<br/>$p.u.$, A, kA|
+|[Carga](load), [Motor de indução](indMotor)<br/>e [Compensador síncrono](syncMotor)|Nome<br/>Potência ativa<br/>Potência reativa|-<br/>$p.u.$, W, kW, MW<br/>$p.u.$, var, kvar, Mvar|
+|[Capacitor](capacitor) e [Indutor](inductor)|Nome<br/>Potência reativa|-<br/>$p.u.$, var, kvar, Mvar|
+
+:::info Informação
+Uma vez inseridos, esses elementos de texto podem ser manipulados da [mesma forma que os elementos de potência](cadTools).
+:::
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/transferFunction.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/transferFunction.md
new file mode 100644
index 0000000..2319b57
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/transferFunction.md
@@ -0,0 +1,95 @@
+---
+id: transferFunction
+title: Função Transferência
+sidebar_label: Função Transferência
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+A Função Transferência é um importante bloco elementar e essencial em um sistema de controle, visto que esse componente possibilita a representação de sistemas dinâmicos por meio de equações algébricas no domínio da frequência.
+
+:::info Informação
+Nesses blocos é utilizado a letra “**s**” para representação no domínio da frequência, pois essa é a terminologia mais utilizada em sistemas de controle.
+:::
+
+## Função Transferência no PSP-UFU
+
+Após a inserção da função transferência, o programa a transforma na representação em espaço de estado, visto que sua solução é sistemática tornando-a, assim, mais eficiente computacionalmente.
+
+Dada uma função transferência genérica abaixo:
+
+$$
+\frac{Y(\bold{s})}{U(\bold{s})} = \frac{b_0 \bold{s}^k + b_1 \bold{s}^{k-1} + \cdots + b_{k-1} \bold{s} + b_k}{\bold{s}^k + a_1 \bold{s}^{k-1} + \cdots + a_{k-1} \bold{s} + a_k}
+$$
+
+A transformação em espaço de estado na *forma canônica controlável* ficará da seguinte forma:
+
+$$
+\begin{bmatrix}
+\bold{s}x_1\\
+\bold{s}x_2\\
+\vdots\\
+\bold{s}x_{k-1}\\
+\bold{s}x_k
+\end{bmatrix}
+=
+\begin{bmatrix}
+0 & 1 & 0 & \cdots & 0\\
+0 & 0 & 1 & \cdots & 0\\
+\vdots & \vdots & \vdots & \ddots & \vdots\\
+0 & 0 & 0 & \cdots & 1\\
+-a_k & -a_{k-1} & -a_{k-2} & \cdots & -a_1
+\end{bmatrix}
+
+\begin{bmatrix}
+x_1\\
+x_2\\
+\vdots\\
+x_{k-1}\\
+x_k
+\end{bmatrix}
++
+\begin{bmatrix}
+0\\
+0\\
+\vdots\\
+0\\
+1
+\end{bmatrix}
+u_n\\
+
+y_n =
+\begin{bmatrix}
+\left( b_n - a_n b_0 \right) & \left( b_{n-1} - a_{n-1} b_0 \right) & \cdots & \left( b_1 - a_1 b_0 \right)
+\end{bmatrix}
+
+\begin{bmatrix}
+x_1\\
+x_2\\
+\vdots\\
+x_k
+\end{bmatrix}
++
+b_0 u_n
+$$
+
+A equação diferencial matricial é resolvida pelo método **Trapezoidal Implícito**. Um processo iterativo é realizado até que o erro entre os cálculos se torne menor que uma tolerância pré-estipulada e caso as iterações se tornem excessivas, ultrapassando um número máximo definido pelo usuário, o processo é interrompido com erro. Com os valores do vetor de estado ($x$) calculados obtém-se o valor da saída do bloco ($y_n$).
+
+## Formulário de edição de dados da Função Transferência
+
+A figura abaixo apresenta o formulário de edição de dados da função transferência.
+
+<div><center><img src={useBaseUrl("images/transferFunctionForm.png")} alt="Formulário de edição de dados da função transferência no PSP-UFU" title="Formulário de edição de dados da função transferência no PSP-UFU" /></center></div>
+
+A função transferência é definida pelos coeficientes do numerador e denominador, **separados por espaço**.
+
+:::caution Atenção!
+Os elementos não presentes são representados como coeficientes de valor zero.
+
+Por exemplo, um parâmetro inserido igual a “$\begin{matrix} 1 & 0{,}5 & 0 & 2 \end{matrix}$” irá gerar: “$s^3+0{,}5s^2+2$”.
+:::
+
+## Referências
+
+1. OGATA, K. Modern Control Engineering. Prentice Hall Inc., New Jersey, 2004.
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/transformer.md b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/transformer.md
new file mode 100644
index 0000000..ead4e06
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-plugin-content-docs/current/transformer.md
@@ -0,0 +1,176 @@
+---
+id: transformer
+title: Transformador
+sidebar_label: Transformador
+---
+import useBaseUrl from "@docusaurus/useBaseUrl";
+
+<link rel="stylesheet" href={useBaseUrl("katex/katex.min.css")} />
+
+import Tabs from '@theme/Tabs';
+import TabItem from '@theme/TabItem';
+
+>Aparelho estático com dois ou mais enrolamentos que, por indução eletromagnética, transforma um sistema de tensão e corrente alternada em outro sistema de tensão e corrente geralmente de valores diferentes e na mesma frequência com o objetivo de transmitir energia elétrica. [*tradução livre* - IEC 60050](
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=421-01-01).
+
+## Transformador no PSP-UFU
+Transformadores com quaisquer conexões e defasagens entre as tensões primárias e secundárias podem ser inseridos no PSP-UFU. Para isso, os transformadores devem ser modelados representando seu tape ($t$) e defasagem ($\phi$), utilizando um transformador ideal de relação de transformação $\overline{t}:1$ em série com sua impedância ($\overline{y}_T = r_T + jx_T$), em que $\overline{t}$ é um número complexo ($t\angle\phi$)
+
+:::caution Atenção
+O modelo utilizado no PSP-UFU **não considera** o ramo magnetizante ou impedâncias mútuas entre as fases.
+:::
+
+A figura abaixo mostra o modelo do transformador implementado no PSP-UFU:
+
+<div><center><img src={useBaseUrl("images/transformerModel.svg")} alt="Modelo PI da linha" title="Modelo PI da linha" /></center></div>
+
+Em que:
+- $\bold{r_T}$	é a resistência total do transformador;
+- $\bold{x_T}$	é a reatância indutiva total do transformador;
+- $\bold{\overline{t}}$	é a o tape complexo do transformador.
+
+A relação entre as tensões e correntes das barras genéricas $i$ e $j$ é obtida pela equação matricial:
+
+$$
+\begin{bmatrix}
+\dot{I}_{ij}\\
+\dot{I}_{ji}
+\end{bmatrix}
+=
+\begin{bmatrix}
+\displaystyle \frac{\overline{y}_T}{t^2} & -\displaystyle \frac{\overline{y}_T}{\overline{t}^*}\\
+-\displaystyle \frac{\overline{y}_T}{\overline{t}} & \overline{y}_T
+\end{bmatrix}
+\times
+\begin{bmatrix}
+\dot{V}_{i}\\
+\dot{V}_{j}
+\end{bmatrix}
+$$
+
+Assim como nas [linhas](line), o transformador de dois enrolamentos deve ser inserido entre barras, porém não há a necessidade das tensões nominais desses barramentos serem idênticas.
+
+:::caution Atenção!
+A primeira barra selecionada será o lado **primário**, o qual é indicado por um círculo. Para alterar o lado primário após a inserção basta desconectar os nós dos transformadores e reconectá-los alternando as barras, utilizando as [ferramentas CAD](cadTools#conexão-e-desconexão-de-elemetos-na-barra).
+:::
+
+### Transformadores no estudo de curto-circuito
+Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas [faltas desbalanceadas](fault) é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.
+
+A matriz admitância de sequência positiva é construída da mesma forma que os estudos de fluxo de carga. Na sequência negativa o ângulo de defasagem entre tensões primária e secundária ($\phi$) deve ser invertido, uma vez que os fasores da sequência negativa **são deslocados na direção oposta**. Com isso, a matriz que representa as admitâncias da equação anterior deve ser substituída pela seguinte expressão:
+
+$$
+\begin{bmatrix}
+\overline{Y}_{ii} & \overline{Y}_{ij}\\
+\overline{Y}_{ji} & \overline{Y}_{jj}
+\end{bmatrix}
+=
+\begin{bmatrix}
+\displaystyle \frac{\overline{y}_T}{t^2} & -\displaystyle \frac{\overline{y}_T}{\overline{t}}\\
+-\displaystyle \frac{\overline{y}_T}{\overline{t}^*} & \overline{y}_T
+\end{bmatrix}
+$$
+
+A impedância de sequência negativa nos transformadores deve ser tratada de maneira específica devido aos diferentes tipos de conexão. Na figura abaixo são mostrados os circuitos equivalentes para cada tipo de conexão de transformadores de dois enrolamentos. As setas indicam os caminhos possíveis para circulação da corrente de sequência zero.
+
+<div><center><img src={useBaseUrl("images/seqZeroTrafo.svg")} alt="Circuitos equivalentes de sequência zero dos transformadores" title="Circuitos equivalentes de sequência zero dos transformadores" /></center></div>
+
+Para os transformadores com conexão estrela aterrado, se a ligação do neutro para a terra apresentar uma impedância $\overline{z}_n$, o circuito equivalente de sequência zero deve ter impedância de $3\overline{z}_n$ em série com a resistência e reatância equivalentes do transformador.
+
+## Formulário de edição dos transformadores
+
+A imagem abaixo apresenta o formulário de inserção/alteração de dados dos transformadores:
+
+<div><center><img src={useBaseUrl("images/trafoForm.png")} alt="Formulário dos transformadores no PSP-UFU" title="Formulário dos transformadores no PSP-UFU" /></center></div>
+
+Esse formulário é subdividido em dois contextos distintos:
+- **Geral**: no qual são inseridas informações gerais dos transformadores, informações do fluxo de carga, sua conexão e defasagem;
+- **Falta**: local onde as impedâncias de sequência zero e impedâncias de aterramento são inseridas.
+
+Além desses dois contextos, pode ser observado o botão "Estabilidade" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do transformador durante o estudo de [estabilidade](stability).
+
+<div><center><img src={useBaseUrl("images/trafoSw.png")} alt="Formulário de chaveamento do transformador" title="Formulário de chaveamento do transformador" /></center></div>
+
+<Tabs
+  groupId="transformer-tabs"
+  defaultValue="general"
+  values={[
+    {label: 'Geral', value: 'general'},
+    {label: 'Falta', value: 'fault'},
+    {label: 'Botão Estabilidade', value: 'stability'},
+  ]
+}>
+<TabItem value="general">
+
+#### Nome
+Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão [Unicode](https://pt.wikipedia.org/wiki/Unicode).
+
+Todos os componentes de potência do PSP-UFU possuem esse campo.
+
+#### Tensão nominal
+Campo de informação *não editável* que apresenta a tensão primária e secundária do transformador. Para alterar esse campo é necessário editar o campo correspondente dos [barramento](bus#geral) conectados.
+
+#### Tensão base
+Tensão utilizada para converter os parâmetros de impedância inseridos em $\Omega$ para $p.u.$ As tensões selecionadas por uma caixa de escolha são editadas nos formulários dos [barramentos](bus#geral) conectados.
+
+:::info Informação
+A tensão base é ignorada caso os dados inseridos estejam em $p.u.$
+:::
+
+#### Potência nominal
+Potência nominal do transformador, inserida em MVA, kVA ou VA.
+
+Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.
+
+#### Impedância do transformador
+Impedância série do transformador (resistência e reatância indutiva), inserida em $p.u.$ ou $\Omega$. Caso as impedâncias sejam inseridas em $\Omega$ será utilizada a tensão base selecionada para conversão para $p.u.$
+
+#### Conexão
+Conexão do transformador utilizada para cálculo das correntes de sequência zero nas faltas desbalanceadas com presença de terra (fase-terra e fase-fase-terra).
+
+As seguintes conexões estão disponíveis:
+1. $Y_{aterrado}-Y_{aterrado}$
+1. $Y-Y_{aterrado}$
+1. $Y_{aterrado}-Y$
+1. $Y-Y$
+1. $\Delta-Y_{aterrado}$
+1. $\Delta-Y$
+1. $Y_{aterrado}-\Delta$
+1. $Y-\Delta$
+1. $\Delta-\Delta$
+
+#### TAP
+Tape do transformador **em relação ao primário**.
+
+:::caution Atenção!
+O tape no PSP-UFU deve ser inserido pela relação entre a tensão primária nominal e a tensão de tape $\left(\text{TAP} = \frac{V_{nominal}}{V_{tape}}\right)$.
+:::
+
+#### Defasagem
+Defasagem do transformador em relação ao primário, em graus. Esse valor independe do tipo de conexão, portanto para conexões diferentes de $Y-Y$(aterrado ou não) e $\Delta-\Delta$, deve-se inserir o ângulo de defasagem correto.
+
+#### Utilizar potência nominal como base
+Caso essa opção seja marcada, o programa irá utilizar a potência nominal do transformador como base para a conversão das unidades, caso contrário será usada a [potência base do sistema](simulationConfig).
+
+</TabItem>
+<TabItem value="fault">
+
+#### Impedância de sequência zero do transformador
+Esses parâmetros são necessários para o correto cálculo das correntes de [falta desbalanceadas](fault) (fase-fase, fase-fase-terra e fase-terra) e devem ser inseridos em $p.u.$
+
+:::warning Cuidado!
+A não inserção desses dados acarretarão em resultados incorretos para [faltas desbalanceadas](fault).
+:::
+
+#### Impedâncias de aterramento
+As impedâncias de aterramento são somente utilizadas para conexão $Y_{aterrado}$ e devem ser inseridas em $p.u.$
+
+</TabItem>
+<TabItem value="stability">
+
+O botão "Estabilidade" irá abrir um formulário, comum a vários outros elementos, que permite a inserção e/ou remoção do transformador durante o estudo de [estabilidade](stability).
+
+Nesse formulário pode ser criada uma lista genérica de inserções e remoções da linha no tempo, personalizada por um contexto de propriedades de chaveamento que são editados o tipo de chaveamento (inserção ou remoção) e o instante (em segundos) do evento. Essas propriedades são atribuídas e retiradas da lista genérica por meio dos botões "Adicionar" e "Remover", respectivamente.
+
+</TabItem>
+</Tabs>
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-theme-classic/footer.json b/docusaurus/i18n/pt/docusaurus-theme-classic/footer.json
new file mode 100644
index 0000000..c204650
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-theme-classic/footer.json
@@ -0,0 +1,38 @@
+{
+  "copyright": {
+    "message": "Copyright © 2021 Thales Lima Oliveira. Construído com Docusaurus.",
+    "description": "The footer copyright"
+  },
+  "link.title.Guide": {
+    "message": "Manual",
+    "description": "The title of the footer links column with title=Guide in the footer"
+  },
+  "link.title.Community": {
+    "message": "Comunidade",
+    "description": "The title of the footer links column with title=Community in the footer"
+  },
+  "link.title.Site Image Credit": {
+    "message": "Crédito das Imagens do Site",
+    "description": "The title of the footer links column with title=Site Image Credit in the footer"
+  },
+  "link.item.label.PSP-UFU Guide": {
+    "message": "Manual do PSP-UFU",
+    "description": "The label of footer link with label=PSP-UFU Guide linking to docs/"
+  },
+  "link.item.label.Code Documentation": {
+    "message": "Documentação do Código",
+    "description": "The label of footer link with label=Code Documentation linking to https://thales1330.github.io/PSP/doxygen/html/index.html"
+  },
+  "link.item.label.Ask a question": {
+    "message": "Faça uma pergunta",
+    "description": "The label of footer link with label=Ask a question linking to https://github.com/Thales1330/PSP/issues"
+  },
+  "link.item.label.Twitter": {
+    "message": "Twitter",
+    "description": "The label of footer link with label=Twitter linking to https://twitter.com/PspUfu"
+  },
+  "link.item.label.Images designed by upklyak / slidesgo / macrovector / Freepik": {
+    "message": "Imagens criadas por upklyak / slidesgo / macrovector / Freepik",
+    "description": "The label of footer link with label=Images designed by upklyak / slidesgo / macrovector / Freepik linking to http://www.freepik.com/"
+  }
+}
\ No newline at end of file
diff --git a/docusaurus/i18n/pt/docusaurus-theme-classic/navbar.json b/docusaurus/i18n/pt/docusaurus-theme-classic/navbar.json
new file mode 100644
index 0000000..2cdae08
--- /dev/null
+++ b/docusaurus/i18n/pt/docusaurus-theme-classic/navbar.json
@@ -0,0 +1,18 @@
+{
+  "title": {
+    "message": "PSP-UFU",
+    "description": "The title in the navbar"
+  },
+  "item.label.User Guide": {
+    "message": "Manual do Usuário",
+    "description": "Navbar item with label User Guide"
+  },
+  "item.label.Changelog": {
+    "message": "Registro de Alterações",
+    "description": "Navbar item with label Changelog"
+  },
+  "item.label.GitHub": {
+    "message": "GitHub",
+    "description": "Navbar item with label GitHub"
+  }
+}
\ No newline at end of file
diff --git a/docusaurus/package-lock.json b/docusaurus/package-lock.json
index 3f8be4f..a10014a 100644
--- a/docusaurus/package-lock.json
+++ b/docusaurus/package-lock.json
@@ -4,432 +4,435 @@
   "lockfileVersion": 1,
   "requires": true,
   "dependencies": {
+    "@algolia/autocomplete-core": {
+      "version": "1.0.0-alpha.44",
+      "resolved": "https://registry.npmjs.org/@algolia/autocomplete-core/-/autocomplete-core-1.0.0-alpha.44.tgz",
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+      "requires": {
+        "@algolia/autocomplete-shared": "1.0.0-alpha.44"
+      }
+    },
+    "@algolia/autocomplete-preset-algolia": {
+      "version": "1.0.0-alpha.44",
+      "resolved": "https://registry.npmjs.org/@algolia/autocomplete-preset-algolia/-/autocomplete-preset-algolia-1.0.0-alpha.44.tgz",
+      "integrity": "sha512-DCHwo5ovzg9k2ejUolGNTLFnIA7GpsrkbNJTy1sFbMnYfBmeK8egZPZnEl7lBTr27OaZu7IkWpTepLVSztZyng==",
+      "requires": {
+        "@algolia/autocomplete-shared": "1.0.0-alpha.44"
+      }
+    },
+    "@algolia/autocomplete-shared": {
+      "version": "1.0.0-alpha.44",
+      "resolved": "https://registry.npmjs.org/@algolia/autocomplete-shared/-/autocomplete-shared-1.0.0-alpha.44.tgz",
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+    },
     "@algolia/cache-browser-local-storage": {
-      "version": "4.3.1",
-      "resolved": "https://registry.npmjs.org/@algolia/cache-browser-local-storage/-/cache-browser-local-storage-4.3.1.tgz",
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+      "version": "4.8.6",
+      "resolved": "https://registry.npmjs.org/@algolia/cache-browser-local-storage/-/cache-browser-local-storage-4.8.6.tgz",
+      "integrity": "sha512-Bam7otzjIEgrRXWmk0Amm1+B3ROI5dQnUfJEBjIy0YPM0kMahEoJXCw6160tGKxJLl1g6icoC953nGshQKO7cA==",
       "requires": {
-        "@algolia/cache-common": "4.3.1"
+        "@algolia/cache-common": "4.8.6"
       }
     },
     "@algolia/cache-common": {
-      "version": "4.3.1",
-      "resolved": "https://registry.npmjs.org/@algolia/cache-common/-/cache-common-4.3.1.tgz",
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+      "version": "4.8.6",
+      "resolved": "https://registry.npmjs.org/@algolia/cache-common/-/cache-common-4.8.6.tgz",
+      "integrity": "sha512-eGQlsXU5G7n4RvV/K6qe6lRAeL6EKAYPT3yZDBjCW4pAh7JWta+77a7BwUQkTqXN1MEQWZXjex3E4z/vFpzNrg=="
     },
     "@algolia/cache-in-memory": {
-      "version": "4.3.1",
-      "resolved": "https://registry.npmjs.org/@algolia/cache-in-memory/-/cache-in-memory-4.3.1.tgz",
-      "integrity": "sha512-bd2Aqn8efGJpR8snjUvBJIONyQ2uqYQSbFH9rTrLPmJPMYdoTKTcVLrtpOhOlmvTTfguhqlv+zIjYdJcraeBvg==",
+      "version": "4.8.6",
+      "resolved": "https://registry.npmjs.org/@algolia/cache-in-memory/-/cache-in-memory-4.8.6.tgz",
+      "integrity": "sha512-kbJrvCFANxL/l5Pq1NFyHLRphKDwmqcD/OJga0IbNKEulRGDPkt1+pC7/q8d2ikP12adBjLLg2CVias9RJpIaw==",
       "requires": {
-        "@algolia/cache-common": "4.3.1"
+        "@algolia/cache-common": "4.8.6"
       }
     },
     "@algolia/client-account": {
-      "version": "4.3.1",
-      "resolved": "https://registry.npmjs.org/@algolia/client-account/-/client-account-4.3.1.tgz",
-      "integrity": "sha512-062Cxw61llvkeHS2bWghufNI0munw5fKGZBhUfDdnC7lsJpzYJwQdkdchzLqqIOXZa8k9vdLlnlKHk8f53E5fQ==",
+      "version": "4.8.6",
+      "resolved": "https://registry.npmjs.org/@algolia/client-account/-/client-account-4.8.6.tgz",
+      "integrity": "sha512-FQVJE/BgCb78jtG7V0r30sMl9P5JKsrsOacGtGF2YebqI0YF25y8Z1nO39lbdjahxUS3QkDw2d0P2EVMj65g2Q==",
       "requires": {
-        "@algolia/client-common": "4.3.1",
-        "@algolia/client-search": "4.3.1",
-        "@algolia/transporter": "4.3.1"
+        "@algolia/client-common": "4.8.6",
+        "@algolia/client-search": "4.8.6",
+        "@algolia/transporter": "4.8.6"
       }
     },
     "@algolia/client-analytics": {
-      "version": "4.3.1",
-      "resolved": "https://registry.npmjs.org/@algolia/client-analytics/-/client-analytics-4.3.1.tgz",
-      "integrity": "sha512-+/gn1z3oAh2CE0xox7/Df9SseHcOuqgm4ngSXGh1cWpldsF+gioA9HWSh/4RSydViASKu3YIk5O61zFzVTKbOA==",
+      "version": "4.8.6",
+      "resolved": "https://registry.npmjs.org/@algolia/client-analytics/-/client-analytics-4.8.6.tgz",
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       "resolved": "https://registry.npmjs.org/xtend/-/xtend-4.0.2.tgz",
       "integrity": "sha512-LKYU1iAXJXUgAXn9URjiu+MWhyUXHsvfp7mcuYm9dSUKK0/CjtrUwFAxD82/mCWbtLsGjFIad0wIsod4zrTAEQ=="
     },
     "y18n": {
-      "version": "4.0.0",
-      "resolved": "https://registry.npmjs.org/y18n/-/y18n-4.0.0.tgz",
-      "integrity": "sha512-r9S/ZyXu/Xu9q1tYlpsLIsa3EeLXXk0VwlxqTcFRfg9EhMW+17kbt9G0NrgCmhGb5vT2hyhJZLfDGx+7+5Uj/w=="
+      "version": "4.0.1",
+      "resolved": "https://registry.npmjs.org/y18n/-/y18n-4.0.1.tgz",
+      "integrity": "sha512-wNcy4NvjMYL8gogWWYAO7ZFWFfHcbdbE57tZO8e4cbpj8tfUcwrwqSl3ad8HxpYWCdXcJUCeKKZS62Av1affwQ=="
     },
     "yallist": {
-      "version": "3.1.1",
-      "resolved": "https://registry.npmjs.org/yallist/-/yallist-3.1.1.tgz",
-      "integrity": "sha512-a4UGQaWPH59mOXUYnAG2ewncQS4i4F43Tv3JoAM+s2VDAmS9NsK8GpDMLrCHPksFT7h3K6TOoUNn2pb7RoXx4g=="
+      "version": "4.0.0",
+      "resolved": "https://registry.npmjs.org/yallist/-/yallist-4.0.0.tgz",
+      "integrity": "sha512-3wdGidZyq5PB084XLES5TpOSRA3wjXAlIWMhum2kRcv/41Sn2emQ0dycQW4uZXLejwKvg6EsvbdlVL+FYEct7A=="
     },
     "yaml": {
       "version": "1.10.0",
@@ -13683,6 +16870,36 @@
         "yargs-parser": "^13.1.2"
       },
       "dependencies": {
+        "find-up": {
+          "version": "3.0.0",
+          "resolved": "https://registry.npmjs.org/find-up/-/find-up-3.0.0.tgz",
+          "integrity": "sha512-1yD6RmLI1XBfxugvORwlck6f75tYL+iR0jqwsOrOxMZyGYqUuDhJ0l4AXdO1iX/FTs9cBAMEk1gWSEx1kSbylg==",
+          "requires": {
+            "locate-path": "^3.0.0"
+          }
+        },
+        "locate-path": {
+          "version": "3.0.0",
+          "resolved": "https://registry.npmjs.org/locate-path/-/locate-path-3.0.0.tgz",
+          "integrity": "sha512-7AO748wWnIhNqAuaty2ZWHkQHRSNfPVIsPIfwEOWO22AmaoVrWavlOcMR5nzTLNYvp36X220/maaRsrec1G65A==",
+          "requires": {
+            "p-locate": "^3.0.0",
+            "path-exists": "^3.0.0"
+          }
+        },
+        "p-locate": {
+          "version": "3.0.0",
+          "resolved": "https://registry.npmjs.org/p-locate/-/p-locate-3.0.0.tgz",
+          "integrity": "sha512-x+12w/To+4GFfgJhBEpiDcLozRJGegY+Ei7/z0tSLkMmxGZNybVMSfWj9aJn8Z5Fc7dBUNJOOVgPv2H7IwulSQ==",
+          "requires": {
+            "p-limit": "^2.0.0"
+          }
+        },
+        "path-exists": {
+          "version": "3.0.0",
+          "resolved": "https://registry.npmjs.org/path-exists/-/path-exists-3.0.0.tgz",
+          "integrity": "sha1-zg6+ql94yxiSXqfYENe1mwEP1RU="
+        },
         "string-width": {
           "version": "3.1.0",
           "resolved": "https://registry.npmjs.org/string-width/-/string-width-3.1.0.tgz",
@@ -13711,6 +16928,11 @@
         }
       }
     },
+    "yocto-queue": {
+      "version": "0.1.0",
+      "resolved": "https://registry.npmjs.org/yocto-queue/-/yocto-queue-0.1.0.tgz",
+      "integrity": "sha512-rVksvsnNCdJ/ohGc6xgPwyN8eheCxsiLM8mxuE/t/mOVqJewPuO1miLpTHQiRgTKCLexL4MeAFVagts7HmNZ2Q=="
+    },
     "zwitch": {
       "version": "1.0.5",
       "resolved": "https://registry.npmjs.org/zwitch/-/zwitch-1.0.5.tgz",
diff --git a/docusaurus/package.json b/docusaurus/package.json
index 9a6e00d..da32722 100644
--- a/docusaurus/package.json
+++ b/docusaurus/package.json
@@ -1,16 +1,17 @@
 {
   "name": "psp-ufu",
-  "version": "0.1.1",
+  "version": "0.1.2",
   "private": true,
   "scripts": {
     "start": "docusaurus start",
     "build": "docusaurus build",
     "swizzle": "docusaurus swizzle",
-    "deploy": "docusaurus deploy"
+    "deploy": "docusaurus deploy",
+    "write-translations": "docusaurus write-translations"
   },
   "dependencies": {
-    "@docusaurus/core": "^2.0.0-alpha.61",
-    "@docusaurus/preset-classic": "^2.0.0-alpha.61",
+    "@docusaurus/core": "^2.0.0-alpha.71",
+    "@docusaurus/preset-classic": "^2.0.0-alpha.71",
     "clsx": "^1.1.1",
     "galleria": "^1.6.1",
     "react": "^16.8.4",
diff --git a/docusaurus/sidebars.js b/docusaurus/sidebars.js
index 2e0c2c2..d1da014 100644
--- a/docusaurus/sidebars.js
+++ b/docusaurus/sidebars.js
@@ -1,36 +1,36 @@
 module.exports = {
   docs: {
-    'Introdução': [
+    'Introduction': [
 		'aboutPSP',
 		'installation',
 		'mainScreen',
 		'cadTools'
 	],
 		
-    'Editor de Elementos de Potência': [
+    'Power Element Editor': [
 		'powerEditor',
 		{
 			type: 'category',
-			label: 'Elementos de Potência',
+			label: 'Power Elements',
 			items: ['bus', 'line', 'transformer', 'load', 'capacitor', 'inductor', 'harmSource', 'syncGenerator', 'syncMotor', 'indMotor'],
 		},
 	],
-    'Editor de Elementos de Controle': [
+    'Control Element Editor': [
 	'controlEditor',
 		{
 			type: 'category',
-			label: 'Elementos de Controle',
+			label: 'Control Elements',
 			items: ['io', 'transferFunction', 'sum', 'multiplier', 'divider', 'exponential', 'limiter', 'rateLimiter', 'const', 'gain', 'mathExpression'],
 		},
 	],
-    'Simulações': [
+    'Simulations': [
 		'powerFlow',
 		'fault',
 		'harmonics',
 		'stability',
 		'simulationConfig',
 	],
-	'Visualização dos Dados' : [
+	'Data Visualization' : [
 		'text',
 		'tabularReport',
 		'graphViewer',
diff --git a/docusaurus/src/pages/index.js b/docusaurus/src/pages/index.js
index 3c60c4d..0a4c7c0 100644
--- a/docusaurus/src/pages/index.js
+++ b/docusaurus/src/pages/index.js
@@ -2,44 +2,45 @@ import React from 'react';
 import clsx from 'clsx';
 import Layout from '@theme/Layout';
 import Link from '@docusaurus/Link';
+import Translate, {translate} from '@docusaurus/Translate';
 import useDocusaurusContext from '@docusaurus/useDocusaurusContext';
 import useBaseUrl from '@docusaurus/useBaseUrl';
 import styles from './styles.module.css';
 
 const features = [
   {
-    title: <>Complete Platform</>,
+    title: <Translate>Complete Platform</Translate>,
     imageUrl: 'img/complete-platform.svg',
     description: (
       <>
-        PSP-UFU is a <b>cross-platform</b>, <b>multilingual</b>, <b>Free and Open-Source Software (FOSS)</b> with <b>advanced GUI (Graphical User Interface)</b> features and <b>CAD (Computer-Aided Design) tools</b> for electrical power system studies.
+        <Translate>PSP-UFU is a</Translate> <b><Translate>cross-platform, multilingual, Free and Open-Source Software (FOSS)</Translate></b> <Translate>with</Translate> <b><Translate>advanced GUI (Graphical User Interface) features</Translate></b> <Translate>and</Translate> <b><Translate>CAD (Computer-Aided Design) tools</Translate></b> <Translate>for electrical power system studies</Translate>.
       </>
     ),
   },
   {
-    title: <>Advanced CAD Features</>,
+    title: <><Translate>Advanced CAD Features</Translate></>,
     imageUrl: 'img/advanced-cad.svg',
     description: (
       <>
-        The software allows the construction of any electric <b>transmission network</b> and <b>control systems</b> through the deployment of visual elements.
+        <Translate>The software allows the construction of any</Translate> <b><Translate>electric transmission network</Translate></b> <Translate>and</Translate> <b><Translate>control systems</Translate></b> <Translate>through the deployment of visual elements</Translate>.
       </>
     ),
   },
   {
-    title: <>Easy Visualization</>,
+    title: <><Translate>Easy Visualization</Translate></>,
     imageUrl: 'img/visualization.svg',
     description: (
       <>
-        For the visualization of results, the program offers linked text elements in the main screen, voltage heatmap display, and also table and graph editors.
+        <Translate>For the visualization of results, the program offers linked text elements in the main screen, voltage heatmap display, and also table and graph editors.</Translate>
       </>
     ),
   },
   {
-    title: <>Application</>,
+    title: <><Translate>Application</Translate></>,
     imageUrl: 'img/application.svg',
     description: (
       <>
-        The PSP-UFU aims to provide efficient computer simulation tools for <b>research</b> and <b>education</b> purposes, in addition to <b>industrial applications</b> in electrical power systems.
+        <Translate>The PSP-UFU aims to provide efficient computer simulation tools for</Translate> <b><Translate>research and education purposes</Translate></b>, <Translate>in addition to</Translate> <b><Translate>industrial applications</Translate></b> <Translate>in electrical power systems</Translate>.
       </>
     ),
   },
@@ -47,46 +48,46 @@ const features = [
 
 const studies = [
 	{
-    title: <>Power Flow</>,
+    title: <><Translate>Power Flow</Translate></>,
     //imageUrl: 'img/undraw_docusaurus_react.svg',
     description: (
       <ul>
         <li>Newton-Raphson</li>
-		<li>Gauss-Seidel</li>
-		<li>Hybrid Newton-Gauss</li>
-		<li>Three-phase induction motors included</li>
+		    <li>Gauss-Seidel</li>
+		    <li><Translate>Hybrid Newton-Gauss</Translate></li>
+		    <li><Translate>Three-phase induction motors included</Translate></li>
       </ul>
     ),
   },
   {
-    title: <>Short-Circuit calculation</>,
+    title: <><Translate>Short-Circuit calculation</Translate></>,
     //imageUrl: 'img/undraw_docusaurus_react.svg',
     description: (
       <ul>
-        <li>Balanced</li>
-		<li>Unbalanced</li>
-		<li>Short-Circuit power in all system buses</li>
+        <li><Translate>Balanced</Translate></li>
+		    <li><Translate>Unbalanced</Translate></li>
+		    <li><Translate>Short-Circuit power in all system buses</Translate></li>
       </ul>
     ),
   },
   {
-    title: <>Harmonics</>,
+    title: <><Translate>Harmonics</Translate></>,
     //imageUrl: 'img/undraw_docusaurus_react.svg',
     description: (
       <ul>
-        <li>Harmonic voltages and THD (Total Harmonic Distortion) calculation</li>
-		<li>Frequency scan</li>
+        <li><Translate>Harmonic voltages and THD (Total Harmonic Distortion) calculation</Translate></li>
+		    <li><Translate>Frequency scan</Translate></li>
       </ul>
     ),
   },
   {
-    title: <>Transient and Dynamic Stability</>,
+    title: <><Translate>Transient and Dynamic Stability</Translate></>,
     //imageUrl: 'img/undraw_docusaurus_react.svg',
     description: (
       <ul>
-        <li>Several synchronous machine models automatically selected</li>
-		<li>Three-phase induction motors</li>
-		<li>User-defined machine controls, exciters and prime moves created using block diagrams (Exciters, AVR, PSS, Hydro and Thermal turbines, Speed Governor, etc.)</li>
+        <li><Translate>Several synchronous machine models automatically selected</Translate></li>
+		    <li><Translate>Three-phase induction motors</Translate></li>
+		    <li><Translate>User-defined machine controls, exciters and prime moves created using block diagrams (Exciters, AVR, PSS, Hydro and Thermal turbines, Speed Governor, etc.)</Translate></li>
       </ul>
     ),
   },
@@ -128,11 +129,11 @@ function Home() {
   return (
     <Layout
       title={`${siteConfig.title}`}
-      description="Power System Platform">
+      description="<Translate>Power System Platform</Translate>">
       <header className={clsx('hero hero--primary', styles.heroBanner)}>
         <div className="container">
           <h1 className="hero__title">{siteConfig.title}</h1>
-          <p className="hero__subtitle">{siteConfig.tagline}</p>
+          <p className="hero__subtitle"><Translate>Power System Platform of Federal University of Uberlândia</Translate></p>
 		  
 		  <p><center>
 		  <iframe src="https://ghbtns.com/github-btn.html?user=thales1330&repo=psp&type=star&count=true&size=large" frameborder="0" scrolling="0" width="130" height="30" title="GitHub"></iframe>
@@ -144,7 +145,7 @@ function Home() {
                 styles.getStarted,
               )}
               to={useBaseUrl('https://github.com/Thales1330/PSP/releases/latest')}>
-              Download
+              <Translate>Download</Translate>
             </Link>
           </div>
 		  <div className={styles.buttons}>
@@ -154,7 +155,7 @@ function Home() {
                 styles.getStarted,
               )}
               to={useBaseUrl('docs/')}>
-              User Guide
+              <Translate>User Guide</Translate>
             </Link>
           </div>
 		  <div className={styles.buttons}>
@@ -164,7 +165,7 @@ function Home() {
                 styles.getStarted,
               )}
               to={useBaseUrl('https://minhaskamal.github.io/DownGit/#/home?url=https://github.com/Thales1330/PSP/tree/master/Project/data/samples')}>
-              Samples
+              <Translate>Samples</Translate>
             </Link>
           </div>
         </div>
@@ -183,7 +184,7 @@ function Home() {
         )}
 		
 		<div className="container">
-		<h2>Studies Performed</h2>
+		<h2><Translate>Studies Performed</Translate></h2>
 		</div>
 		
 		{studies && studies.length > 0 && (
@@ -199,8 +200,8 @@ function Home() {
         )}
 		
 		<div className="container">
-		<h2>Published Papers</h2>
-			Further details can be found in the published papers:
+		<h2><Translate>Published Papers</Translate></h2>
+			<Translate>Further details can be found in the published papers:</Translate>
 			<blockquote cite="https://doi.org/10.1002/2050-7038.12185">
 			Oliveira, T. L., Guimarães, G. C., & Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. <i>International Transactions on Electrical Energy Systems</i>, e12185. doi: <a href="https://doi.org/10.1002/2050-7038.12185">10.1002/2050-7038.12185</a>
 			</blockquote>
@@ -210,17 +211,17 @@ function Home() {
 		</div>
 		
 		<div className="container">
-		<h2>Code Documentation</h2>
+		<h2><Translate>Code Documentation</Translate></h2>
 		<p>
-			All detailed descriptions of the source-code can be found at <b><a href="https://thales1330.github.io/PSP/doxygen/html/index.html">Online Documentation</a></b>, generated by <a href="http://www.doxygen.org/">Doxygen</a>.
+			<Translate>All detailed descriptions of the source-code can be found at</Translate> <b><a href="https://thales1330.github.io/PSP/doxygen/html/index.html"><Translate>Online Documentation</Translate></a></b>, <Translate>generated by</Translate> <a href="http://www.doxygen.org/">Doxygen</a>.
 		</p>
 		</div>
 		
 		<div className="container">
-		<h2>Screenshots</h2>
+		<h2><Translate>Screenshots</Translate></h2>
 			<center>
 			<iframe src="gallery.html" style={{position: 'relative', width: '100%', height: '56.25vw', display: 'block', overflow: 'hidden', border: 'none'}}> 
-				Your browser doesn’t support the this feature. 
+				<Translate>Your browser doesn’t support the this feature.</Translate> 
 			</iframe>
 			</center>